uk: Математика, природничі науки

description: The strengths of the schemes we will present are their high order of accuracy in both space and time combined with their ability to march in time with a time step at the domain of dependence limit independent of the order. Additionally, the methods are globally super-convergent, i.e. the number of degrees of freedom per cell is (m+1)^d but the methods achieve orders of accuracy 2m. We note that the L2 super-convergence holds globally in space and time, unlike most other spatial discretizations, where super-convergence is limited to a few specific points and often rely on the use of negative norms. Our primary interest of these schemes are as highly efficient building blocks in hybrid methods where most of the mesh can be taken to be rectilinear and where geometry is handled by more flexible (but less efficient) methods close to physical boundaries. In this work we restrict our consideration to square geometries with boundary conditions of Dirichlet, Neumann or periodic type. We provide stability and convergence results for one dimensional periodic domains. The analysis of the conservative method is quite different from the analysis of previous dissipative Hermite methods and introduces a, to our knowledge, novel technique for analyzing conservative schemes for wave equations in second order form. This is joint work with Thomas Hagstrom (SMU) and Arturo Vargas (Rice, LLNL).

uk: вхід без авторизації

name: Globally Super-Convergent Conservative Hermite Methods for the Scalar Wave Equation


Ehud Deshalit

type: Person

keywords:
Mathematics
Dynamical systems and ergodic theory
Difference and functional equations
Dynamical systems

cs: Matematika

de: Mathematik

fi: matematiikka

uk: Математика

en: Mathematics

fr: Mathématiques

da: Matematik

es: Matemáticas

nl: Wiskunde

de: Mathematik, Naturwissenschaften

uk: Математика, природничі науки

en: Mathematics, Natural Sciences

description: Adamczewski and Bell proved in 2017 a 30-year old conjecture of Loxton and van der Poorten, asserting that a Laurent power series, which simultaneously satisfies a p-Mahler equation and a q-Mahler equation for multiplicatively independent integers p and q, is a rational function. Similar looking theorems have been proved by Bezivin-Boutabaa and Ramis for pairs of difference, or difference-differential equations. Recently, Schafke and Singer gave a unified treatment of all these theorems. In this talk we shall discuss a similar theorem for (p,q)-difference equations over fields of elliptic functions. Despite having the same flavor, there are substantial differences, having to do with issues of periodicity, and with the existence of non-trivial (p,q)-invariant vector bundles on the elliptic curve.

TIB AV-Portal

type: Service

type: WebContent

type: MediaObject

@language: en

cs: přihlášení není vyžadováno

de: keine Anmeldung erforderlich

uk: вхід без авторизації

en: no login required

nl: geen login vereist

name: Difference equations over fields of elliptic functions

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery

type: Organization

cs: Video

de: Video

fi: Video

uk: Відео

en: Video

fr: Vidéo

da: Video

nl: Video

es: Video


uk: Математика, природничі науки

uk: вхід без авторизації

name: Billiards and the arithmetic of non-arithmetic groups


uk: Математика, природничі науки

uk: вхід без авторизації

name: Genomic tracking of SARS-CoV-2 evolution and spread


uk: Математика, природничі науки

uk: вхід без авторизації

name: How different is the flu you transmit from the flu you received


uk: Математика, природничі науки

uk: вхід без авторизації

name: The evolution of multi-strain bacterial populations


uk: Математика, природничі науки

uk: вхід без авторизації

name: High-resolution characterization of global Salmonella subpopulations


uk: Математика, природничі науки

uk: вхід без авторизації

name: Bayesian inference of parameters in transmission models


uk: Математика, природничі науки

uk: вхід без авторизації

name: Mapping the shared evolutionary trajectories of resistance genes and other (possibly!) bad actors


On the Zariski dense orbit conjecture

uk: Математика, природничі науки

description: We prove the following theorem. Let f be a dominant endomorphism of a projective surface over an algebraically closed field of characteristic 0. If there is no nonconstant invariant rational function under f, then there exists a closed point whose orbit under f is Zariski dense. This result gives us a positive answer to the Zariski dense orbit conjecture for endomorphisms of projective surfaces. We define a new canonical topology on varieties over an algebraically closed field which has finite transcendence degree over Q. We call it the adelic topology. This topology is stronger than the Zariski topology and an irreducible variety is still irreducible in this topology. Using the adelic topology, we propose an adelic version of the Zariski dense orbit conjecture, which is stronger than the original one and quantifies how many such orbits there are. We also prove this adelic version for endomorphisms of projective surfaces, for endomorphisms of abelian varieties, and split polynomial maps. This yields new proofs of the original conjecture in the latter two cases.

uk: вхід без авторизації


uk: Математика, природничі науки

description: By a Schwarzian differential equation, we mean an equation of the form $S_{\frac{d}{dt}}(y) +(y')^2 R(y) =0,$ where $S_{\frac{d}{dt}}(y)$ denotes the Schwarzian derivative and $R$ is a rational function with complex coefficients. The equation naturally appears in the study of automorphic functions (such as the modular $j$-function): if $j_{\Gamma}$ is the uniformizing function of a genus zero Fuchsian group of the first kind, then $j_{\Gamma}$ is a solution of some Schwarzian equation. In this talk, we discuss recent work towards the proof of a conjecture/claim of P. Painlev\â e (1895) about the irreducibility of the Schwarzian equations. We also explain how, using the model theory of differentially closed fields, this work on irreducibility can be used to tackle questions related to the study of algebraic relations between the solutions of a Schwarzian equation. This includes, for example, obtaining the Ax-Lindemann-Weierstrass Theorem with derivatives for all Fuchsian automorphic functions.

uk: вхід без авторизації

name: Schwarzian equation, automorphic functions and functional transcendence


Elliptic surfaces and R-divisors

uk: Математика, природничі науки

description: I will discuss some unlikely-intersection problems for curves in elliptic surfaces, defined over a number field. We introduce a new tool: an equidistribution theorem for heights associated to R-divisors, extending known results for Q-divisors of Chambert-Loir, Thuillier, and Yuan (2008). As applications, we obtain a new proof of a result of Barroero and Capuano (2016) and prove a case of a conjecture of Zhang (1998).

uk: вхід без авторизації


Large-scale phylogeography of hCoV-19

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Genomic epidemiology of SARS-CoV-2 in Quebec


uk: Математика, природничі науки

uk: вхід без авторизації

name: Estimating the age groups that sustain resurging COVID-19 epidemics in the United States


uk: Математика, природничі науки

uk: вхід без авторизації

name: Decomposing the sources of SARS-CoV-2 fitness variation using phylodynamics


uk: Математика, природничі науки

uk: вхід без авторизації

name: SARS-CoV-2 Phylodynamic inference - with and without phylogenies


uk: Математика, природничі науки

uk: вхід без авторизації

name: Bayesian modeling and data integration in infectious disease phylodynamics


uk: Математика, природничі науки

description: The Dynamical Mordell-Lang Conjecture predicts the structure of the intersection between a subvariety $V$ of a variety $X$ defined over a field $K$ of characteristic $0$ with the orbit of a point in $X(K)$ under an endomorphism $\Phi$ of $X$. The Zariski dense conjecture provides a dichotomy for any rational self-map $\Phi$ of a variety $X$ defined over an algebraically closed field $K$ of characteristic $0$: either there exists a point in $X(K)$ with a well-defined Zariski dense orbit, or $\Phi$ leaves invariant some non-constant rational function $f$. For each one of these two conjectures we formulate an analogue in characteristic $p$; in both cases, the presence of the Frobenius endomorphism in the case $X$ is isotrivial creates significant complications which we will explain in the case of algebraic tori.

uk: вхід без авторизації

name: A couple of conjectures in arithmetic dynamics over fields of positive characteristic


uk: Математика, природничі науки

uk: вхід без авторизації

name: Equivariant currents and heights on the boundary of the ample cone of a K3 surface


uk: Математика, природничі науки

uk: вхід без авторизації

name: Additive uncorrelated relaxed clock models


uk: Математика, природничі науки

uk: вхід без авторизації

name: Hierarchical population genomic structure of bacterial pathogens in EnteroBase

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


uk: Математика, природничі науки

uk: вхід без авторизації

name: Estimating population-level HIV transmission dynamics from deep-sequence data


uk: Математика, природничі науки

uk: вхід без авторизації

name: Phylogenetic inference with distribution continuum


uk: Математика, природничі науки

uk: вхід без авторизації

name: Improving the interpretability of k-mer-based signatures for antibiotic resistance prediction


uk: Математика, природничі науки

uk: вхід без авторизації

name: mbImpute: an accurate and robust imputation method for microbiome data


uk: Математика, природничі науки

uk: вхід без авторизації

name: Definable Galois theory and holomorphic vector bundles


uk: Математика, природничі науки

description: I will present in detail a new twist in the subject of arithmetic algebraization theorems. It comes out of a joint work in progress with Frank Calegari and Yunqing Tang on irrational periods, and bears also a relation to a variation by Zudilin around the classical Polya-Bertrandias determinantal criterion for the rationality of a formal function on the projective line. Time permitting, I will sketch an application to an irrationality proof of the 2-adic avatar of $\zeta(5)$.

uk: вхід без авторизації

name: On an arithmetic criterion for holonomicity


Model theory of group actions on fields

uk: Математика, природничі науки

description: For a fixed group G, we study the model theory of actions of G by field automorphisms. The main question here is to characterize the class of groups G for which the theory of such actions has a model companion (a first-order theory of "large" actions). In my talk, I will discuss several classes of groups G in this context. The case of finite groups is joint work with Daniel Hoffmann ("Existentially closed fields with finite group actions", Journal of Mathematical Logic, (1) 18 (2018), 1850003). The case of finitely generated virtually free groups is joint work with Ã zlem Beyarslan ("Model theory of fields with virtually free group actions", Proc. London Math. Soc., (2) 118 (2019), 221-256). The case of commutative torsion groups is joint work with Ã zlem Beyarslan

