uk: Математика, природничі науки

description: We present a continuum-level description of dynamics of a thin electrolyte film on substrate which is characterized by spatially periodic variation of surface properties.The model couples together the electrostatic effects and viscous flow in the liquid. Linear stability analysis is carried out using a combination of numerical techniques for finding the eigenvalues of the discretized stability problem, asymptotic methods valid for small charge density variation, and Floquet theory. Substrate non-uniformity can have either stabilizing or destabilizing effect. For the important practical case of a liquid film with oppositely charged boundaries and thickness comparable to the Debye length, transition from stabilizing to destabilizing influence is observed as the patterning wavelength is decreased. Numerical simulations of the strongly nonlinear evolution of the film are conducted, with emphasis on competition between patterns induced by substrate nonuniformity and by the intrinsic nonlinearity present even for uniform substrate. The topic of motion of contact line over patterned surface will also be discussed briefly. (Joint work with M. Jutley.)

uk: вхід без авторизації

name: Stability and break-up of thin electrolyte films on patterned surfaces


Satish Kumar

type: Person

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: When evaporated on a substrate, a droplet containing solutes usually deposits the solutes onto the substrate in a ``coffee-ring'' pattern. Recent experiments have shown that substrate permeability can suppress the coffee-ring pattern and promote more uniform solute deposition. Motivated by these observations, we have developed a lubrication-theory-based model to describe imbibition and evaporation of droplets of colloidal suspensions on permeable substrates. The model consists of a system of one-dimensional partial differential equations accounting for the changing droplet shape and depth-averaged concentration of colloidal particles. We also incorporate a precursor film, disjoining pressure, and substrate topography to control contact-line motion of the droplet. Solvent evaporation is described using the well-known one-sided model, and imbibition of solvent by the substrate is assumed to only depend on the excess pressure on the liquid side. The governing equations are solved with finite-difference methods. Our results reveal that solvent evaporation and solvent imbibition have the same qualitative effect on the final particle deposition pattern. For the case where the substrate is smooth, we find that increasing imbibition or evaporation leads to a transition from a cone-shaped deposition pattern to a ring-shaped deposition pattern. For the case where the substrate is rough, the droplet contact line is pinned at a defect on the substrate, and the pinning-depinning transition leads to the ``bullseye'' deposition pattern often observed in experiments. Finally, we also find that particle adsorption onto the substrate can promote more uniform particle deposition patterns for both smooth and rough substrates, and solvent imbibition can indirectly suppress the coffee-ring pattern by inducing more particle adsorption.

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: Imbibition and Evaporation of Droplets of Colloidal Suspensions on Permeable Substrates

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


The Drying of Liquid Droplets

uk: Математика, природничі науки

description: The drying of liquid droplets is a common daily life phenomenon that has long held a special interest in scientific research. We propose an Onsager variational principle theory that describes the droplet shape evolution and predicts the deposit distribution of nonvolatile components on the substrate. It is shown that for the drying of a single droplet, the deposition pattern changes continuously from a coffee ring to volcano-like and to mountain-like depending on the mobility of the contact line and the evaporation rate. When drying of two neighbouring droplets, asymmetrical ring-like deposition patterns are formed, including fan-like and eclipse-like deposition patterns. The same theoretical model is also used to explain the multi-ring patterns of the deposit that are formed when colloidal suspensions are dried on a substrate. Using a standard model for the stick-slip motion of the contact line, the theory predicts (a) the multi-ring patterns are not observed at high evaporation rate, (b) the spacing between rings decreases with the decrease of the ring radius, and (c) the multi-ring pattern is taken over by a disk pattern near the center. These results are in qualitative agreement with existing experiments, and the predictions of the theory about how the evaporation rate, droplet radius and receding contact angle affects the pattern can be tested experimentally.

uk: вхід без авторизації


Thin films of viscoplastic fluid

uk: Математика, природничі науки

description: Given suitable constitutive laws, thin-film equations can be derived for non-Newtonian fluids. In this talk, I will review the procedure for fluids with a yield stress, and then confront the resulting theory with experiments and computations that avoid the thin-layer approximation. The comparison with the computations illustrates that the theory can be effective, but the comparison with experiments highlights how additional physical considerations are often needed in realistic settings.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The formation of iterated structures, such as satellite and subsatellite drops, filaments, and bubbles, is a common feature in interfacial hydrodynamics. Here we undertake a computational and theoretical study of their origin in the case of thin films of viscous fluids that are destabilized by long-range molecular or other forces. We demonstrate that iterated structures appear as a consequence of discrete self-similarity, where certain patterns repeat themselves, subject to rescaling, periodically in a logarithmic time scale. The result is an infinite sequence of ridges and filaments with similarity properties. The character of these discretely self-similar solutions as the result of a Hopf bifurcation from ordinarily self-similar solutions is also described. Joint work with M. Dallaston, D. Tseluiko and S. Kalliadasis.

uk: вхід без авторизації

name: Discrete self-similarity in thin film equations and the formation of iterated structures


uk: Математика, природничі науки

description: The presence of a free-surface between air and liquid can lead to nonlinear behaviour, even in the absence of fluid inertia. Consequently, multiple distinct flow states may occur under the same driving conditions. A natural question to ask is whether simplified models of the system can capture the correct nonlinear behaviour, compared to the full field equations. For the (canonical) flow of a liquid film on the inner, or outer, surface of a rotating cylinder, we find multiple solutions and other complex behaviours can be accurately described over a wider range of parameters using a simplified model system derived via gradient dynamics rather than strict asymptotic arguments.

uk: вхід без авторизації

name: Multiple solutions in free surface flows


uk: Математика, природничі науки

description: The tear film has a heterogeneous structure. Next to the ocular surface is a region hundreds of nanometers thick that is rich in mucin polymers. Atop this so-called mucus layer lies the aqueous layer, which is up to 5 microns thick. The outer surface of the aqueous layer is covered by a thin lipid layer that is exposed to the ambient air. The mucin protects the ocular epithelium against pathogens, and clinical evidence points to progressive destruction of the membrane-associated mucins (MAM) in eye infection. From a fluid mechanical viewpoint, the MAM modifies the wetting condition on the solid substrate, and modulates the van der Waals forces between the interfaces. We hypothesize that a change in the configuration of the MAM, say from complete coverage to partial coverage, induces a stronger van der Waals attraction and enhances slip on the ocular surface. Both factors should accelerate the breakup of the tear film. This will explain clinical observations of faster tear-film breakup in various eye diseases. We numerically simulate the tear-film breakup process to study the effect of the MAM through an elevated Hamaker constant and a modified slip length. Results show that the loss of MAM indeed precipitates tear-film breakup, and suggest that the tear-film breakup time can be a potential diagnostic of eye diseases.

uk: вхід без авторизації

name: Tear-film breakup: the role of membrane-associated mucin polymers


uk: Математика, природничі науки

description: Crystal growth is a model system for non-equilibrium physics. As a consequence, it has been widely studied, and in particular the nonlinear dynamics giving rise to complex shapes such as those of snow flakes have attracted much interest. However until now, most of these studies have been devoted to free growth, i.e. gowth in infinite systems. Confinement, which breaks translation invariance, gives rise to a novel class of nonlinear dynamics and morphologies, which are relevant in natural sciences such as geology –where crystal growth is often confined in small pores or faults of rocks, and for biomineralization, the process by which living organisms produce minerals (e.g. for their skeleton) –where growth is confined into a soft and complex environment. We propose a model for growth and dissolution of a crystal in confinement describing the non-equilibrium dynamics within the contact region using a continuum thin film equation [1]. Our model accounts self-consistently (in the lubrication regime) for surface tension effects, for the disjoining pressure, and for non-equilibrium transport processes such as diffusion and liquid convection. Based on this model, we study dissolution under a macroscopic load (pressure solution) [1] and growth under an applied supersaturation (crystallisation force) [2,3]. In pressure solution [1], the functional form of the crystal-substrate interaction potential appears to influence strongly the dynamics. For example, a divergent repulsion leads to flat contacts, and to a dissolution rate which increases indefinitely with the applied load. In contrast, a finite repulsion implies a sharp pointy contact shape, and a dissolution rate independent from the applied load. In confined growth it is well known that crystals can produce forces on their environment, and can exhibit a rim in the contact region. Our model shows the generic formation and growth of a precursor cavity which ultimately leads to the formation of the rim. The results are supported by experiments on NaClO3 in the University of Oslo [2]. We show that the formation of the cavity can be supercritical or subcritical, depending on the functional form of disjoining pressure [3]. Finally, the production of forces close to and far from equilibrium is also discussed [4]. References: [1] Thin film modeling of crystal dissolution and growth in confinement L Gagliardi, O Pierre-Louis, Physical Review E 97 (1), 012802 (2018) [2] Cavity formation in confined growing crystals F Kohler, L Gagliardi, O Pierre-Louis, DK Dysthe PHYSICAL REVIEW LETTERS 121, 096101 (2018) [3] Crystal growth in nano-confinement: Subcritical cavity formation and viscosity effects L Gagliardi, O Pierre-Louis, to be published in Journal of Crystal Growth (2019) [4] The non-equilibrium crystallization force L Gagliardi, O Pierre-Louis, preprint (2019).