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Increased power from conditional bacterial genome-wide association identifies macrolide resistance mutations in Neisseria gonorrhoeae


uk: Математика, природничі науки

uk: вхід без авторизації

name: Adaptation to the cervical environment is associated with increased antibiotic susceptibility in Neisseria gonorrhoeae


uk: Математика, природничі науки

uk: вхід без авторизації

name: Population genomics of A. baumannii reveals a licentious and highly mobile resistome


uk: Математика, природничі науки

uk: вхід без авторизації

name: Synthetic biology approaches to suppression of antibiotic resistance


uk: Математика, природничі науки

uk: вхід без авторизації

name: Identifying prognostic pairwise relationships among bacterial species in microbiome studies


uk: Математика, природничі науки

uk: вхід без авторизації

name: Bayesian community modelling of the microbiome


uk: Математика, природничі науки

uk: вхід без авторизації

name: Interpretable machine learning methods for predicting drug resistance


uk: Математика, природничі науки

description: This work is a collaboration with B. Adamczewski (ICJ, France), T. Dreyfus (IRMA, France) and M. Wibmer (Graz University of Technology, Austria). In this talk, we will consider pairs of automorphisms $(\phi,\sigma)$ acting on fields of Laurent or Puiseux series: pairs of shift operators, of $q$-difference operators and of Mahler operators. Assuming that the operators $\phi$ and $\sigma$ are "independent", we show that their solutions are also "independent" in the sense that a solution $f$ to a linear $\phi$-equation and a solution $g$ to a linear $\sigma$-equation are algebraically independent over the field of rational functions unless one of them is a rational function. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of questions and is based on a suitable Galois theory: the $\sigma$-Galois theory of linear $\phi$-equations developed by Ovchinnikov and Wibmer.

uk: вхід без авторизації

name: Algebraic independence of solutions of linear difference equations


uk: Математика, природничі науки

description: Consider a complex projective surface $X$, with a non-abelian free group $G$ acting faithfully and regularly on $X$. It may happen that $G$ has infinitely many periodic orbits: this is the case when $X$ is an abelian surface and all torsion points are $G$-periodic. In this talk, I will describe recent results obtained with Romain Dujardin aiming at a complete classification of all such examples. The main players will be canonical heights, arithmetic equidistribution, and rigidity results in ergodic theory.

uk: вхід без авторизації

name: Finite orbits and canonical heights for large groups of automorphisms


uk: Математика, природничі науки

uk: вхід без авторизації

name: Considerations for estimating Rt in Alberta, Canada


uk: Математика, природничі науки

uk: вхід без авторизації

name: Inference of Multiscale Selection using a Continuous Type Birth Death Model

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


uk: Математика, природничі науки

uk: вхід без авторизації

name: Using phylodynamics and decision trees to estimate the effects of mutation on viral fitness


uk: Математика, природничі науки

uk: вхід без авторизації

name: Combining sequencing technologies and data mining to study (mega)plasmid families


uk: Математика, природничі науки

uk: вхід без авторизації

name: Case fatality imputation using machine learning


uk: Математика, природничі науки

uk: вхід без авторизації

name: Pop-PUNK-web: In-browser genome sketching for secure real-time genomic epidemiology


uk: Математика, природничі науки

uk: вхід без авторизації

name: Do machine learning predictors of microbial phenotype from genotype identify causal variants


uk: Математика, природничі науки

uk: вхід без авторизації

name: ggCaller: a gene caller for bacterial pangenome graphs


uk: Математика, природничі науки

uk: вхід без авторизації

name: Deep clustering of bacterial tree images


uk: Математика, природничі науки

uk: вхід без авторизації

name: PseudotimeDE: inference of differential gene expression along cell pseudotime with well-calibrated p-values from single-cell RNA sequencing data


uk: Математика, природничі науки

uk: вхід без авторизації

name: Benchmarking computational doublet-detection methods for single-cell RNA sequencing data


keywords:
Partial Differential Equations
Fluid Mechanics
Fluid Dynamics

uk: Математика, природничі науки

description: We establish long-time and large-data existence of a suitable weak solution to three-dimensional internal unsteady flows described by Kolmogorovâ s two-equation model of turbulence. The governing system of equations is completed by initial and boundary conditions; concerning the velocity we consider generalized stickâ slip boundary conditions. The fact that the admissible class of boundary conditions includes various types of slipping mechanisms on the boundary makes the result robust from the point of view of possible applications.

uk: вхід без авторизації

name: Large data analysis for Kolmogorovâ s two-equation model of turbulence


uk: Математика, природничі науки

description: We discuss the validity of the ergodic hypothesis (convergence of ergodic means) for solutions of the compressible Navier-Stokes systems with either non-homogeneous boundary conditions or driven by a stochastic forcing. In both cases we show convergence of the classical Krylov-Bolyubov method that gives rise to a statistical (stochastic) stationary solution of the problem.

uk: вхід без авторизації

name: Ergodic hypothesis for open fluid systems


uk: Математика, природничі науки

description: In this talk, we will discuss the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. We will explain why the prediction of such a blow-up, via direct numerical experiments, is so difficult. We will describe how, in such a scenario, the solution becomes unstable as time approaches the blow-up time. The method use the relation between the vorticity of the solution, and the bi-characteristic amplitude solutions, which describe the evolution of the linearized Euler equation at high frequency. In the axisymmetric case, we can also study the instability of blow-up profiles. This work was partially supported by the NSFDMS-1907981.

uk: вхід без авторизації

name: Instability of finite time blow-ups solutions for incompressible Euler


uk: Математика, природничі науки

description: We consider the equations of a linear Maxwell fluid with spatially varying coefficients. Pure stress modes are solutions with zero velocity but nonzero stresses. We derive equations to characterize such solutions. In two dimensions, we find that under generic hypotheses only certain "trivial" solutions exist. In three dimensions, on the other hand, there exist nontrivial solutions. To get them, we derive a system of partial differential equations whose type (elliptic or hyperbolic) depends on the sign of the Gauss curvature of level surfaces of the relaxation time.

uk: вхід без авторизації

name: Pure stress modes for linear viscoelastic flows with variable coefficients


uk: Математика, природничі науки

description: We shall review recent results on the mathematical derivation of Einstein's formula and higher order corrections for the effective viscosity of a dilute suspension of neutrally buoyant particles.

uk: вхід без авторизації

name: On the effective viscosity of suspensions


uk: Математика, природничі науки

description: We show that weak solutions of degenerate Navier-Stokes equations converge to the strong solutions of the pressureless Euler system with linear drag term, Newtonian repulsion and quadratic confinement. The proof is based on the relative entropy method using the artificial velocity formulation for the one-dimensional Navier-Stokes system.

uk: вхід без авторизації

name: From compressible Naveir-Stokes with nonlocal forces to Euler


uk: Математика, природничі науки

description: We are interested in finding solutions of nonlinear differential equations describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models. To describe the response of viscoelastic solids we assume a nonlinear relationship among the linearized strain, the strain rate and the Cauchy stress. We consider the corresponding Cauchy problem for the stress variable. Under the monotonicity assumption of the nonlinear constitutive function, we convert the problem to a new form for the strain variable and prove local-in-time existence of solutions.

uk: вхід без авторизації

name: Local-in-time existence of solutions to strain-limiting viscoelasticity


uk: Математика, природничі науки

description: We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant $T > 0$. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and SzÃ©kelyhidi and we construct infinitely many solutions.

uk: вхід без авторизації

name: Non-uniqueness of admissible weak solutions to the compressible Euler equations with smooth initial data

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


uk: Математика, природничі науки

description: In multiscale models for polymeric fluids, the evolution of the polymer chain is usually modeled using an entropic force, computed from the free energy associated with the end-to-end vector. We will present results which aim at justifying under which circumstances such a dynamics is indeed close to the original dynamics based on the full-atom chain. References: F. Legoll and T. LeliÃ¨vre, Effective dynamics using conditional expectations, Nonlinearity, 23, 2131-2163, (2010). F. Legoll, T. LeliÃ¨vre and S. Olla, Pathwise estimates for an effective dynamics, Stochastic Processes and their Applications, 127, 2841-2863, (2017). F. Legoll, T. LeliÃ¨vre and U. Sharma, Effective dynamics for non-reversible stochastic differential equations: a quantitative study, Nonlinearity, 32(12), 4779, (2019)

uk: вхід без авторизації

name: Effective dynamics for stochastic differential equations


uk: Математика, природничі науки

description: We analyse the system of the form \begin{align*} {\partial}_t{\rho} +{\rm div \,}_x(\rho u) = 0\\ {\partial}_t(\rho u) +{\rm div \,}_x(\rho u\otimes u) + \nabla_x p = {\rm div \,}_x {\mathbb{S}}\label{secondequation}\\ {\rm div \,}_x(u) = 0 \end{align*} where $\rho$ is the mass density, $u$ denotes velocity field, ${\mathbb{S}}$ the stress tensor and $p$ is the pressure. We are interested in the measure-valued solutions to those equations and prove the mv-strong uniqueness property. This work bases its assumptions on the recent paper by Abbatiello and Feireisl [1], but differs from it in density dependency. Surprisingly the solutions are not defined by the Young measures, but by the similar tool to the so-called defect measure. BIBLIOGRAPHY [1] A. Abbatiello and E. Feireisl. On a class of generalized solutions to equations describing incompressible viscous fluids. Ann. Mat. Pura Appl. (4), 199(3):1183â 1195, 2020.

uk: вхід без авторизації

name: Mv-strong uniqueness for density dependent, incompressible, non-Newtonian fluids


uk: Математика, природничі науки

description: We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a~priori bounded but the heat flux provides merely $L^1$-coercivity. We use the concept of renormalized solutions and higher differentiability techniques to prove existence and uniqueness of weak solution with $L^1$-integrable flux for all values of a positive model parameter $a$. If this parameter is smaller than $2/(d+1)$, where $d$ denotes the spatial dimension, we obtain higher integrability of the flux. We also relate the studied problem to problems in fluid mechanics.

uk: вхід без авторизації

name: On evolutionary problems with a-priori bounded gradients


uk: Математика, природничі науки

description: Analysis of finite amplitude stability of fluid flows is a challenging task even if the fluid of interest is described using the classical mathematical models such as the Navier--Stokes--Fourier model. The issue gets more complicated when one has to deal with complex models for coupled thermomechanical behaviour of non-Newtonian fluids; in particular the viscoelastic rate-type fluids. We show that the knowledge of thermodynamical underpinnings of these complex models can be gainfully exploited in the stability analysis. First we introduce general concepts that allow one to deal with thermodynamically isolated systems, and then we proceed to thermodynamically open systems. Next we document the applications of these concepts in the case of container flows (thermodynamically isolated systems), and in the case of flows in containers with non-uniformly heated walls (mechanically isolated but thermally open system). We end up with mechanically driven systems such as the Taylor--Couette flow.

uk: вхід без авторизації

name: Thermodynamics of viscoelastic rate-type fluids and its implications for stability analysis


uk: Математика, природничі науки

description: We discuss long time behavior of solutions to a non-linear second order integrodifferential convolution equation, in particular we focus on the speed of convergence to equilibrium. The key assumptions are that the convolution kernel is small and the non-linear operator satisfies the Lojasiewicz inequality.

uk: вхід без авторизації

name: Decay of solutions to integrodifferential equations


uk: Математика, природничі науки

description: The rheology of complex fluids such as polymeric liquids is highly non-Newtonian in nature and manifests itself as an extra stress component in the Cauchy stress tensor. At the purely macroscopic level, the extra stress tensor is linked to the velocity field through, say, a partial differential equation. An alternative approach consists in finding an expression for the macroscopic extra stress tensor in terms of the microscopic dynamics of the polymer chains. We present a thermodynamically based approach to the design of a class of such micro-macro models for dilute polymeric liquids and show that the thermodynamic background of the model naturally yields stability of the steady state when the fluid occupies an isolated vessel.

uk: вхід без авторизації

name: Thermodynamically consistent derivation of a micro-macro model for dilute polymeric fluids


Compressible flow or rough denisty

uk: Математика, природничі науки

uk: вхід без авторизації


uk: Математика, природничі науки

description: Classical models describing the motion of Newtonian fluids, such as water, rely on the assumption that the Cauchy stress is a linear function of the symmetric part of the velocity gradient of the fluid. This assumption leads to the Navier-Stokes equations. It is known however that the framework of classical continuum mechanics, built upon an explicit constitutive equation for the Cauchy stress, is too narrow to describe inelastic behavior of solid-like materials or viscoelastic properties of materials. Our starting point in this work is therefore a generalization of the classical framework of continuum mechanics, called the implicit constitutive theory, which was proposed recently in a series of papers by K.R. Rajagopal. The underlying principle of implicit constitutive theory in the context of viscous flows is the following: instead of demanding that the Cauchy stress is an explicit (and, in particular, linear) function of the symmetric part of the velocity gradient, one may allow a nonlinear, implicit and not necessarily continuous relationship between these quantities. The resulting general theory therefore admits non-Newtonian fluid flow models with implicit and possibly discontinuous power-law-like rheology. We develop the analysis of finite element approximations of implicit power-law-like models for viscous in-compressible fluids. The Cauchy stress and the symmetric part of the velocity gradient in the class of models under consideration are related by a, possibly multi-valued, maximal monotone graph. Using a variety of weak compactness techniques, we show that a subsequence of the sequence of finite element solutions converges to a weak solution of the problem as the discretisation parameter, measuring the granularity of the finite element triangulation, tends to zero. A key new technical tool in our analysis is a finite element counterpart of the Acerbi-Fusco Lipschitz truncation of Sobolev functions. The talk is based on a series of recent papers with Lars Diening and Tabea Tscherpel (Bielefeld), Christian Kreuzer (Dortmund), Alexei Gazca Orozco (Erlangen) and Patrick Farrell (Oxford).

uk: вхід без авторизації

name: Analysis and approximation of implicitly constituted fluid flow models


uk: Математика, природничі науки

description: Mathematical modelling and numerical simulations of phase separation becomes much more involved if one component is a macromolecular compound. In this case, the large molecular relaxation time gives rise to a dynamic coupling between intra-molecular processes and the unmixing on experimentally relevant time scales, with interesting new phenomena, for which the name â viscoelastic phase separationâ has been coined. Our model of viscoelastic phase separation describes time evolution of the volume fraction of a polymer and the bulk stress leading to a strongly coupled (possibly degenerate) cross-diffusion system. The evolution of volume fraction is governed by the Cahn-Hilliard type equation, while the bulk stress is a parabolic relaxation equation. The system is further combined with the Navier-Stokes-Peterlin system, describing time evolution of the velocity and (elastic) conformation tensor. Under some physically relevant assumptions on boundedness of model parameters we have proved that global in time weak solutions exist. Further, we have derived a suitable notion of the relative energy taking into account the non-convex nature of the energy law for the viscoelastic phase separation. This allows us to prove the weak-strong uniqueness principle and consequently the uniqueness of a weak solution in special cases. Our extensive numerical simulations confirm robustness of the analysed model and the convergence of a suitable numerical scheme with respect to the relative energy.

uk: вхід без авторизації

name: Viscoelastic phase separation: analysis and numerics


uk: Математика, природничі науки

description: In this talk we consider the interaction of a Stokes/Navier-Stokes flow with a viscoelastic body. The elastic body is allowed to undergo large deformations (but no self-collisions). In order to handle this situation correctly, we devise a variational approximation scheme in the spirit of DeGiorgi to the combined problem. Moreover, by using a two-scale scheme, we also extend this approach to the hyperbolic regime including inertia of the solid body. These variational approaches allow us to prove proper energetic estimates while also controling the geometric restictions posed on the solid body and, eventually, to establish existence of weak solutions.

uk: вхід без авторизації

name: A variational approach to fluid-structure interaction


uk: Математика, природничі науки

description: In tis talk I will present a derivation of macroscopic model of interacting particles. The population of N particles evolve according to a diffusion process and interacts through a dynamical network. In turn, the evolution of the network is coupled to the particles' positions. In contrast with the mean-field regime, in which each particle interacts with every other particle, i.e. with O(N) particles, we consider the a priori more difficult case of a sparse network; that is, each particle interacts, on average, with O(1) particles. We also assume that the network's dynamics is much faster than the particles' dynamics. The derivation combines the stochastic averaging (over time-scale parameter) and the many particles ($N\to \infty$) limits.

uk: вхід без авторизації

name: On the dynamical network of interacting particles: from micro to macro


uk: Математика, природничі науки

description: Several notions of weak or 'very weak' solutions have been suggested for the incompressible and compressible Euler systems, motivated by the lack of a satisfactory well-posedness theory for these equations in turbulent regimes. Surprisingly, the speaker and L. SzÃ©kelyhidi showed in 2012 that dis- tributional and measure-valued solutions are in a sense the same, although the latter had been expected to be a much weaker notion. In this talk, we turn to the isentropic compressible Euler system, where the situation is fundamentally different.

uk: вхід без авторизації

name: Weak and measure-valued solutions for the compressible Euler equations


uk: Математика, природничі науки

description: We concern the system consisting of a moving body filled with a compressible fluid. We present several existence proofs, however, our main aim is to deal with the long-time behavior of the whole system.

uk: вхід без авторизації

name: On a body with a cavity filled with compressible fluid


uk: Математика, природничі науки

description: We consider the barotropic Navier-Stokes system describing the motion of a compressible viscous fluid confined to a bounded domain driven by time periodic inflow/outflow boundary conditions. We show that the problem admits a time-periodic solution in the class of weak solutions satisfying the energy inequality.

uk: вхід без авторизації

name: On the motion of a compressible viscous fluid driven by time-periodic inflow/outflow boundary conditions


uk: Математика, природничі науки

description: We study a two-species model of tissue growth describing dynamics under mechanical pressure and cell growth. The pressure is incorporated into the common fluid velocity through an elliptic equation, called Brinkmanâ s law, which accounts for viscosity effects in the individual species. Our aim is to establish the incompressible limit as the stiffness of the pressure law tends to infinity - thus demonstrating a rigorous bridge between the population dynamics of growing tissue at a density level and a geometric model thereof.

uk: вхід без авторизації

name: Incompressible limit for a two-species tumour growth model


uk: Математика, природничі науки

description: The nonlocal Mullins-Sekerka flow can be seen as the $H^{-\frac12}$-gradient flow of the so called sharp-interface Ohta-Kawaski energy. In this talk we will show that three-dimensional periodic configurations that are strictly stable with respect to this energy are exponentially stable also for the nonlocal Mullins-Sekerka flow.

uk: вхід без авторизації

name: Stability results for the nonlocal Mullins-Sekerka flow


uk: Математика, природничі науки

uk: вхід без авторизації

name: Weak Solutions for Compressible Navier-Stokes equations with inhomogenous and thermodynmaically unstable pressure laws


uk: Математика, природничі науки

description: We present global-in-time existence results for two cross-diffusion systems modeling heat-conducting fluid mixtures. Both models consist of the balance equations for the mass densities and temperature. The key difficulty is the nonstandard degeneracy in the diffusion (Onsager) matrices, i.e., ellipticity is lost when the fluid density or temperature vanishes. This problem is overcome in the first model by exploiting the volume-filling property of the mixture, leading to gradient estimates for the square root of the partial densities, and in the second model by compensated compactness and renormalization techniques from mathematical fluid dynamics. The first model is joint work with C. Helmer, the second one with G. Favre, C. Schmeiser, and N. Zamponi.

uk: вхід без авторизації

name: Analysis of degenerate cross-diffusion systems for heat-conducting fluid mixtures

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


On the dynamic slip boundary condition

uk: Математика, природничі науки

description: In the talk, we discuss a response of the fluid on the boundary, which acts as a delayed slip due to material properties. In the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier-Stokes-like problems with dynamic slip boundary condition, which requires a proper generalisation of the Gelfand triplet and the corresponding function spaces setting.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Amyloid fibrils are important biological structures associated with devastating human diseases such as Alzheimer disease, as well as have vital biological functions such as adhesion and biofilm formation. The division of amyloid protein fibrils is required for the propagation of the amyloid state and is an important contributor to their stability, pathogenicity, and normal function. We apply asymptotic results on the fragmentation equation to develop an inverse problem approach, allowing us to compare the division stability of amyloid fibrils and estimate their division features (fragmentation rate and kernel).