uk: вхід без авторизації

name: Dissolution and Growth in confinement: dynamics within contacts


uk: Математика, природничі науки

description: Interfacial instabilities and pattern formation can occur in films on partially wetting surfaces. Lubrication models asymptotically reduce the governing equations for the free-surface flow to a fourth-order nonlinear parabolic partial differential equation for the evolution of the film height. Extensive studies have examined dynamics over a range of time-scales including finite-time rupture singularities and long-time droplet coarsening cascades when the fluid mass is conserved. We show that for volatile fluids, where the mass changes due to evaporation or condensation, the behaviors observed and the analysis needed yield new challenges and important differences from the non-volatile case. Some discussion will be given on dynamics that can occur when the form of the evaporative flux violates an energy dissipating structure. This is joint work with Hangjie Ji (UCLA).

uk: вхід без авторизації

name: Dynamics of non-conservative fluid thin films


uk: Математика, природничі науки

description: The subject of this talk is the electrostatic stabilization of a viscous thin liquid film wetting the underside of a horizontal surface in the presence of an electric field applied parallel to the undisturbed interface. The formulated asymptotic model includes the effect of bounding solid dielectric regions typically found in experiments above and below the fluids. The competition between gravitational forces, surface tension, and the non-local effect of the applied electric field is captured analytically in the form of a nonlinear evolution equation. State-of-the-art computational tools based on the volume-of-fluid method are also implemented to both assess the range of validity of the derived model and guide these arguments towards practical (and highly nonlinear) contexts involving mixing, pumping and directed polymer assembly. Joint work with T.G. Anderson (Caltech), D.T. Papageorgiou (Imperial College London) and P.G. Petropoulos (NJIT).

uk: вхід без авторизації

name: Thin (and not so thin) liquid films under the action of electric fields


Flow in a Hele-Shaw Cell

uk: Математика, природничі науки

description: The invasion of one fluid into a fluid occupying the narrow gap between parallel plates is a model for the displacement of a resident fluid in a porous medium by injecting a second fluid. In this paper, I describe equations formulated by Ruben Juanes and Luis Cueto-Felguereso based on a phase-field formulation. Numerical solutions of plane waves exhibit unusual behavior that is explored mathematically. Connections are made to finite speed propagation of a small amplitude wave solution of the porous medium equation, and to traveling waves that in a singular limit would be anomalous shock wave solutions of the equations. This research is documented in the thesis of Melissa Strait.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The Hele-Shaw problem is a classical example for studying interface dynamics or systems driven out of equilibrium. In this talk, I will first discuss numerical issues related to long-time computation of moving interfaces. Then I will present a time adaptive scheme for efficiently computing the dynamics of a moving interface. The idea is to design a time-space mapping such that in the new time scale, the interfaces can evolve at arbitrarily fast or slow speed. We then show numerical examples and present largest (expanding interface) and smallest (shrinking interface) Hele-Shaw simulation up to date.

uk: вхід без авторизації

name: Extremely large or small Hele-Shaw bubble: computational challenges and discoveries


uk: Математика, природничі науки

description: The notion of a holomorphic bundle and Hermitian-Yang-Mills connections is one that proved to be very fruitful in complex geometry. There are some natural generalizations of these notions in almost complex geometry such as those of pseudo-holomorphic and pseudo-Hermitian-Yang-Mills connections. In this talk, I will focus on a system of partial differential equations, the DT-instanton equations, whose solutions give a further generalization of the notion of a Hermitian-Yang-Mills connection in the setting of real 6 dimensional almost Hermitian manifolds.

uk: вхід без авторизації

name: Generalizing holomorphic bundles to almost complex manifolds


uk: Математика, природничі науки

description: I shall discuss recent work with Andrew Swann that proposes a framework for studying exceptional holonomy manifolds with torus symmetry, much similar to the study of hyperkaehler manifolds with a tri-Hamiltonian action of a torus.

uk: вхід без авторизації

name: Toric geometry of exceptional holonomy manifolds


uk: Математика, природничі науки

description: In this talk we explain how to describe pure types of Spin(7) structures in terms of spinors and focus on the construction of balanced examples. An 8-dimensional Riemannian manifold admitting a Spin(7) structure determined by a 4-form Ω is spin and the structure can also be described in terms of a spinor η. Balanced Spin(7) structures are a pure class and are characterized by the equation (∗dΩ)∧Ω=0 or, equivalently, by the condition that η is harmonic, that is, Dη=0 where D is the Dirac operator. For our purposes, the description of balanced structures in terms of spinors turns out to be much simpler. Our examples are products (N×T,g+gk), where (N,g) is a k-dimensional nilmanifold endowed with a left-invariant metric, (T,gk) is an (8−k)-dimensional flat torus, and k=5,6. Under these assumptions, the presence of a left-invariant balanced Spin(7) structure on the product is equivalent to the fact that (N,g) admits a left-invariant non-zero harmonic spinor. For this reason we search left-invariant metrics on N that admit left-invariant harmonic spinors. The results of our investigation are a list of 5 and 6-dimensional nilmanifolds that verify this condition, and the description of the set of left-invariant metrics with left-invariant harmonic spinors in the particular case k=5.

uk: вхід без авторизації

name: A spinorial approach to the construction of balanced Spin(7) manifolds


uk: Математика, природничі науки

description: The most natural maps into nearly Kaehler manifolds are pseudoholomorphic curves. When taking the cone of a nearly Kaehler manifold one gets a torsion-free G2 manifold, and the cone of a pseudoholomorphic curve will be an associative submanifold. Most of the work on this topic has been done for specific examples of ambient manifolds since compared to pseudoholomorphic curves in symplectic manifolds little is known about them in the general setting. After reviewing the relevant background I will show how holomorphic data can be used to construct integer invariants and examples of these curves, as done by Bryant for S6 and by Xu for CP3. I will then show how the latter construction can be combined via the Eells-Salamon correspondence with a generalisation of a formula of Friedrich to compute the Euler number of a conformal and harmonic map into S4. In the end, I will present open problems I am working on such as finding new examples of pseudoholomorphic curves.

uk: вхід без авторизації

name: Pseudoholomorphic Curves in Nearly Kähler Manifolds


uk: Математика, природничі науки

description: It is known that higher codimensional mean curvature flow appears naturally in the G2-Laplacian flow of coassociative fibrations. With this motivation, we consider entire higher codimensional mean curvature flow in Rn,m of n-dimensional spacelike manifolds and prove a long time existence theorem starting from arbitrary spacelike initial data. We will see that the key to the proof is to demonstrate local spacelike gradient estimates, and to get around difficulties with cutoff functions in Rn,m. This yields new long time existence results for the G2-Laplacian flow.

uk: вхід без авторизації

name: G2 Laplacian flow and higher codimensional spacelike mean curvature flow


uk: Математика, природничі науки

description: In this talk we will discuss the deformation theory for G2 instantons on 7-dimensional manifolds with nearly parallel G2-structure. We prove that the space of infinitesimal deformations of a G2 instanton on such manifolds can be identified with the eigenspaces of a 1-parameter family of Dirac operators and as a result prove that the abelian instantons are rigid, yielding a different proof of a recent result of Ball-Oliveira. This identification along with some other properties of these instantons can then be further used to gather more information about the deformation space in the non abelian case. This talk is based on a work in progress and we aim in future to determine the deformation space of some particular instantons on some known examples of nearly G2 manifolds.