uk: вхід без авторизації

name: Estimating the division of amyloid fibrils


uk: Математика, природничі науки

description: We deal with the flows of non-Newtonian fluids in three dimensional setting subjected to the homogeneous Dirichlet boundary condition. Under the natural monotonicity, coercivity and growth condition on the Cauchy stress tensor expressed by a critical power index $p=\frac{11}{5}$ we show that a Gehring type argument is applicable which allows to improve regularity of any weak solution. Improving further the regularity of weak solutions along a regularity ladder allows to show that actually solution belongs to a uniqueness class provided data of the problem are sufficiently smooth. We also briefly discuss if the similar technique is applicable to critical Convective Brinkman-Forchheimer equation.

uk: вхід без авторизації

name: Uniqueness and regularity of flows of non-Newtonian fluids with critical power-law growth


uk: Математика, природничі науки

description: Viscoelastic fluids often exhibit high sensitivity of material properties on temperature changes. Nevertheless, the available mathematical theory for these fluids concerns only models that are isothermal or that are simplified in other ways. For example, one can find existence theories in 2D, for small data, with only the corotational derivative, with only the spherical part of the elasticity tensor etc. In the talk, we introduce an existence theory without any of these assumptions and treat a rather general class of Johnson-Segalman-like models including full thermal evolution. To avoid the well-known ill-posedness of the corresponding PDE system, we modify the ``elastic part'' of the dissipation of the fluid far from the equilibrium, while preserving thermodynamic compatibility of the model. This way, we are able to prove the existence of a global-in-time weak solution for any initial datum with finite total energy and entropy.

uk: вхід без авторизації

name: Analysis of an unsteady flow of an incompressible heat-conductive rate-type viscoelastic fluid with stress diffusion


uk: Математика, природничі науки

description: The Keller-Segel model for bacterial chemotaxis supports travelling wave solutions which have been described in the literature as both linearly stable and unstable and in the case of linear consumption (conditionally) nonlinearly stable. We reconcile this apparent contradiction by locating the essential spectrum, absolute spectrum and point spectrum of the linear operators associated with the travelling wave solutions. We derive conditions for the spectral (in)stability of the travelling wave solutions and the critical parameters that indicate a transition from a transient to absolute instability. Furthermore, we show that the absolute spectrum deforms as the consumption is changed, illustrating a connection between the constant, sublinear and linear cases.

uk: вхід без авторизації

name: Absolute Instabilities of Travelling Waves Solutions in a KellerSegel Model


uk: Математика, природничі науки

description: We analyse fast reaction limit in the reaction-diffusion system \begin{align*} \partial_t u^{\varepsilon} &= \frac{v^{\varepsilon} - F(u^{\varepsilon})}{\varepsilon}, \\ \partial_t v^{\varepsilon} &= \Delta v^{\varepsilon} + \frac{F(u^{\varepsilon}) - v^{\varepsilon}}{\varepsilon}, \end{align*} with nonmonotone reaction function $F$. As speed of reaction tends to infinity, the concentration of non-diffusing component $u^{\varepsilon}$ exhibits fast oscillations. We identify precisely its Young measure which, as a by-product, proves strong convergence of the diffusing component $v^{\varepsilon}$, a result that is not obvious from a priori estimates. Our work is based on analysis of regularization for forward-backward parabolic equations by Plotnikov [2]. We rewrite his ideas in terms of kinetic functions which clarifies the method, brings new insights, relaxes assumptions on model functions and provides a weak formulation for the evolution of the Young measure.

uk: вхід без авторизації

name: Fast reaction limit with nonmonotone reaction function


uk: Математика, природничі науки

description: Rate-type fluid models involving the stress and its observer-invariant time derivatives of higher order are used to describe a large class of viscoelastic mixtures - geomaterials like asphalt, biomaterials such as vitreous in the eye, synthetic rubbers such as SBR. A standard model that belongs to the category of viscoelastic rate-type fluid models of the second order is the model due to Burgers, which can be viewed as a mixture of two Oldroyd-B models of the first order. This viewpoint allows one to develop the whole hierarchy of generalized models of a Burgers type. We study one such generalization. Carrying on the study by Masmoudi (2011), who briefly proved the weak sequential stability of weak solutions to the Giesekus model, we prove long time and large data existence of weak solutions to a mixture of two Giesekus models in two spatial dimensions.

uk: вхід без авторизації

name: On planar flows of viscoelastic fluids of the Burgers type


uk: Математика, природничі науки

description: We consider the Kelvin-Voigt model for viscoelasticity and prove propagation of $H^1$-regularity for the deformation gradient of weak solutions in two and three dimensions assuming that the stored energy satisfies the Andrews-Ball condition, in particular allowing for a non-monotone stress. By contrast, a counterexample indicates that for non-monotone stress-strain relations (even in 1-d) initial oscillations of the strain lead to solutions with sustained oscllations. In addition, in two space dimensions, we prove that the weak solutions with deformation gradient in $H^1$ are in fact unique, providing a striking analogy to the 2D Euler equations with bounded vorticity.

uk: вхід без авторизації

name: Existence and uniqueness for a viscoelastic Kelvin-Voigt model with nonconvex stored energy


uk: Математика, природничі науки

description: In this talk, I will discuss with the small porosity asymptotic behavior of the coupled Stokes-Brinkman system in the presence of a curved interface between the Stokes region and the Brinkman region. In particular, we derive a set of approximate solutions, validated via rigorous analysis, to the coupled Stokes-Brinkman system. Of particular interest is that the approximate solution satisfies a generalized Beavers-Joseph-Saffman-Jones interface condition (1.9) with the constant of proportionality independent of the curvature of the interface.

uk: вхід без авторизації

name: Vanishing porosity limit of the coupled Stokes-Brinkman system


uk: Математика, природничі науки

description: We present large-data existence result for weak solutions to a steady compressible Navier-Stokes-Fourier system for chemically reacting fluid mixtures. General free energies satisfying some structural assumptions are considered, with a pressure containing a $\gamma$-power law. The model is thermodynamically consistent and contains the Maxwell-Stefan cross-diffusion equations in the Fick-Onsager form as a special case. Compared to previous works, a very general model class is analyzed, including cross-diffusion effects, temperature gradients, compressible fluids, and different molar masses. A priori estimates are derived from the entropy balance and the total energy balance. The compactness for the total mass density follows from an estimate for the density in $L^{\gamma}$ with $\gamma>3/2$, the effective viscous flux identity, and uniform bounds related to Feireisl's oscillations defect measure. These bounds rely heavily on the convexity of the free energy and the strong convergence of the relative chemical potentials.

uk: вхід без авторизації

name: Existence analysis of a stationary compressible fluid model for heat-conducting and chemically reacting mixtures


uk: Математика, природничі науки

description: We consider pressureless compressible Euler equations driven by nonlocal repulusion-attraction and alignment forces. Our attention is directed to measure-valued solutions, i.e., very weak solutions described by a classical Young measure together with appropriate concentration defects. We investigate the evolution of a relative energy functional to compare a measure-valued solution to a regular solution emanating from the same initial datum. This leads to a weak-strong uniqueness principle.

uk: вхід без авторизації

name: Dissipative measure-valued solutions for the Euler-Poisson equation


uk: Математика, природничі науки

description: In continuum models for non-perfect fluids, viscoelastic stresses have often been introduced as extra-stresses of purely-dissipative (entropic) nature, similarly to viscous stresses that induce motions of infinite propagation speed. A priori, it requires only one to couple an evolution equation for the (extra-)stress with the momentum balance. In many cases, the apparently-closed resulting system is often not clearly well-posed, even locally in time. The procedure also raises questions about how to encompass transition toward alastic solids. A noticeable exception is K-BZK theory where one starts with a purely elastic fluids. Viscoelasticity then results from dissipative (entropic) stresses due to the relaxation of the fluids'"memory". That K-BKZ approach is physically appealing, but mathematically quite difficult because integrals are introduced to avoid material ('natural') configurations. We propose to introduce viscoelastic stress starting with hyperelastic fluids (like K-BKZ) and evolving material configurations (unlike K-BKZ). At the price of an enlarged system with an additional material-metric variable, one can define well-posed (compressible) motions with finite propagation speed through a system of conservation laws endowed with a "contingent entropy" (like in standard polyconvex elastodynamics).

uk: вхід без авторизації

name: Viscoelastic motions of Maxwell fluids with finite propagation speed


uk: Математика, природничі науки

description: The Navier-Stokes-Voigt model of viscoelastic incompressible fluid has been proposed as a regularization of the three-dimensional Navier-Stokes equations for the purpose of direct numerical simulations. Besides the kinematic viscosity parameter, $\nu>0$, this model possesses a regularizing parameter, $\alpha> 0$, a given length scale parameter, so that $\frac{\alpha^2}{\nu}$ is the relaxation time of the viscoelastic fluid. In this talk I will derive several statistical properties of the invariant measures associated with the solutions of the three-dimensional Navier-Stokes-Voigt equations. Moreover, I will show that, for fixed viscosity, $\nu>0$, as the regularizing parameter $\alpha$ tends to zero, there exists a subsequence of probability invariant measures converging, in a suitable sense, to a strong stationary statistical solution of the three-dimensional Navier-Stokes equations, which is a regularized version of the notion of stationary statistical solutions - a generalization of the concept of invariant measure introduced and investigated by Foias. This fact is also supported by numerical observations, which provides an additional evidence that, for small values of the regularization parameter $\alpha$, the Navier-Stokes-Voigt model can indeed be considered as a model to study the statistical properties of the three-dimensional Navier-Stokes equations and turbulent flows via direct numerical simulations.

uk: вхід без авторизації

name: Statistical Properties of the Navier-Stokes-Voigt Model


uk: Математика, природничі науки

uk: вхід без авторизації

name: Hardware-accelerated genome sketching enables real-time genomic epidemiology of pathogens


Monopoles and hyper-Poisson bivectors

keywords:
Mathematik
Mathematics
Algebraic Geometry
Differential Geometry

uk: Математика, природничі науки

description: It is well-known that the Riemannian geometry of the moduli space of Euclidean SU(2)-monopoles of charge $k$ is determined by spectral curves of monopoles. I had often wondered whether there is a purely differential-geometric explanation of this fact, i.e. whether there exists an infinitesimal object on the moduli space which makes it so. I shall show that the answer is yes, and that the object in question is what I call a hyper-Poisson bivector, i.e. a bivector which induces, for each complex structure, a Poisson structure on holomorphic functions, compatible with the respective complex-symplectic form.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The Hitchin-Simpson equations defined over a KÃ¤hler manifold are first order, non-linear equations for a pair of a connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. We will describe the behavior of solutions to the Hitchinâ Simpson equations with norms of these 1-forms unbounded. We will also discuss the deformation problem of Taubes' Z2 harmonic 1-form.

uk: вхід без авторизації

name: The compactness problem for the Hitchin-Simpson equations


Elliptic zastava

uk: Математика, природничі науки

description: For a semisimple group G and a smooth curve C, open zastava space Z(G,C) is a smooth variety, affine over a configuration space of C. In case C is the additive or multiplicative group, Z(G,C) is isomorphic to a moduli space of euclidean or periodic monopoles. It carries a natural symplectic form, and the projection to the configuration space is an integrable system (open Toda lattice for G=SL(2)). I will explain what happens when C is an elliptic curve. This is a joint work with Mykola Matviichuk and Alexander Polishchuk.

uk: вхід без авторизації


Spectral curves in surfaces

uk: Математика, природничі науки

description: An embedded curve in a Poisson surface \Sigma\subset X defines a smooth deformation space \mathcal{B} of nearby embedded curves. In this talk we will describe a key idea of Kontsevich and Soibelman to equip the Poisson surface X with a foliation in order to study the deformation space \mathcal{B}. For example, X=TP^1\to P^1 is a Poisson surface surface foliated by its fibres. The foliation, together with a vector space V_\Sigma of meromorphic differentials on \Sigma, endows an embedded curve \Sigma with the structure of the initial data of topological recursion, which defines a collection of symmetric tensors on V_\Sigma. These tensors produce a formal series, which turns out to be a formal Seiberg-Witten differential, that descends under a quotient to an analytic series.

uk: вхід без авторизації


keywords:
Mathematics
Algebraic Geometry
Differential Geometry

uk: Математика, природничі науки

uk: вхід без авторизації

name: Asymptotics of monopole moduli spaces (Survey)


uk: Математика, природничі науки

description: Magnetic monopoles and quantum theory have been joined at the hip from the beginning. Over the years monopoles have motivated, or at least had front row seats at, many of the most fundamental advances in quantum field theory. In recent times, monopoles have become a primary conduit through which quantum field theory directly impacts mathematics and vice versa. In this talk I will highlight some of these developments and connections.

uk: вхід без авторизації

name: Monopoles and Quantum Field Theory (Survey)


uk: Математика, природничі науки

description: The moduli spaces M_k of SU(2) monopoles on R^3 of charge k are among the oldest studied objects in gauge theory, yet open questions still remain, such as Sen's conjecture for their L^2 cohomology. I will discuss compactifications of these moduli spaces as manifolds with corners, with respect to which their hyperKahler metrics are of "quasi fibered boundary" (QFB) type, a metric structure which generalizes the quasi asymptotically locally euclidean (QALE) and quasi asymptotically conic (QAC) structures introduced by Joyce and others. This geometric structure, which is best understood by comparison to the simpler moduli space of point clusters on R^3, systematically organizes the various asymptotic regions of the moduli space in which charge k monopoles decompose into widely separated monopoles of charges summing to k. This is joint work with M. Singer and K. Fritzsch.

uk: вхід без авторизації

name: The quasi fibered boundary (QFB) compactification of monopole moduli spaces

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


Monopoles and difference modules

uk: Математика, природничі науки

description: We shall discuss equivalences between various types of monopoles and difference modules. They are variants of Kobayashi-Hitchin correspondences between algebro-geometric objects and differential geometric objects. We may also regard them as equivalences between monopoles and their underlying scattering data, which have been pursued in various contexts. Though it is difficult to construct monopoles explicitly, we would like to explain that some asymptotic properties can be easily understood through the corresponding difference modules.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The advances of the last 20 years of nonperturbative supersymmetric gauge theory led to the double deformation of complex geometry of Hitchin systems and moduli spaces of periodic monopoles. I will review these results and sketch a speculative path towards construction of hyperkahler metrics on these moduli spaces.

uk: вхід без авторизації

name: From instanton counting to hyperkahler geometry of monopole moduli


uk: Математика, природничі науки

description: I will discuss the effect of negative curvature and girth on singularity formation in Yang Mills and harmonic maps. This work is joint with Luca DiCerbo.

uk: вхід без авторизації

name: Bubbling threshholds for Yang Mills and harmonic maps in fat negatively curved spaces


uk: Математика, природничі науки

description: I will discuss my recent work constructing a non-conical singular Hermitian Yang-Mills connection on a homogeneous reflexive sheaf over \mathbb{C}^3, which is supposed to model the generic situation of bubbling phenomenon when the Fueter section has a zero. This example in particular shows that the uniqueness part of the Hitchin-Kobayashi correspondence does not extend naively to noncompact manifolds. A variant of this construction gives a sequence of HYM connections on the unit ball in \mathbb{C}^3 with uniformly bounded L^2 curvature, but the number of codimension 6 singularities tends to infinity along the sequence. This illustrates the substantial difficulty of the compactification problem in higher dimensional gauge theory.

uk: вхід без авторизації

name: High codimension phenomena for Hermitian Yang-Mills connections


uk: Математика, природничі науки

description: The moduli space of the Bogomolny Equations on R3 with certain asymptotic conditions has been fully studied by S.Donaldson, based on an algebraic geometry approach developed by N.Hitchin. An alternative analytical approach towards studying the moduli space was established by C.Taubes. An adaption of C.Taubes' method can also be used to study the moduli space of the Bogomolny equations on R3 with a knot singularity. Moreover, it is natural to ask, whether it is possible to study the equations with a knot singularity by algebraic geometry method If this is possible, then it may have the potential to bring knot theory into algebraic geometry.

uk: вхід без авторизації

name: Bogomolny Equations on R3 with a Knot Singularity


uk: Математика, природничі науки

description: Modular Origami is the art of constructing shapes, usually polyhedral ones, by putting together easy-to-fold modules or units that securely lock together without cuts or glue. The art enables construction of interesting and complex structures using simple units which look rather uninteresting just by themselves. During the workshop, we will make a simple six-unit modular origami cube with patterns on the faces. We will then discuss other possible constructions with the same kinds of units. If time permits, we can even construct a three-unit hexahedron. However, we will not have time to construct the other possible polyhedral structures during workshop hour, though you will leave with enough basic knowledge and resources to construct on your own at home. Recommended materials: 6 sheets of 4-6â squares of origami paper, preferably in three colors, two sheets of each. Also, have at least about a dozen extra such sheets handy for learning about the additional possible structures. Note that regular origami paper is colored on one side and white on the other. You could use any other paper that has different colors on either side, as long as you can cut them yourselves into precise squares before the workshop begins.

uk: вхід без авторизації

name: Exploring Polyhedral Constructions with Modular Origami


uk: Математика, природничі науки

description: It is vital for preservice teachers to experience innovative and creative mathematics learning and teaching if they are to be successful in their future classrooms. The Geometry as Arts project was a vehicle to explore some mathematical ideas in an elementary preservice teachersâ classroom. This presentation is a qualitative probing slice of a longitudinal study concerned with pre-service teacher beliefs, values, and attitudes toward kinds of pedagogical approaches that engage mathematics learners and lead to desirable outcomes.

uk: вхід без авторизації

name: Creativity, Change and Innovation: Geometry as Art in Preservice Teachers Classroom


uk: Математика, природничі науки

description: The sixth Huzita-Justin origami axiom, first discovered by Italian mathematician Margherita Beloch, allows for geometric constructions not possible with a compass and straightedge. Some of the problems that stumped early geometers but are solvable with this move include trisecting the angle, doubling the cube, solving cubic equations, and constructing regular heptagons. Utilizing origami, we will demonstrate how this so-called Beloch move is equivalent to finding the mutual tangent to two parabolas and unlock the mystery of a sevenfold Islamic pattern. Recommended materials: Plain printer paper and pencil/pen

uk: вхід без авторизації

name: Mini- Course #1 (Part1): Folding Sevens: The Power of Origami


uk: Математика, природничі науки

description: We give a visual introduction to several geometric techniques for analyzing data. These include both unsupervised learning (clustering, dimensionality reduction, topic modeling), and supervised learning (k-nearest neighbors, support vector machines), though we don't expect you to know what any of those words mean! The goal is to distill the methods down to visual and oral description without mathematical notation. The performance of data analysis techniques will be illustrated on real-world image and text datasets. Mini-course participants will be encouraged to develop their own purely visual explanations

uk: вхід без авторизації

name: Mini Course #2 (Part1): A Visual Introduction to Geometric Data Analysis


uk: Математика, природничі науки

description: In this lecture, I will discuss the process of creating, implementing, and accessing the impact of an Indigenous beadwork assignment in a second-year undergraduate linear algebra course at my university. Emphasis is placed on the process behind the project, including the motivation, context, and relationship building, and I will report our findings.

uk: вхід без авторизації

name: Indigenous Beadwork in a Mathematics Classroom


uk: Математика, природничі науки

description: We begin with a blank sheet of paper and step by step draw layers of lines and circles using a compass and straightedge. From this underlying grid, we can derive a series of related patterns. These are patterns that would have been drawn in the same way across the Islamic Lands for many centuries. They have adorned architecture and objects and illuminated Manuscripts from Spain across to China from the 8th century till now. To begin the session Samira will outline the process, tools and potential outcomes. She will then guide you through drawing a single unit of a repeating pattern step by step so participants may draw along at the same time. Samira will then demonstrate how to draw a slightly more complex pattern derived from the same underlying grid. As this will be at a slightly quicker pace, participants will be sent a PDF step by step guide to aid them to continue drawing a range of related patterns after the session. Materials required: A4 plain paper x 2, a compass, a sharp pencil, a ruler, fineliner pens, sharpener, eraser, colouring pens & pencils.