uk: вхід без авторизації

name: Deformations of G2 instantons on nearly G2 manifolds


Closed G2-structures with symmetry

uk: Математика, природничі науки

description: G2-structures defined by a closed positive 3-form constitute the starting point in known and potentially effective methods to obtain holonomy G2 metrics on seven-dimensional manifolds. Currently, no general results guaranteeing the existence of such structures on compact 7-manifolds are known. Moreover, the construction of new explicit examples requires substantial efforts. In the first part of this talk, I will discuss the properties of the automorphism group of a compact 7-manifold endowed with a closed G2-structure, showing how they impose strong constraints on the construction of examples with high degree of symmetry (e.g. homogeneous, cohomogeneity one). Then, I will focus on the non-compact case. Here, motivated by known results on symplectic Lie groups, I will discuss some structure theorems for Lie groups admitting left invariant closed G2-structures. This is based on joint works with F. Podestà (Firenze), A. Fino (Torino), and M. Fernández (Bilbao).

uk: вхід без авторизації


Quadratic closed G2-structures

uk: Математика, природничі науки

description: I will talk about a special class of closed G2-structures, those satisfying the `quadratic' condition. This is a second order PDE system first written down by Bryant that can be interpreted as a condition on the Ricci curvature of the induced metric, that includes the extremally Ricci-pinched (ERP) condition as a special case. I will talk about various constructions of quadratic closed G2-structures, including the first examples of ERP closed G2-structures that are not locally homogeneous and the first examples of quadratic closed G2-structures that are not ERP. I will discuss the relationship with the Laplace flow, and give new examples of Laplace solitons.

uk: вхід без авторизації


uk: Математика, природничі науки

description: All known examples of instantons on (non-trivial) asymptotically conical G2 manifolds asymptote to nearly K\"ahler instantons which live on the link of the cone. I will use this observation to develop a framework for studying the moduli space of such a G2 instanton, showing that the expected dimension of this space is the index of a twisted Dirac operator on a weighted Sobolev space. I will focus on the example of R7 where the link is the homogeneous space G2/SU(3) and show how one can use representation theoretic methods to calculate the expected dimension of the moduli space.

uk: вхід без авторизації

name: Instantons on Asymptotically Conical G2 Manifolds


S1 invariant Laplacian flow

uk: Математика, природничі науки

description: The Laplacian flow is an evolution equation of closed G2-structures arising as the gradient flow of the so-called Hitchin volume functional. In this talk, we shall consider the flow of those G2 structures admitting S1 symmetry and derive explicitly the evolution equations of the SU(3)-structure on the quotient manifold together with a connection 1-form. We describe these equations in a couple of examples and mention some partial results of ongoing work.

uk: вхід без авторизації


uk: Математика, природничі науки

description: One of the most important ideas in the study of differential equations is the use of symmetries to cut down the number of variables. As manifolds with exceptional holonomy cannot be homogeneous the most symmetric case are group actions with cohomogeneity one, i.e. where a generic orbit has codimension one. In this case the PDE system is reduced to an ODE system. I will give an overview of recent progress in the construction of cohomogeneity one metrics with holonomy G2 and Spin(7). All complete examples have a asymptotically locally conical (ALC) or asymptotically conical (AC) geometry at infinity.

uk: вхід без авторизації

name: Cohomogeneity one manifolds with holonomy G2 and Spin(7)


Toric G2 and nearly Kaehler geometry

uk: Математика, природничі науки

description: We will discuss analogues of toric geometry in the G2 and nearly Kaehler settings using a multisymplectic generalization of the moment map called the multi-moment map. We will then present recent work on complete toric nearly Kaehler manifolds, demonstrating some information about their global structure and giving evidence to the conjecture that the homogeneous nearly Kaehler structure on the product of two three spheres is the only example.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The first compact examples of G2-manifolds were constructed by D. Joyce by desingularising G2-orbifolds with singularities modeled on flat T3×(C2/±1). T. Walpuski constructed G2-instantons on these compact G2-manifolds. D. Joyce and S. Karigiannis generalised the original construction of G2-manifolds by resolving orbifold singularities modeled on L×(C2/±1), where L is an associative submanifold in a G2-manifold. The obvious question is how the original instanton construction can be adapted to the new setting. In the talk I will explain (1) the construction by Joyce and Karigiannis of compact G2-manifolds, (2) the construction by Walpuski of G2-instantons on the original construction, and (3) the construction that I hope will produce G2-instantons on the Joyce-Karigiannis G2-manifolds. I will explain possible sources of examples using desingularisations of G2-orbifolds of the form T3×(K3-surface/involution).

uk: вхід без авторизації

name: G2-instantons on Joyce-Karigiannis G2-manifolds


uk: Математика, природничі науки

description: In a hyper-Kaehler 4-manifold, holomorphic curves are stable minimal surfaces. One may wonder whether those are all the stable minimal surfaces. Micallef gave an affirmative answer in many cases. However, this cannot be true in general. The minimal sphere in the Atiyah--Hitchin manifold is a counter-example. In this talk, we will first recall the hyper-Kaehler geometry of Atiyah--Hitchin manifold. We will then explain that the minimal sphere is quite rigid in various senses. This is based on a joint work with Mu-Tao Wang.

uk: вхід без авторизації

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


uk: Математика, природничі науки

description: We will talk about a flow of isometric G2 structures. We consider the negative gradient flow of the energy functional restricted to the class of G2 structures inducing a given Riemannian metric. We will discuss various an- alytic aspects of the flow including global and local derivative estimates, a compactness theorem and a local monotonicity formula for the solutions. We also study the evolution equation of the torsion and show that under a modification of the gauge and of the relevant connection, it satisfies a nice reaction-diffusion equation. After defining an entropy functional we will prove that low entropy initial data lead to solutions that exist for all time and converge smoothly to a G2 structure with divergence free torsion. We will also discuss finite time singularities and show that at the singular time the flow converges to a smooth G2 structure outside a closed set of finite 5- dimensional Hausdorff measure. Finally, we will prove that if the singularity is Type I then a sequence of blow-ups of a solution has a subsequence which converges to a shrinking soliton of the flow. This is a joint work with Pana- giotis Gianniotis (University of Athens) and Spiro Karigiannis (University of Waterloo).

uk: вхід без авторизації

name: A gradient flow of isometric G2 structures


uk: Математика, природничі науки

description: Seven-manifolds with a G2-structure possess two distinguished classes of submanifolds: associative 3-folds and coassociative 4-folds. When the G2-structure is torsion-free, such submanifolds are minimal (in fact, calibrated) and exist locally. We are led to ask: For which classes of G2-structures is it the case that associative 3-folds (respectively, coassociative 4-folds) are always minimal submanifolds? We will answer this by deriving a simple formula for the mean curvature, in the process uncovering new obstructions to the local existence of coassociatives. Time permitting, we will discuss the analogous results for special Lagrangian 3-folds (respectively Cayley 4-folds) in 6-manifolds with SU(3)-structures (respectively 8-manifolds with Spin(7)-structures). This is joint work with Gavin Ball.

uk: вхід без авторизації

name: Minimality of and Local Obstructions to Associative and Coassociative Submanifolds


Photons walking the line

uk: Математика, природничі науки

description: Light is a fascinating medium. It is interesting on its own but is equally well suited to be the tool for mimicking other physical systems. Recently considerable attention was given to processes called quantum walks. The quantum walk [1-2] is an excellent tool for modeling, simulating and testing a wide range of physical processes and effects. Quantum walks are defined as specific generalizations of classical (random) walks. The simplest model of a walk-the one dimensional discrete quantum walk (on a line)-is based on the combination of the dynamics of the internal degree of freedom defined by the coin operator and the conditioned shift in position space (step operator). The evolution of the walk is given by the repeated application of the resulting evolution operator. The coin as well as the step operator can suffer from imperfections and this leads to deviations from the ideal situation. The way how the ideal situation is alternated leads to additional interesting effects. We present results on theoretical and experimental studies of ideal and perturbed quantum walks [3-8] based on the all optical implementation of quantum walks. We point out the main results and future trends. References: [1] Y. Aharonov, L. Davidovich, N. Zagury, Phys. Rev. A 48, 1687 (1993). [2] D. A. Meyer, J. Stat. Phys. 85, 551 (1996). [3] A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, P.J. Mosley, E. Andersson, I. Jex, Ch. Silberhorn, Phys. Rev. Lett. 104, 050502 (2010). [4] A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gábris, E, I. Jex, Ch. Silberhorn, Phys. Rev. Lett. 104, 050502 (2011). [5] A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefanák, V. Potocek, C. Hamilton, I. Jex, Ch. Silberhorn,. Science 336, 55 (2012). [6] F. Elster, S. Barkhofen, T. Nitsche, J. Novotný, A. Gábris, I. Jex, Ch. Silberhorn, Sci. Rep. 5 (2015) 13495. [7] T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, Ch. Silberhorn, New J. Phys. 18 (2016) 063017. [8] T. Nitsche, S. Barkhofen, R. Kruse, L. Sansoni, M. Štefanák, A. Gábris, V. Potocek, T. Kiss, I. Jex, Ch. Silberhorn, Science Adv. 4, eaar6444 (2018).