uk: вхід без авторизації

name: Drawing Islamic Geometric Patterns using a Compass and Straightedge


uk: Математика, природничі науки

description: Strictly speaking, a wallpaper pattern lives in the plane. When we attempt to weave a wallpaper pattern from bands, as in the image shown, the result is no longer a planar object. The symmetry group of this pattern is one of the eighty layer groups. The main point of this presentation is to familiarize the audience with these less well-known groups and highlight their importance in the study of symmetry. For many of the wallpaper groups, there are two ways to extend the action on the plane to an action on space that preserve the base plane. For instance, in the example, the mirror symmetry in the wallpaper group becomes a flip symmetry of the woven pattern in space. This leads to a rich set of examples, which we find beautiful as well as mathematically interesting.

uk: вхід без авторизації

name: Wallpaper Patterns from Looping Strands: The Layer Groups


The Math Behind the Pop-up Spinner

uk: Математика, природничі науки

description: Pop-up books and cards have been around since the 18th century, and recently have seen a surge in popularity through the elaborate designs of pop-up masters like Robert Sabuda and Matthew Reinhart. But the most stunning and elegant pop-up effect I have encountered is the Pop-Up Spinner card invented by an anonymous Japanese student. How it functions turns out to depend on a geometric theorem concerning linkages proven in an undergraduate thesis. I will explain the connection and prove the theorem.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Thurston's geometrization conjecture, proved by Perelman in 2003, states that every three-dimensional manifold can be cut into a collection of pieces, each of which has one of eight geometries. These "Thurston geometries" include the euclidean, spherical, and hyperbolic geometries, the products of the two-dimensional spherical and hyperbolic geometries with the euclidean line, and three other, stranger geometries. In this talk, I'll describe joint work with RÃ©mi Coulon, Sabetta Matsumoto, and Steve Trettel, in which we simulate the "inside view" from within each of the Thurston geometries. That is we generate images assuming that light travels along geodesics in the geometry of the manifold.

uk: вхід без авторизації

name: Viewing the Thurston geometries from the inside


uk: Математика, природничі науки

uk: вхід без авторизації

name: Mini-Course #1 (Part2): Folding Sevens: The Power of Origamiâ


uk: Математика, природничі науки

description: This hands-on workshop will explore the Wurzelschnecke ("root snail"), a simple and elegant spiral construction based on right angle triangles, first attributed to Plato's tutor Theodorus of Cyrene in the 5th C BCE and still studied by mathematicians in our time. We will make and view versions of the Wurzelschnecke at a variety of scales and materials, and play with its possibilities in embodied geometric design in architecture, playground equipment, jewelry, mathematical millinery and more. Our focus will be on its potential use in education, supporting understanding of irrational numbers and trigonometry, and in the geometry of design. I'll share examples from earlier workshops with middle school and high school students, math teachers, grad students and faculty. Workshop participants should have the following materials at hand, if possible: paper, pencil or pen, ruler or straightedge with a square corner, scissors, corrugated cardboard. A protractor and/or carpenter's square are optional but might be helpful.

uk: вхід без авторизації

name: The Wurzelschnecke (AKA Spiral of Theodorus): Understanding number and creating geometric design


uk: Математика, природничі науки

description: We give a visual introduction to several geometric techniques for analyzing data. These include both unsupervised learning (clustering, dimensionality reduction, topic modeling), and supervised learning (k-nearest neighbors, support vector machines), though we don't expect you to know what any of those words mean! The goal is to distill the methods down to visual and oral description without mathematical notation. The performance of data analysis techniques will be illustrated on real-world image and text datasets. Mini-course participants will be encouraged to develop their own purely visual explanations.

uk: вхід без авторизації

name: Mini-Course #2 (Part2): A Visual Introduction to Geometric Data Analysis


uk: Математика, природничі науки

uk: вхід без авторизації

name: Panel Discussion: Education + Art + Research


keywords:
Mathematik
Mathematics
Relativity And Gravitational Theory
Quantum Theory
Mathematical Physics

uk: Математика, природничі науки

description: I will present an argument that if a theory of quantum gravity is physically discrete at the Planck scale and the theory recovers General Relativity as an approximation, then, at the current stage of our knowledge, causal sets must arise within the theory, even if they are not its basis.

uk: вхід без авторизації

name: Recovering General Relativity from a Planck scale discrete theory of quantum gravity

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


uk: Математика, природничі науки

description: When a massive quantum body is put into a spatial superposition, it is of interest to consider the quantum aspects of the gravitational field sourced by the body. In this talk, I will discuss a Gedankenexperiment where Alice and Bob control masses in quantum superposition. I will show that the analysis of this experiment does not lead to any inconsistency provided the gravitational radiation is quantized and that vacuum fluctuations limit the localization of a particle to no better than a Planck length. This provides support for the view that (linearized) gravity should have a quantum field description and that table-top experiments testing entanglement of systems interacting via gravity do probe the quantum nature of gravity, even if no ``gravitons'' are emitted during the experiment.

uk: вхід без авторизації

name: Quantum Superposition of Massive Objects and the Quantization of Gravity


uk: Математика, природничі науки

description: We are building an experiment in which a nitrogen-vacancy-centre electron spin would be used to put a levitated nanodiamond into a spatial quantum superposition [1-3]. This would be able to test theories of spontaneous wavefunction collapse and is the first step of a much more ambitious experiment to test if gravitational effects can be in a quantum superposition [4, 5]. This talk will describe our current experimental design [6-11], and our latest experimental progress: the first steps on a long journey. [1]A. T. M. A. Rahman, A. C. Frangeskou, M. S. Kim, S. Bose, G. W. Morley & P. F. Barker, Sci. Rep. 6, 21633 (2016). [2]A. T. M. A. Rahman, A. C. Frangeskou, P. F. Barker & G. W. Morley, RSI 89, 023109 (2018). [3]A. C. Frangeskou, A. T. M. A. Rahman, L. Gines, S. Mandal, O. A. Williams, P. F. Barker & G. W. Morley, NJP 20, 043016 (2018). [4]S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. ToroÅ¡, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim & G. Milburn, PRL 119, 240401 (2017). [5]C. Marletto & V. Vedral, PRL 119, 240402 (2017). [6]S. Bose & G. W. Morley, arXiv:1810.07045 (2018). [7]M. Scala, M. S. Kim, G. W. Morley, P. F. Barker & S. Bose, PRL 111, 180403 (2013). [8]C. Wan, M. Scala, G. W. Morley, A. T. M. A. Rahman, H. Ulbricht, J. Bateman, P. F. Barker, S. Bose & M. S. Kim, PRL 117, 143003 (2016). [9]J. S. Pedernales, G. W. Morley & M. B. Plenio, PRL 125, 023602 (2020). [10] R. J. Marshman, A. Mazumdar, G. W. Morley, P. F. Barker, S. Hoekstra & S. Bose, NJP 22, 083012 (2020). [11] Z.-q. Yin, T. Li, X. Zhang & L. M. Duan, PRA 88, 033614 (2013).

uk: вхід без авторизації

name: Levitating nanodiamond experiments: towards a test of quantum gravity


uk: Математика, природничі науки

description: In a quantum world, reference frames are ultimately quantum systems too -- but what does it mean to "jump into the perspective of a quantum particle" In this work, we show that quantum reference frame (QRF) transformations appear naturally as symmetries of simple physical systems. This allows us to rederive and generalize known QRF transformations within an alternative, operationally transparent framework, and to shed new light on their structure and interpretation. We give an explicit description of the observables that are measurable by agents constrained by such quantum symmetries, and apply our results to a puzzle known as the `paradox of the third particle'. We argue that it can be reduced to the question of how to relationally embed fewer into more particles, and give a thorough physical and algebraic analysis of this question. This leads us to a generalization of the partial trace (`relational trace') which arguably resolves the paradox, and it uncovers important structures of constraint quantization within a simple quantum information setting, such as relational observables which are key in this resolution. While we restrict our attention to finite Abelian groups for transparency and mathematical rigor, the intuitive physical appeal of our results makes us expect that they remain valid in more general situations.

uk: вхід без авторизації

name: Quantum reference frame transformations as symmetries and the paradox of the third particle


Thermality of circular motion

uk: Математика, природничі науки

description: An observer in uniform linear acceleration responds to the Minkowski vacuum thermally, in the Unruh temperature T_U =\frac{proper\, acceleration}{2 \pi}. An observer in uniform circular motion experiences a similar Unruh-type temperature T_c, with better prospects of detection in analogue spacetime laboratory experiments, but T_c depends not just on the proper acceleration but also on the orbital radius and on the excitation energy. We establish a range of analytic and numerical results for T_c for a massless scalar field in 3+1 and 2+1 spacetime dimensions, the latter being motivated by proposed condensed matter experiments. In particular, we find that the circular motion analogue Unruh temperature grows arbitrarily large in the near-sonic limit, encouragingly for the experimental prospects, but less quickly in effective spacetime dimension 2+1 than in 3+1. [Based on Biermann et al, Phys. Rev. D 102, 085006 (2020)]

uk: вхід без авторизації


uk: Математика, природничі науки

description: Entanglement harvesting has been studied extensively using the Unruh-deWitt (UdW) detector. In this detector the matter systems are modelled as two-level quantum detector systems while the classical centre of mass degrees of freedom are described using a smearing profile function. As has been extensively shown, entanglement harvesting depends very sensitively on the detector details. In this talk we consider two quantum delocalized detectors in their respective ground states [1], and we ask how their ability to become entangled with each other is affected by their mass and their initial centre of mass delocalization. For comparison we consider entanglement harvesting from two UdW detectors with classical centre of mass and Gaussian smearing profile. We will show that the process of entanglement harvesting is affected by the coherent delocalization of matter and, in particular, that delocalized detectors harvest less entanglement than detectors whose centre of mass degrees of freedom are assumed to behave classically. We will also identify the limit in which the results for entanglement harvesting for coherently delocalized detectors reduce to the results for detectors with classical external degrees of freedom [2]. [1] N. Stritzelberger and A. Kempf, â Coherent delocalization in the light-matter interactionâ , Phys. Rev. D, 101, 036007, 2020. [2] Nadine Stritzelberger, Laura J. Henderson, Valentina Baccetti, Nicolas C. Menicucci, and Achim Kempf, â Entanglement harvesting with coherently delocalized matterâ , Phys. Rev. D 103, 016007, 2021.

uk: вхід без авторизації

name: Vacuum entanglement harvesting with delocalized matter


uk: Математика, природничі науки

description: Quantum mechanics and general relativity are two fundamentally different theories and have both been tested independently with very high precision. However, even after a century of research, the interplay of those two very different theories has never been tested experimentally. Within this talk I will present the experimental research aiming to explore this interface between quantum mechanics and general relativity by performing high-precision experiments at the level of single quanta of light, the photons. Such quantum systems allow one to examine the influence of gravity on interference effects. For this purpose, a high-precision interferometer whose paths are subject to different gravitational potentials will be used.

uk: вхід без авторизації

name: Shining light on the interface of gravity and quantum physics: precision measurements using photonic quantum interferometry


uk: Математика, природничі науки

description: I will discuss the notion of time-delocalised quantum systems and its relevance to the question of realisability of processes with indefinite causal order in standard quantum theory and quantum gravity. I will explain how given a description of an experiment in the form of a (generally cyclic) circuit, the experiment can be described with respect to an arbitrary alternative choice of (sub)systems. This provides a very general notion of transformation between different equivalent descriptions of an experiment. I will review how the quantum SWITCH can be seen as realisable on time-delocalised systems in standard quantum mechanics and will show that all unitarily extendible tripartite processes admit such realisations. Notably, this includes processes violating causal inequalities, among which some purely classical ones. I will discuss the meaning of causal inequality violations with such time-delocalised variables. Finally, I will speculate on the link between time-delocalised systems and quantum reference frames.

uk: вхід без авторизації

name: Quantum processes on time-delocalised systems


uk: Математика, природничі науки

description: The notorious Wigner's friend thought experiment has in recent years received renewed interest especially due to new arguments that force us to question some of the fundamental assumptions of quantum theory. In this work we formulate a no-go theorem for the persistent reality of Wigner's friend's perception, which allows us to conclude that the perceptions that the friend has of her own measurement outcomes at different times cannot "share the same reality", if seemingly natural quantum mechanical assumptions are met. We show that there is no joint probability distribution for the friend's perceived measurement outcomes at two different times, that depends linearly on the initial state of the measured system and whose marginals reproduce the predictions of unitary quantum theory. This theorem entails that one must either (1) propose a nonlinear modification of the Born rule for two-time predictions, (2) sometimes prohibit the use of present information to predict the future --thereby reducing the predictive power of quantum theory-- or (3) deny that unitary quantum mechanics makes valid single-time predictions for all observers. We briefly discuss which of the theorem's assumptions are more likely to be dropped within various popular interpretations of quantum mechanics.

uk: вхід без авторизації

name: A no-go theorem for the persistent reality of Wigner's friend's perception


uk: Математика, природничі науки

uk: вхід без авторизації

name: Analog Quantum Simulation of Strongly-Coupled Field Theories with a Parametric Cavity


uk: Математика, природничі науки

description: Do astrophysical black holes observed by LIGO and EHT actually trap light We argue that properties of the near-horizon regions, such as an atmosphere of exotic matter violating the null energy condition or a firewall produced by an accreting black hole, combined with our current understanding of physics make this unlikely. If, however, a distant observer does detect an apparent horizons this will indicate the onset of radically new physics.

uk: вхід без авторизації

name: Why astrophysical black holes do not have horizons, and what it means if they do


uk: Математика, природничі науки

description: In this talk I will discuss the criteria for constructing a covariant quantum dynamics for causal sets using the Rideout-Sorkin sequential growth paradigm. In the histories-formulation, the growth process determines the complex weight or quantum measure, with the covariant observables or be-ables given by the covariant measurable sets. The existence of covariant observables in turn puts constraints on the allowed growth dynamics. I will present some recent results with Stav Zalel on a simplified â abelianâ model which demonstrates the existence of classes of covariant quantum dynamics. I will end the talk with a broad discussion of this observer independent formulation of quantum gravity.

uk: вхід без авторизації

name: Growing a Quantum Discrete Universe, Causally


uk: Математика, природничі науки

description: We will discuss how relativistic causality and covariance play a role in the measurement problem in quantum fields and even in our ability to put information into a quantum field. We will compare the Fewster-Verch formalism with the Unruh-DeWitt-like (UDW) particle detector models. We will discuss the limitations of both measurement frameworks as well as the typical abuses that are often made with Unruh-DeWitt-like detector models and how they matter in general relativistic scenarios. Finally we will discuss the claim in [arXiv:2103.13400] that weakly coupled detectors cannot harvest entanglement and show what is the effect of that claim, if any, on the feasibility of entanglement harvesting with particle detectors.

uk: вхід без авторизації

name: A tale of two detector models: Causal structure and measurements in quantum fields


uk: Математика, природничі науки

description: Understanding and characterising causal structures which posses provably non-classical features is of direct relevance for the research on quantum theory with indefinite causal order as well as for quantum gravity. Due to diffeomorphism invariance of general relativity (GR), causal structure of spacetime in GR requires an operational description â in terms of physical systems and their ability to exchange information. In quantum theory such systems can be taken as composite quantum particles, modelling ideal clocks as well as particle detectors. I will discuss how such particles allow insights into quantum causal structures by allowing us to construct quantum spacetimes and quantify non-classical features of their causal structures. Surprisingly, this approach also shows that operational means may fail to distinguish a genuine superposition of different (non-diffeomorphic) spacetimes from a single classical spacetime. This opens a question: under what conditions non-classical spacetimes can be operationally distinguished from the classical ones.

uk: вхід без авторизації

name: Clocks and detectors for characterising quantum causal structures


uk: Математика, природничі науки

description: The process matrix formalism formalises non-classical causal structures, which are expected to emerge in a theory that combines quantum indeterminacy with the dynamical causal structure of general relativity. The framework relies on the notion of local laboratoriesâ an abstraction of spacetime events or regionsâ where local operations and measurements can be performed. In a theory of quantum gravity, we expect that it should not be possible to label laboratories a priori: this requires a background reference frame (although possibly non-classical), while general relativity is background independent. I will present a background independent formulation of the process matrix formalism that incorporates a discrete version of background independence: invariance under permutations. Although the formalism abandons an absolute notion of â local operationâ , such a notion re-emerges as relative to a physical reference frame. Permutation symmetry also implies some type of superselection rule, although, for quantum processes, this cannot be interpreted as implying the existence of a conserved charge with a definite value. Finally, I will comment on the possibility to extend the approach to include quantum coordinate transformations.

uk: вхід без авторизації

name: Background independent quantum causal structures


uk: Математика, природничі науки

description: Quantum gravity is yet to be tested in a laboratory. I will provide criteria for testing the quantum fluctuation of a graviton in a laboratory by colliding two non-relativistically Schrödinger' s kittens and study how the final states would be entangled in this process. Realising such an experiment will be filled with various challenges, from developing new technologies to logistics to financial costs. I will motivate the community, despite all these challenges, why doing such an experiment is necessary to understand both the foundations of quantum mechanics and gravity.

uk: вхід без авторизації

name: Quantum test of gravity by colliding Schrödinger's kittens


uk: Математика, природничі науки

description: There is no empirical evidence yet as to â whetherâ gravity has a quantum mechanical origin. Motivated by this, I will present a feasible idea for testing the quantum origin of the Newtonian interaction based on the simple fact that two objects cannot be entangled without a quantum mediator. I will show that despite its weakness, gravity can detectably entangle two adjacent micron sized test masses held in quantum superpositions even when they are placed far apart enough to keep Casimir-Polder forces at bay. A prescription for witnessing this entanglement through spin correlations is also provided. Further, I clarify the assumptions underpinning the above proposal such as our reasonable definition of â classicalityâ , as well as relativistic causality. We note a few ways to address principal practical challenges: Decoherence, Screening EM forces and Inertial noise reduction. I will also describe how unprecedented compact sensors for classical gravity (including meter scale sensors for low frequency gravitational waves) will arise on the way to the above grand goal.

uk: вхід без авторизації

name: Quantum Nature of Gravity in the Lab: Assumptions, Implementation and Applications on the Way


uk: Математика, природничі науки

description: Quantum field fluctuations are the more strongly correlated the smaller their spacetime distance. As a consequence, the very notion of distance can be replaced by the notion of correlation strength. This suggests a picture in which all degrees of freedom are described by the same abstract structure, namely (multi-point) correlators, a picture which is essentially information theoretic. At low energies, these abstract correlators possess a mathematical representation as the correlation functions of matter fields that live on a curved spacetime with a definite causal structure. As one approaches the Planck scale, the abstract correlators may no longer possess such a representation, thereby superseding the notions of matter and of spacetime, but the abstract correlators can still be examined information theoretically.

uk: вхід без авторизації

name: Spacetime and causality when the notion of distance is replaced by the notion of correlation.


uk: Математика, природничі науки

description: In physics, a reference frame is an abstract coordinate system that specifies observations within that frame. Recently, it has been proposed to consider reference frames as physical objects that ultimately obey quantum mechanical laws. The transformations between such quantum reference frames can be seen as an extension of the transformations between classical reference frames. In my talk, I will show how the notion of quantum reference frames can help us formulate and test the weak and Einstein equivalence principles when a "free-falling laboratory" is in a quantum state (which can be spatiotemporally extended, for example), or when it falls into a superpositon of gravitational fields. This gives rise to the quantum Einstein equivalence principle, which could pave a fruitful way to the conceptual foundation of a future theory of quantum gravity.