uk: вхід без авторизації


The quantum first detection problem

uk: Математика, природничі науки

description: We are considering a quantum system initially prepared in a pure state, that is repeatedly projectively measured in some detection state with a fixed inter-detection period. We investigate the distribution and the statistics of the time of the first successful detection event. The such obtained first detection time gives an operational definition to the time-of-arrival in the detection state. The probability of first detection can be obtained by means of a renewal equation. For systems with an absolutely continuous energy spectrum, we demonstrate how the asymptotics of this first detection probability are dominated by singularities in the spectral measures. For generic initial and detection states, these singularities can be identified with the system's van-Hove singularities. Furthermore, we give a detailed discussion of the transition problem in the tight-binding model on the infinite line, where the initial and detection position of a single quantum walker are separated.

uk: вхід без авторизації


uk: Математика, природничі науки

description: In this talk we discuss the model of quantum Markov chains, due to S. Gudder, and most particularly the subset of open quantum walks, due to S. Attal et al., acting on finite graphs. As an iterative process, we use a monitoring procedure to determine the mean time for a quantum walker to visit some chosen vertex for the first time. We are interested in ways of calculating such hitting times besides making direct use of its definition and here we notice algebraic similarities and differences with the classical case. The case of unitary quantum walks remains an interesting open problem for which a solution could have potential applications to the associated theory of Schur functions.

uk: вхід без авторизації

name: Mean hitting times of open quantum walks in terms of generalized inverses


Power series in stochastic processes

uk: Математика, природничі науки

description: This talk is about a class of classical random processes on graphs that include the discrete Bak-Sneppen process, introduced in 1993, and the several versions of the contact process. These processes are parametrized by a probability 0≤p≤1 that controls a local update rule. Numerical simulations reveal a phase transition when p goes from 0 to 1, which I will discuss in the talk. Analytically little is known about the phase transition threshold, even for one-dimensional chains. In this talk we consider a power-series approach based on representing certain quantities, such as the survival probability or expected hitting times, as a power-series in p. We prove that the coefficients of those power series stabilize as the length n of the chain grows, and I will give a sketch of this proof in the talk. This stabilization of coefficients is a phenomenon that has been used in the physics community but was not yet proven. We show that for local events A, B of which the support is a distance d apart we have cor(A,B)=O(pd). The stabilization is useful because it allows for the (exact) computation of coefficients for arbitrary large systems which can then be analyzed using the wide range of existing methods of power series analysis.

uk: вхід без авторизації


Quantum walks and coarse structure

uk: Математика, природничі науки

description: Locality is the key property that ties a quantum walk to the underlying lattice. By this we mean that matrix elements of the unitary one-step operator become small between distant sites. The extreme case of this is a nearest neighbour walk, where matrix elements become zero for distance >1. We are interested here in the notions of locality for infinite lattices, allowing some decay of matrix elements, and focusing on the large scale propagation behaviour. The mathematical notion for a locality structure, which takes due note of composition properties, is called a coarse structure. I will describe this and show how even in one dimension there are different natural choices, which subtly differ even in the translation invariant case. I will then go to higher dimension, and discuss the relation to compactifications, C*-algebras, and their classification via K-theory.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Quantum informational protocols involve coherent evolution, measurement, and postselection of qubits. A typical example is entanglement distillation. The resulting conditional dynamics is nonlinear, in contrast to the usual evolution of both closed and open quantum systems. Already the simplest types of such protocols may result in rich, complex chaotic dynamics when applied iteratively. An important property of these iterated dynamical systems is that initially pure quantum states remain pure throughout the evolution. For single-qubit systems, there is a one to one correspondence of the pure-state quantum dynamics to the iterated dynamics of quadratic rational maps with one complex variable. For two-qubit systems LOCC operations may lead to dynamics, where the evolution of entanglement is chaotic in the sense of crucially depending infinitely fine details of the initial state. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. They provide for example a solution to the quantum state matching problem, i.e. the task of deciding whether an unknown qubit state falls in a prescribed neighborhood of a reference state. We determine that such protocols may exhibit sensitive, quasi-chaotic evolution not only for pure initial states but also for mixed states, i.e., the complex dynamical behavior is not destroyed by small initial uncertainty. We show that the appearance of sensitive, complex dynamics associated with a fractal structure in the parameter space of the system has the character of a phase transition. The purity of the initial state plays the role of the control parameter, and the dimension of the fractal structure is independent of the purity value after passing the phase transition point. References: [1] A. Gilyén, T. Kiss, and I. Jex, Sci. Rep. 6, 20076 (2016). [2] O. Kálmán and T. Kiss, Phys. Rev. A 97, 032125 (2018). [3]Martin Malachov, Igor Jex, Orsolya Kálmán, and Tamás Kiss, Chaos 29, 033107 (2019).

uk: вхід без авторизації

name: Measurement induced complex chaos in quantum protocols


uk: Математика, природничі науки

description: Topological insulators have Hamiltonians with bulk topological invariants, which control the interesting processes at the surface of the system, but are hard to measure directly. We have found a way to measure the bulk topological invariant (winding number) of one-dimensional topological insulator Hamiltonians and quantum walks with chiral symmetry: it is given by the displacement of a single particle, observed via losses [1]. Losses represent the effect of repeated weak measurements on one sublattice only, which interrupt the dynamics periodically. When these do not detect the particle, they realize negative measurements. In our repeated measurement scheme these losses occur at the end of every timestep. In the limit of rapidly repeated, vanishingly weak measurements, this corresponds to non-Hermitian Hamiltonians, such as the lossy Su-Schrieffer-Heeger model [2]. Contrary to intuition, the time needed to detect the winding number can be made shorter by decreasing the efficiency of the measurement. Our scheme has since been used to measure the bulk topological invariants of a discrete-time quantum walk on photons [3]. References: [1] T Rakovszky, JK Asboth, and A Alberti, Phys. Rev. B 95, 201407 (2017). [2] MS Rudner and LS Levitov, Phys. Rev. Lett. 102, 065703 (2009). [3] X Zhan et al, Phys. Rev. Lett. 119, 130501 (2017).

uk: вхід без авторизації

name: Topological phases of quantum walks and how they can be detected


uk: Математика, природничі науки

description: We show that the evolution of two-component particles under a continuous-time quantum walk can reveal topological phases. A kink in the mean width of the walker distribution signals the closing of the energy gap, a prerequisite for a quantum phase transition between topological phases. For realistic and experimentally motivated Hamiltonians, the distribution in topologically non-trivial phases displays characteristic rings in the vicinity of the origin that are absent in topologically trivial phases. The results are expected to have immediate application to systems of ultracold atoms and photonic lattices, and should aid in the realization of topological states suitable for quantum computation.

uk: вхід без авторизації

name: Detecting 2D topological order by quantum walk


uk: Математика, природничі науки

description: A quantum walk algorithm can detect the presence of a marked vertex on a graph quadratically faster than the corresponding random walk algorithm (Szegedy, FOCS 2004). However, quantum algorithms that actually find a marked element quadratically faster than a classical random walk were only known for the special case when the marked set consists of just a single vertex, or in the case of some specific graphs. We present a new quantum algorithm for finding a marked vertex in any graph, with any set of marked vertices, that is (up to a log factor) quadratically faster than the corresponding classical random walk.