uk: вхід без авторизації

name: Quantum reference frames and the weak and Einstein equivalence principles

name: Banff International Research Station (BIRS) for Mathematical Innovation and Discovery


Quantum Relativity of Subsystems

uk: Математика, природничі науки

description: One of the most basic notions in physics is the partitioning of a system into subsystems, and the study of correlations among its parts. In this talk, we will explore these notions in the context of quantum reference frame (QRF) covariance, in which this partitioning is subject to a symmetry constraint. We will see that different reference frame perspectives induce different sets of subsystem observable algebras, which leads to a gauge-invariant, frame-dependent notion of subsystems and entanglement. We will then see that subalgebras which commute before imposing the symmetry constraint can translate into non-commuting algebras in a given QRF perspective after symmetry imposition. Such a QRF perspective does not inherit the distinction between subsystems in terms of the corresponding tensor factorizability of the kinematical Hilbert space and observable algebra. Since the condition for this to occur is contingent on the choice of QRF, the notion of subsystem locality is frame-dependent. Finally I will comment on the relevance of these results for existing programs in quantum gravity, as well as relate them to existing results on the relativity of quantum systems within approaches in quantum information. This talk is based on joint work with Philipp A. Hoehn, Maximilian P. E. Lock, Shadi Ali Ahmad and Alexander R. H. Smith which can be found at arxiv:abs/2103.01232.

uk: вхід без авторизації


uk: Математика, природничі науки

description: I'll briefly overview the basic notion that one can look for entanglement generation via gravity in experiments. In particular I will present a recent proposal (joint with H. Muller and J. M. Taylor) using an atom interferometer coupled to a high-Q mechanical mass. The central idea relies on a new method of entanglement verification in time-dependent systems. Time permitting, I'll also make some comments about the interpretation of these entanglement experiments more broadly, particularly the relation to gravitons.

uk: вхід без авторизації

name: Tests and implications of gravitational entanglement


uk: Математика, природничі науки

description: Quantum field theory is completely characterized by the field correlations between spacetime points. In turn, some of these can be accessed by locally coupling to the field simple quantum systems, a.k.a. particle detectors. In this work, we consider what happens when a quantum- controlled superposition of detectors at different space-time points is used to probe the correlations of the field. We show that, due to quantum interference effects, two detectors can gain information on field correlations which would not be otherwise accessible. This has relevant consequences for information theoretic quantities, like entanglement and mutual information harvested from the field. In particular, the quantum control allows for extraction of entanglement in scenarios where this is otherwise provably impossible.

uk: вхід без авторизації

name: Entanglement Harvesting with a Temporal Superposition


Quantum Superpositions of Graphs

uk: Математика, природничі науки

description: Graphs are a discrete topological canvas that in many applications can completely replace a continuous manifold. Nevertheless, invariance of dynamics defined on graphs under changes of reference frames is typically attempted by embedding the graph in an ambient manifold. This seems superfluous as node names already label points similarly to a choice of coordinates in continuous space. As node names are fiducial, graph renamings can be seen as a change of coordinates on the graph. Thus, graph renamings correspond to a natively discrete analogue of diffeomorphisms. In quantum theory, node names become even more important. We first provide a robust notion of quantum superpositions of graphs and argue with a simple example that in quantum theory in order to avoid instantaneous signalling it is necessary to use node names to define the â localisationâ of a node, rather than values of a physical field. We propose renaming invariance as a symmetry principle of similar weight to diffeomorphism invariance and show how to impose it at the level of quantum superpositions of graphs.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Transformations between reference frames play a crucial role in our understanding of physical processes. In practice, reference frames are realised by physical systems, which are standardly treated as classical. However, assuming that every physical system is ultimately quantum, it is interesting to ask how a theory of transformations wrt quantum reference frames would look like, and what implications it would have for our description of spacetime. Recently, there has been a lot of effort towards developing a quantum generalisation of reference frame transformations, unveiling novel phenomena that are absent in the classical treatment of reference frames. Here, we develop a first-principles framework for quantum reference frame transformations which clarifies important conceptual issues of previous treatments. Based on the algebra of relative observables between a system and a reference frame, our operational perspective leads naturally to a mixed-state approach (incoherent twirling), in contrast to current pure-state approaches (coherent twirling). Within our framework, the full invariant quantum subsystem contains not only the algebra of relative observables between the system and the reference frame but also an â extra particle,â related to the invariant degrees of freedom of the reference frame itself. Importantly, this extra particle contains information about the â quantumnessâ of the reference frame and is essential to the unitarity of quantum reference frame transformations. Our approach is general, in the sense that it can be applied to a vast set of symmetry groups and to any type of system. We illustrate the physical meaning of the concepts developed by analysing quantum reference frame transformations with respect to the (centrally extended) Galilei group.

uk: вхід без авторизації

name: Relative subsystems and quantum reference frame transformations


uk: Математика, природничі науки

uk: вхід без авторизації

name: Discussion session, June 1, 2021 09:32 - 10:21


uk: Математика, природничі науки

uk: вхід без авторизації

name: Discussion session, June 1, 2021 17:41 - 18:15


uk: Математика, природничі науки

uk: вхід без авторизації

name: Discussion session, June 2, 2021 17:51 - 18:20


uk: Математика, природничі науки

uk: вхід без авторизації

name: Discussion session, June 3, 2021 09:30 - 10:08


uk: Математика, природничі науки

uk: вхід без авторизації

name: Discussion session, June 4, 2021 09:23 - 10:01


uk: Математика, природничі науки

uk: вхід без авторизації

name: Discussion session, June 4, 2021 17:55 - 18:07


uk: Математика, природничі науки

description: Ribbon categories are 3-dimensional algebraic structures that control quantum link polynomials and that give rise to 3-manifold invariants known as skein modules. I will describe how to use Khovanov-Rozansky link homology, a categorification of the gl(N) quantum link polynomial, to obtain a 4-dimensional algebraic structure that gives rise to vector space-valued invariants of smooth 4-manifolds. The technical heart of this construction is the functoriality of Khovanov-Rozansky homology in the 3-sphere. Based on joint work with Scott Morrison and Kevin Walker.

uk: вхід без авторизації

name: Invariants of 4-manifolds from Khovanov-Rozansky link homology


uk: Математика, природничі науки

description: What do annular Khovanov homology, Ozsvath-Szabo's "correction terms", Kapustin-Witten equations, and enumerative BPS invariants have in common The goal of the talk will be to explain, from multiple perspectives, how this structure makes a somewhat surprising appearance in a problem of generalizing Khovanov homology to homology of knots in arbitrary 3-manifolds.

uk: вхід без авторизації

name: From knot homology to 3-manifold homology


Annular link Floer homology and gl(1|1)

uk: Математика, природничі науки

description: The Reshetikhin-Turaev construction for the quantum group U_q(gl(1|1)) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. Tangle Floer homology is a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. In earlier work with Ellis and Vertesi, we show that tangle Floer homology categorifies a Reshetikhin-Turaev invariant arising naturally in the representation theory of U_q(gl(1|1)); we further construct bimodules \E and \F corresponding to E, F in U_q(gl(1|1)) that satisfy appropriate categorified relations. After a brief summary of this earlier work, I will discuss how the horizontal trace of the \E and \F actions on tangle Floer homology gives a gl(1|1) action on annular link Floer homology that has an interpretation as a count of certain holomorphic curves. This is based on joint work in progress with Andy Manion and Mike Wong.

uk: вхід без авторизації


uk: Математика, природничі науки

description: We will discuss properties of the stable homotopy refinement of Khovanov homology and some aspects of its construction. We will focus on features that also appear for other Floer-type invariants, and on gaps in our understanding. The results are joint with Tyler Lawson and Sucharit Sarkar (or are due to other people).

uk: вхід без авторизації

name: Khovanov stable homotopy type and friends


uk: Математика, природничі науки

description: We describe invariant counts of holomorphic curves in a Calabi-Yau 3-fold with boundary in a Lagrangian in the skein module of that Lagrangian. We show how to turn this into concrete counts for the toric brane in the resolved conifold. This leads to a notion of basic holomorphic disks for any knot conormal in the resolved conifold. These basic holomorphic disks seem to generate HOMFLY homology in the basic representation. We give a conjectural description of similar holomorphic object generating parts of higher symmetric representation HOMFLY homology and verify some predictions coming from this conjecture in examples.

uk: вхід без авторизації

name: Skein valued curve counts, basic holomorphic disks, and HOMFLY homology


uk: Математика, природничі науки

description: We define a new family of commuting operators F_k in Khovanov-Rozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that F_2 satisfies "hard Lefshetz property" and hence exhibits the symmetry in Khovanov-Rozansky homology conjectured by Dunfield, Gukov and Rasmussen. This is a joint work with Matt Hogancamp and Anton Mellit.

uk: вхід без авторизації

name: Tautological classes and symmetry in Khovanov-Rozansky homology


uk: Математика, природничі науки

description: I will survey link concordance invariants coming from Khovanov homology, particularly those similar in spirit to OzsvÃ¡th-Stipsicz-SzabÃ³'s Upsilon, a 1-parameter family of invariants coming from knot Floer homology. This is related to my joint work with Linh Truong on annular link concordance invariants as well as ongoing work with Ross Akhmechet.

uk: вхід без авторизації

name: Upsilon-like invariants from Khovanov homology


uk: Математика, природничі науки

description: A well-known strategy to disprove the smooth 4D Poincare conjecture is to find a knot that bounds a disk in a homotopy 4-ball but not in the standard 4-ball. Freedman, Gompf, Morrison and Walker suggested that Rasmussenâ s invariant from Khovanov homology could be useful for this purpose. I will describe three recent results about this strategy: that it fails for Gluck twists (joint work with Marengon, Sarkar and Willis); that an analogue works for other 4-manifolds (joint work with Marengon and Piccirillo); and that 0-surgery homeomorphisms provide a large class of potential examples (joint work with Piccirillo).

uk: вхід без авторизації

name: Khovanov homology and the search for exotic 4-spheres