uk: вхід без авторизації

name: Quadratic speedup for finding marked vertices by quantum walks


uk: Математика, природничі науки

description: We provide a new spatial search algorithm by continuous-time quantum walk which can find a marked node on any ergodic, reversible Markov chain P, in a time that is quadratically faster than the corresponding classical random walk on P. In the scenario where multiple nodes are marked, the running time of our algorithm scales as the square root of a quantity known as the extended hitting time. This solves an open problem concerning the difference between the running time of spatial search by discrete-time and continuous time quantum walk. We also show that the widely used Childs and Goldstone algorithm for spatial search by continuous-time quantum walk is quite restrictive: we identify limitations in its applicability whenever P is not state-transitive. We subsequently improve and extend this algorithm to be applicable for any P. Our generalizations imply that most hitherto published results on the performance of quantum spatial search in the Childs and Goldstone framework on specific graphs are particular cases of our result. However, we prove that the running time of the Childs and Goldstone algorithm and its subsequent improvement is suboptimal: our spatial search algorithm outperforms it. Our results can be adapted to a number of Markov chain-based quantum algorithms and will lead to exploring other connections between discrete-time and continuous-time quantum walks.

uk: вхід без авторизації

name: Finding a marked node on any graph by continuous time quantum walk


uk: Математика, природничі науки

uk: вхід без авторизації

name: Embezzlement-based nonlocal game (Bell inequality) that cannot be won (violated) optimally with finite amount of entanglement (i.e., proving the set of quantum correlations is not closed using embezzlement)


uk: Математика, природничі науки

description: I will present several open problems in the context of continuous-time quantum walks in graphs. To each problem, I will try to outline their technical nature, and I will speculate which techniques could be applied to solve them.

uk: вхід без авторизації

name: Open problems in continuous-time quantum walks


Average mixing matrix of quantum walks

uk: Математика, природничі науки

description: A quantum walk is governed by its transition matrix which is unitary; since this process is necessarily non-ergodic and one cannot speak of a stationary distribution, we study the average behaviour of the quantum walk. The average of the mixing matrices contains relevant information about the quantum walk and about the graph. There has been a considerable amount of success in approaching questions about continuous-time quantum walks with tools in linear algebra and algebraic graph theory and we will discuss several recent works in this area, based on joint work with Chris Godsil, Gabriel Coutinho, Harmony Zhan and John Sinkovic.

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Fractional revival of threshold graphs under Laplacian dynamics


uk: Математика, природничі науки

description: Some quantum walks can be modeled using weighted graphs, where each vertex represents a qubit, each weighted edge indicates the coupling strength between two qubits, and each weighted loop indicates the strength of the magnetic field of a qubit. In this talk, I will discuss two analytic approaches to quantum walks: orthogonal polynomials, which have been applied mostly to weighted paths, and spectral graphs theory, which has been applied mostly to simple unweighted graphs. I will also talk about some interesting relations between quantum walks on weights paths and quantum walks on simple unweighted graphs.

uk: вхід без авторизації

name: Quantum walks, orthogonal polynomials, and spectral graph theory


uk: Математика, природничі науки

description: It is known that Krawtchouk polynomials play a central role in the analysis of (continuous-time) quantum walk on hypercube. Such walk also gives the spin lattice Hamiltonian where perfect state transfer occurs. In this talk (or presentation), we will start with multivariate Krawtchouk polynomials. We will introduce the associated graph and show that some spin lattice Hamiltonian in multi-dimension can be described as projections of quantum walk on this graph. In the model, fractional revival in addition to perfect state transfer is shown to take place.

uk: вхід без авторизації

name: Multivariate Krawtchouk polynomials and quantum walks on the associated graphs


uk: Математика, природничі науки

description: The subject starts with C. Shannon questions at Bell Labs and the amazing answers found in the 60's by Slepian, Landau and Pollak. The crux of the matter is that in some naturally appearing problems in signal processing a certain integral operator admits an explicit commuting second order differential one, or a full matrix admits a commuting tridiagonal one. These operators are parametrized by the duration of the signal and its bandwidth. I have been trying to understand and extend this miracle for a long time, mainly in "non translation invariant cases". All the examples from Bell Labs involve Fourier analysis in different setups. I have managed to extend this to a few other cases. My "new motivation" comes from looking at two papers: "Maximal violations of Bell inequalities by position measurements" , J. Kiukas and R. Werner 2009 and "Properties of the entanglement Hamiltonian for finite free-fermion chains", V. Eisler and I. Peschel 2018. Both of these papers exploit the commutativity property in some physically important cases. My hope is that some members of the audience may point out a few more cases where this very exceptional mathematical phenomenon plays a physically relevant role.

uk: вхід без авторизації

name: Time and band limiting: a bouquet of commuting miracles (some old, some very recent), and a new motivation to look for more of them


uk: Математика, природничі науки

description: Tensor network states provide successful descriptions of strongly correlated quantum systems with applications ranging from condensed matter physics to cosmology. Any family of tensor network states possesses an underlying entanglement structure given by a graph of maximally entangled states along the edges that identify the indices of the tensors to be contracted. Recently, more general tensor networks have been considered, where the maximally entangled states on edges are replaced by multipartite entangled states on plaquettes. Both the structure of the underlying graph and the dimensionality of the entangled states influence the computational cost of contracting these networks. Using the geometrical properties of entangled states, we provide a method to construct tensor network representations with smaller effective bond dimension. We illustrate our method with the resonating valence bond state on the kagome lattice.

uk: вхід без авторизації

name: Tensor network representations from the geometry of entangled states


Evolution in space

uk: Математика, природничі науки

description: Most populations are spread over spatial ranges much bigger than any one individual will explore in its lifetime. How does the simple fact of this spatial structure affect adaptive evolution and genetic diversity? We quantify how space interacts with recombination (sex) to slow down or speed up adaptation by changing the amount of clonal interference among beneficial mutations, and to change how adaptation reshapes neutral diversity through genetic hitchhiking. We also show how the spatial and temporal variation in genetic diversity revealed by sequencing can be used as a lens to watch how organisms move, capturing rare jumps that have large effects on populations' evolution.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The rapid global increase of multidrug-resistant organisms presents a major global health threat that will dramatically reduce the efficacy of antibiotics and thus constrain the number of effective treatments available to patients. As opposed to analogous efforts in viral epidemiology, accurate reconstruction of the pandemic spread of antibiotic resistance remains intractable for reasonable sample sizes due, in large part, to the high rate of homologous recombination and horizontal gene transfer that prevents the application of traditional phylogenetic approaches. Lastly, complete assemblies are a prerequisite to such quantitative study. In this talk, I will present a novel computational framework for bacterial evolution that generalizes the traditional linear reference genome to a pan-genomic non-planar graph. Using this framework, I will analyze 120 antibiotic resistant genomes collected in Basel, Switzerland over the course of a decade, sequenced with both ONT and Illumina data. We show that the evolution of antibiotic resistance exhibits a nested doll structure in which genetic transposition, homologous recombination, and clonal expansion occur at similar time-scales.

uk: вхід без авторизації

name: Observing bacterial pathogen evolution with long read sequencing


uk: Математика, природничі науки

description: The dynamics of some systems are directional, meaning that the state of a system can be characterized by a function that increases in time. This includes, for example, the growth of entropy in physical systems, or the growth of fitness in certain formulations of Darwinian evolution. Yet even when directionality holds, it might be sensitive to perturbations, such as interactions in evolutionary dynamics. I'll describe a model of ecological community dynamics which features a phase-transition, from a regime where directionality is sensitive to a regime where it is robust. This has far reaching consequences: In the latter phase, the system admits many alternative community states, that are able to expand in space, forming (exact or approximate) copies of themselves. This leads to community-level selection, in analogy with Darwinian selection, with the increasing function acting as a fitness.

uk: вхід без авторизації

name: Directionality and community-level selection

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


uk: Математика, природничі науки

description: Microbes are everywhere. For every human cell of our body, there is at least one bacterial cell living inside us. Due to the advancement of sequencing techniques, there has been an explosion of data that track the composition of microbial communities. In a traditionally data-poor discipline as ecology, this novel richness of data represents a unique opportunity to understand quantitatively how different ecological forces shape diversity. In this talk, I will present three independent, emergent statistical patterns of distribution of abundances across species and communities, which are conserved across different ecosystems. I will then discuss how these "laws" can inform us about the fundamental mechanisms that are shaping the composition of these communities.

uk: вхід без авторизації

name: Laws of diversity and variation in microbial communities


uk: Математика, природничі науки

description:
Ecological species can spread their extinction risk in an uncertain environment by adopting a bet-hedging strategy, i.e, by diversifying individual phenotypes. I will present a theory of bet-hedging for populations colonizing an unknown environment that fluctuates either in space or time. We find that diversification is more favorable for range expansion than in the well-mixed case, supporting the view that range expansions promote diversification. For slow rates of variation, spatial fluctuations open more opportunities for bet-hedging than temporal variations. Opportunities for bet-hedging reduce in the limit of frequent environmental variations. These conclusions are robust against demographic stochasticity induced by finite population sizes.

Ref. P. Villa-Martin, M.A. Muñoz, S. Pigolotti, Plos Comp. Biol. 15(4): e1006529 (2019).

uk: вхід без авторизації

name: Bet-hedging strategies in expanding populations


uk: Математика, природничі науки

description: A recently proposed mechanism suggests that transient compartmentalization could have preceded cell division in prebiotic scenarios. Here, we study various classes of transient compartmentalization dynamics. We show that two regimes are possible: In a diffusion-limited regime (e.g. simple autocatalysis), a large noise is generated at the population level due to asynchronous growth. In contrast, in a replication-limited regime with many steps (e.g. polymerization), a low noise is generated at the population level. Since strong noise will yield many unviable population compositions, polymerization can present a strong fitness advantage. For deterministic growth dynamics, we introduce mutations that turn functional replicators into parasites. This can either lead to coexistence or parasite dominance, and we derive the phase boundary separating these two phases as a function of relative growth, inoculation size and mutation rate. We show that transient compartmentalization allows coexistence beyond the classical error threshold.

uk: вхід без авторизації

name: Transient compartmentalization and its associated error thresholds


Phenotype evolution as optimization

uk: Математика, природничі науки

description: Biological evolution can be described as a population climbing a fitness landscape, and has inspired a variety of derivative-free optimization algorithms. Here we describe how phenotype evolution has sophisticated optimization properties. In particular, natural selection approximates second order gradient descent (Newton's method), and recombination is efficient in generating diversity. We use these insights to design a new type of derivative-free optimization algorithm for continuous problems.

uk: вхід без авторизації


uk: Математика, природничі науки

description: Living systems evolve one mutation at a time, but a single mutation can alter the effect of subsequent mutations. The underlying mechanistic determinants of such epistasis are unclear. Here, we argue that the physical properties of a biological system can generically and easily constrain its epistasis. We analyze the interaction between mutations in generic models of proteins and biochemical networks. In each case, a slow, collective physical mode is actuated upon mutation, reducing the dimensionality of mutational effects and thus the rank of the epistatic matrix. This, in turn, reduces the ruggedness of the sequence-to-function map. By providing a mechanistic basis for experimentally observed global epistasis, these results suggest that slow collective physical modes can make biological systems more evolvable.

uk: вхід без авторизації

name: Physical constraints on epistasis in proteins and gene regulatory networks


uk: Математика, природничі науки

description: Frequencies of synonymous codons are typically non-uniform, despite the fact that such codons correspond to the same amino acid in the genetic code. This phenomenon, known as codon bias, is broadly believed to be due to a combination of factors including genetic drift, mutational effects, and selection for speed and accuracy of codon translation; however, quantitative modeling of codon bias has been somewhat elusive. I will present a biophysical model which explains genome-wide codon frequencies observed across 20 organisms. Our model implements detailed codon-level treatment of mutations and includes two contributions to codon fitness which describe codon translation speed and accuracy. We find that the observed patterns of genome-wide codon usage are consistent with a strong selective penalty for mistranslated amino acids, while the dependence of codon fitness on translation speed is much weaker on average. Treating the translation process explicitly in the context of a finite ribosomal pool has allowed us to highlight the biophysical underpinnings of codon-level selective pressures. Overall, our approach offers a unified biophysical and population genetics framework for understanding the origin of codon bias.

uk: вхід без авторизації

name: A biophysical model of codon bias evolution


uk: Математика, природничі науки

description: Cancer is the result of a stochastic evolutionary process characterized by the accumulation of mutations that are responsible for tumor growth, immune escape, and drug resistance, as well as mutations with no effect on the phenotype. Stochastic modeling can be used to describe the dynamics of tumor cell populations and to obtain insights into the hidden evolutionary processes leading to cancer. I will present recent approaches that use branching process models of cancer evolution to quantify intra-tumor heterogeneity and the development of drug resistance, and their implications for interpretation of cancer sequencing data and the design of optimal treatment strategies.

uk: вхід без авторизації

name: Evolutionary dynamics of cancer and its response to treatment


The Evolutionary "Design" of Proteins

uk: Математика, природничі науки

description: From the minute 28:28 of video does not have audio. Proteins can fold spontaneously into well-defined three-dimensional structures and can carry out complex biochemical reactions such as binding, catalysis, and long-range information transfer. The precision required for these properties is achieved while also preserving evolvability – the capacity to adapt in response to fluctuating selection pressures in the environment. What is the basic design of proteins that supports all of these properties? Recent work suggests that rather than direct physical analysis, statistical analysis of genome sequences provides a powerful and general approach to this problem. Using different methodologies, this approach has revealed both direct structural contacts as well as collective functional modes within protein structures. In this talk, I will present the current state of these approaches and the possibility of unifying them into a single theoretical framework for representing the evolutionary design of proteins.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The advent of a new modeling paradigm known as ‘differentiable programming’ makes possible bespoke machine-learned models of biological phenomena that are partly learned from data and partly informed by human-derived biophysical knowledge. In this talk I will describe three instantiations of this new approach for (i) de novo protein structure prediction, (ii) elucidation of the combinatorial grammar underlying metazoan signaling networks, and (iii) design of new protein function. In all cases qualitative improvements in model accuracy or speed, or both, are achieved using differentiable programming, enabling new scientific insights into biological macromolecules and the networks they comprise.

uk: вхід без авторизації

name: End-to-end differentiable learning of protein structure


uk: Математика, природничі науки

description: Antibodies mediate immunity to viruses and other pathogens by binding to specific antigenic targets. Our bodies deploy an evolvable arsenal of naïve antibodies capable of responding to diverse antigens. Upon exposure to a new antigen, potent and mature antibodies emerge from this arsenal through the Darwinian process of affinity maturation, consisting of iterative rounds of mutation and selection for improved binding to the antigen. This work seeks to understand the underlying relationship between sequence and function that guides this somatic evolutionary process by combining bioinformatic analysis of antibody repertoire sequencing data and high-throughput mutational scanning approaches. This framework is being applied to the broadly neutralizing HIV antibody VRC01, a representative of an important class of antibodies that are the subject of ongoing efforts in HIV vaccine design and antiviral therapy. By combining phylogenetic reconstruction of the VRC01 lineage with high-throughput ligand-binding assays, we are uncovering the relationship between sequence and function that guides VRC01 affinity maturation, revealing how epistasis, stochasticity, and historical contingency contribute to the complex maturation trajectory of this antibody. The results will highlight the landscape over which this broad class of therapeutically-important antibodies develop, with key implications for antibody engineering and HIV vaccine design.

uk: вхід без авторизації

name: The sequence-function landscape of antibody affinity maturation


uk: Математика, природничі науки

description: Over the last ten years high-throughput sequencing has enabled increasingly quantitative measurements of the diversity of lymphocyte receptor repertoires. A striking finding of these sequencing efforts has been that the clone sizes of cells sharing the same receptor are heavy-tail distributed. Here, we present a simple neutral birth-death model for immune repertoire formation in which all cells compete for a global resource. Homeostatic control of proliferation leads to a founder effect, in which large clones emerge early when there is less competition. We show that this mechanism produces a transient but long-lived regime of power-law scaling of clone sizes. A reanalysis of a cohort study shows that indeed early founded T cell clones are over-represented among the most abundant clones. We use data about how the founder effect diminishes over time to constrain how much peripheral selection impacts immune repertoire dynamics. Overall, our work suggests that dynamical processes early in life have a strong and long-lasting influence on the structure of the immune repertoire.

uk: вхід без авторизації

name: Immune repertoire dynamics out of steady state


uk: Математика, природничі науки

description:
Features of the CRISPR-Cas system, in which bacteria integrate small segments of phage genome (spacers) into their DNA to neutralize future attacks, suggest that its effect is not limited to individual bacteria but may control the fate and structure of whole populations [1]. In our model, we find that early dynamics of large phage clones is largely independent of bacterial dynamics but crucially depends on the burst-size of phage infections. In contrast, the fates of early phage mutants are strongly influenced by the feedback from bacterial population that creates a time-dependent fitness landscape for that phage type. Taken together, we quantify the role of population parameters in maintaining phage and bacterial diversity where CRISPR-cas is in the play.
[1])

uk: вхід без авторизації

name: How adaptive immunity governs co-evolution in microbes


uk: Математика, природничі науки

uk: вхід без авторизації

name: Tumors versus T-cells, a simple co-evolutionary model


uk: Математика, природничі науки

uk: вхід без авторизації

name: The McKay correspondence and noncommutative resolutions of discriminants of reflection groups 1


uk: Математика, природничі науки

description: Following work of Buchweitz, one defines Tate-Hochschild cohomology of an algebra A to be the Yoneda algebra of the identity bimodule in the singularity category of bimodules. We show that Tate-Hochschild cohomology is canonically isomorphic to the ordinary Hochschild cohomology of the singularity category of A (with its canonical dg enrichment). In joint work with Zheng Hua, we apply this to prove a weakened version of a conjecture by Donovan-Wemyss which states that a complete isolated cDV singularity is determined by the derived equivalence class of the contraction algebra associated with a resolution.

uk: вхід без авторизації

name: Tate-Hochschild cohomology, the singularity category and applications


uk: Математика, природничі науки

description: Exceptional collections frequently arise in algebraic and symplectic geometry. Since the work of Beilinson on coherent sheaves on projective space, exceptional collections have been used to construct derived equivalences. In representation theory, exceptional collections appear for the class of quasi-hereditary algebras. In the first talk I will give an overview over the theory of exceptional collections and quasi-hereditary algebras. In a second talk, I will focus on an approach to quasi-hereditary algebras using exact Borel subalgebras, corings, and A-infinity Koszul duality.

uk: вхід без авторизації

name: Introduction to quasi-hereditary algebras 1


uk: Математика, природничі науки

description: Given an A-module M, and a dimension vector d, one can define a quiver Grassmannian, a projective algebraic variety parametrising the d-dimensional A-submodules of M. A famous result in geometric representation theory, obtained by several different authors, states that every projective variety X (over an algebraically closed field of characteristic zero) is isomorphic to such a quiver Grassmannian. In this talk I will explain how, at least in certain cases, one can use this algebraic description to construct a desingularisation of X. The construction is representation-theoretic, involving a tilt of an endomorphism algebra in mod(A), and the desingularising variety is again described in terms of quiver Grassmannians. This talk is based on joint work with Julia Sauter, in which we extend methodology of Crawley-Boevey and Sauter, and of Cerulli Irelli, Feigin and Reineke.

uk: вхід без авторизації

name: Desingularising quiver Grassmannians via tilting


uk: Математика, природничі науки

@language: de

uk: вхід без авторизації

name: The McKay correspondence and noncommutative resolutions of discriminants of reflection groups 2


uk: Математика, природничі науки

description: We report on a work jointly started with Ragnar Buchweitz about the pull back of a tilting bundle T to the total space of the canonical line bundle Y. Let X be an algebraic variety with a tilting bundle T, then we have a criterion when its pull back to Y is also a tilting bundle. This is closely related to my previous work on distinguished tilting sequences and generalizes the results therein. Morover, we can compute the endomorphism ring of the pull back tilting bundle as the higher preprojective algebra. This leads to a geometric construction of those algebras. The construction needs tilting bundles T with endomorphism algebra of global dimension dimX. In this talk we consider several examples for surfaces and compute the possible global dimensions of A.

uk: вхід без авторизації

name: Tilting bundles, BGG-correspondence, higher preprojective algebras and the Serre functor

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


uk: Математика, природничі науки

description: From the minute 49:44 of video does not have audio. Quasi-hereditary algebras were introduced by L. Scott in the context of the work of E. Cline, B. Parshall and L. Scott on highest weight categories arising in the representation theory of Lie algebras and algebraic groups. The notion of a quasi-hereditary algebra depends on a partial order given to the set of simple modules. In particular an algebra may be quasi-hereditary for one partial order but not for another one, even in the hereditary case. After recalling basic definitions, we will introduce an equivalence relation on the set of all partial orders giving a quasi-hereditary algebra, calling the equivalence classes quasi-hereditary structures. In the case of the path algebra of an equioriented quiver of type A, we will classify all its quasi-hereditary structures in terms of tilting modules, highlighting its nice combinatorial properties. Then we will generalise this classification to any orientation. As a complementary example we will discuss a class of quiver algebras with a unique quasi-hereditary structure. Time permitting, we will introduce a partial order on the set of all quasi-hereditary structures and give some examples. This is joint work in progress with Yuta Kimura and Baptiste Rognerud.

uk: вхід без авторизації

name: Classifying quasi-hereditary structures of some quiver algebras


uk: Математика, природничі науки

description: We show that the bounded derived category of many singular projectve varieties (with isolated singularities) does not admit tilting objects (and more generally no semiorthogonal decompositions of "Kawamata-type“). Using singularity categories, most of our obstructions come from cluster theory via NCCRs and also from negative K-theory. Our approach is inspired by recent work of Kawamata. This is joint work with N.Pavic and E.Shinder.

uk: вхід без авторизації

name: Singularity categories and NCCRs as obstructions to tilting on singular projective varieties


uk: Математика, природничі науки

description: Consider a finitely generated normal commutative algebra R over a field K. A non-commutative resolution of singularities of Spec R is a (non-commutative) R-algebra A with finite global dimension of the form End(M) where M is some finitely generated reflexive R-module. The existence of a non-commutative resolution for a commutative ring R places strong conditions on R, such as rational singularities. In this talk, we discuss how in prime characteristic, the Frobenius can be used to construct non-commutative resolutions of nice enough rings. We conjecture that for a strongly F-regular ring R, End(F_*R) is a non-commutative resolution of R, where F_*R denotes R viewed as an R-module via restriction of scalars from Frobenius. We prove this conjecture when R is the coordinate ring of an affine toric variety. We also show that for toric rings, the ring of differential operators D(R) has finite global dimension (joint with Eleonore Faber and Greg Muller).

uk: вхід без авторизації

name: Non-Commutative Resolution of Singularities for Toric varieties


Tale of two moduli stacks

uk: Математика, природничі науки

description: Tilting theory provides a fascinating link between the representation theory of finite dimensional algebras and algebraic geometry. Traditionally, it is approached from the algebraic geometry side by seeking tilting bundles on projective stacks. However, in studying representation theory, it is much more natural to start with a finite dimensional algebra and ask how one might attempt to construct a projective stack which is derived equivalent to it. In this talk, we look at two moduli stacks which address this question, the moduli of refined representations and tensor stable representations. The key is to incorporate data corresponding to the monoidal structure of the category of coherent sheaves on the derived equivalent stack. This is joint work with Tarig Abdelgadir and Boris Lerner.

uk: вхід без авторизації


Moduli of A-infinity structures

uk: Математика, природничі науки

description: The triangulated category of graded matrix factorizations for an exceptional unimodal singularity is known to have a tilting object by Kajiura-Saito-Takahashi and Lenzing-de la Pena. If we deform the singularity, then we lose the grading, which can be recovered by adding one more variable to the defining polynomial. The triangulated category of graded matrix factorizations of the resulting four-variable polynomial no longer has a tilting object, but has a classical generator, whose endomorphism algebra is the degree 2 trivial extension of the endomorphism algebra of the tilting object of the original category. In the talk, we will discuss the moduli space of A-infinity structures on this graded algebra, and its relation to 1. the positive part of the universal unfolding of the exceptional unimodal singularity, 2. the moduli space of K3 surfaces, and 3. homological mirror symmetry. If the time permits, we also discuss higher-dimensional generalizations and iterated singularity categories (i.e., singularity categories of singularity categories of ...) of non-isolated singularities. This is a joint work with Yanki Lekili.

uk: вхід без авторизації


uk: Математика, природничі науки

description: We introduce the notion of a Nakayama functor relative to an adjunction, generalizing the classical Nakayama functor for a finite-dimensional algebra. We show that it can be characterized in terms of an ambidextrous adjunction of monads and comonads. We also study this concept from the viewpoint of Gorenstein homological algebra. In particular we obtain a generalization of the equality of the left and right injective dimension for a finite-dimensional Iwanaga-Gorenstein algebra, and for a module category we show that this property can also be characterized by the existence of a tilting module.

uk: вхід без авторизації

name: A generalization of the Nakayama functor


uk: Математика, природничі науки

description: Our goal is to find conditions on a noetherian AS-regular algebra A and an idempotent e∈A for which the graded singularity category Singgr(eAe) admits a tilting object. Of particular interest is the situation in which A is a graded skew-group algebra S#G, where S is the polynomial ring in n variables and G<SL(n,k) is finite, and e=1|G|∑g∈Gg, so that eAe≅SG. A tilting object was found by Amiot, Iyama and Reiten in the case where A has Gorenstein parameter 1. Generalizing the work of Iyama and Takahashi, Mori and Ueyama obtained a tilting object in Singgr(SG), provided that S is a noetherian AS-regular Koszul algebra generated in degree 1 and G has homological determinant 1. In this talk, we will discuss certain silting objects and then specialise to the setting in which the Beilinson algebra is a levelled algebra, giving a generalisation of the result of Mori and Ueyama.

uk: вхід без авторизації

name: Graded singularity category of Gorenstein algebras with levelled Beilinson algebras


Stability conditions for 3-fold flops

uk: Математика, природничі науки

description: For a 3-fold flopping contraction from X to the spectrum Spec(R) of a complete local Gorenstein ring (R,m) with terminal singularity at m, we give a description of a distinguished connected component of the (normalized) space of Bridgeland stability conditions on certain triangulated categories associated to the flopping contraction. More precisely, we show that the connected component is a regular covering space of the complement of the complexification of a hyperplane arrangement associated to the 3-fold flop. We also determine the autoequivalence groups of the triangulated categories. As an application of these results, we determine the Stringy K ̈ahler Moduli Space (SKMS) for all smooth irreducible 3-fold flops. This is a joint work with Michael Wemyss.

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: On a cubical generalization of preprojective algebras


uk: Математика, природничі науки

description: We study the scaling dimension (or the similarity dimension) of the perfect derived category of a smooth compact dg algebra called the Serre dimension. It is expected that the infimum of the Ikeda-Qiu’s global dimsion function on the space of stability conditions also gives another “good” notion of dimension. One of our results is that its infimum is always greater than or equal to the Serre dimension. Motivated by the ADE classification of the 2-dimensional N=2 SCFT with \widehat{c}<1, we also give a characterization of the derived category of Dynkin quivers in terms of the Serre dimension and the global dimension function. This is a joint work in progress with Kohei Kikuta and Genki Ouchi.

uk: вхід без авторизації

name: Serre dimension and stability conditions


uk: Математика, природничі науки

uk: вхід без авторизації

name: Atmosphere models I: total energy errors in dynamics and physics-dynamics coupling


uk: Математика, природничі науки

uk: вхід без авторизації

name: Atmosphere models II: energy conversions and internal entropy production in a moist atmosphere


uk: Математика, природничі науки

uk: вхід без авторизації

name: Eldred - Atmosphere breakout 1: energy conserving numerics and physical consistency


uk: Математика, природничі науки

uk: вхід без авторизації

name: Dubos - Atmosphere breakout 1: energy conserving numerics and physical consistency


uk: Математика, природничі науки

uk: вхід без авторизації

name: Johansen - Atmosphere breakout 1: energy conserving numerics and physical consistency


uk: Математика, природничі науки

uk: вхід без авторизації

name: Total Energy and Water Conservation in Atmospheric Models


uk: Математика, природничі науки

uk: вхід без авторизації

name: Dry mass adjustment for a fixed pressure coordinate


uk: Математика, природничі науки

description: Including cloud and turbulence parameterizations.

uk: вхід без авторизації

name: An energy fixer for CLUBB in the E3SM climate model


uk: Математика, природничі науки

uk: вхід без авторизації

name: Parallel coupling of physics and dynamics in a global atmosphere model


uk: Математика, природничі науки

uk: вхід без авторизації

name: Evaluating physics on a separate grid: physics-dynamics coupling (pdf) with element based high-order Galerkin methods

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


uk: Математика, природничі науки

uk: вхід без авторизації

name: Atmosphere breakout 3: missing processes in current models


uk: Математика, природничі науки

description: With Christiane Jablonowski, Peter Lauritzen, Florian Lemarié.

uk: вхід без авторизації

name: Atmosphere breakout 4: better test cases for assessing physics-dynamics coupling errors


Wrap up and future directions: group

uk: Математика, природничі науки

description: Group discussion of recommendations and priorities for future research.

uk: вхід без авторизації


uk: Математика, природничі науки

uk: вхід без авторизації

name: Group discussion of recommendations and priorities for future research


Torelli theorems for derived categories

uk: Математика, природничі науки

description: I will present joint work with Martin Olsson on a conjectural Torelli-type statement for derived categories of arbitrary proper smooth varieties. The key is a replacement for the Hodge structure by “filtrations” that naturally arise from geometry.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The talk will start by recalling essential dimension and looking at some elementary examples. For certain kinds of Deligne-Mumford stacks, P. Brosnan Z. Reichstein and A. Vistoli have proved a very powerful genericity theorem which reduces the calculation of essential dimension to that of gerbes. For Gm-gerbes there is a conjectural formula of J.L Colliot-Thelene, N. Karpenko and A. Merkurjev for the essential dimension of a gerbe that relates the essential dimension to Brauer invariants such as the index. In joint work with I. Biswas and N. Hoffmann, modulo this conjecture, the essential dimension of the moduli stack of vector bundles on a smooth projective curve was computed. Recently these results were extended to orbifold curves with D. Valluri.

uk: вхід без авторизації

name: Essential dimension of stacks of bundles


uk: Математика, природничі науки

description: Topological Hochschild homology gives rise to a number of intrinsic derived invariants of a variety in positive characteristic. We will explain how to compute these quantities in terms of more classical invariants of X, such as de Rham and crystalline cohomology. As a consequence, we obtain new restrictions on the Hodge numbers of derived equivalent varieties in positive characteristic. We will also present an example of two derived equivalent 3-folds in characteristic 3 with different Hodge numbers. This is joint work with Benjamin Antieau and Nick Addington.

uk: вхід без авторизації

name: Derived invariants of varieties in positive characteristic


Rational points and derived equivalence

uk: Математика, природничі науки

description: It is natural to ask which properties of a smooth projective variety are recovered by its derived category. In this talk, I will consider the question: is the existence of a rational point preserved under derived equivalence? In recent joint work with Nicolas Addington, Benjamin Antieau, and Katrina Honigs, we show that over Q, the answer is no. We give two examples: an abelian variety and a torsor over it, and a pair of hyperkaehler fourfolds. The latter is independently interesting as a new example of a transcendental Brauer-Manin obstruction to the Hasse principle.

uk: вхід без авторизації


Gluing Periods for DHT Mirrors

uk: Математика, природничі науки

description: Let X be a Calabi–Yau manifold that admits a Tyurin degeneration to a union of two quasi-Fano varieties X1 and X2 intersecting along a smooth anticanonical divisor D. The “DHT mirror symmetry conjecture” implies that the Landau–Ginzburg mirrors of (X1,D) and (X2,D) can be glued to obtain the mirror of X. Initial motivation came from considering the bounded derived categories of X, X1, and X2 and symplectomorphisms on the Landau-Ginzburg models mirror to (X1,D) and (X2,D). In this talk, flipping the roles of the two categories, I will explain how periods on the Landau-Ginzburg mirrors of (X1,D) and (X2,D) are related to periods on the mirror of X. The relation among periods relates different Gromov-Witten invariants via their respective mirror maps. This is joint work with Fenglong You and Jordan Kostiuk.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The failure of the integral Hodge conjecture has been known since the famous counterexamples of Atiyah and Hirzebruch. Currently, several methods exist for producing other counterexamples. For instance, a seminal result of Colliot-Thélène and Voisin relates the failure of the integral Hodge conjecture in degree 4 to degree 3 unramified cohomology. After giving a quick overview of this method, I will discuss how it can be used to obtain new counterexamples to the integral Hodge conjecture.

uk: вхід без авторизації

name: Unramified cohomology and the integral Hodge conjecture


uk: Математика, природничі науки

description: The problem of determining conditions under which a rational map can exist between a pair of twisted flag varieties plays an important role in the study of algebraic groups and their torsors over general fields. A non-trivial special case arising in the theory of quadratic forms amounts to studying the extent to which an anisotropic quadratic form can become isotropic after extension to the function field of a quadric. To this end, let p and q be anisotropic quadratic forms over an arbitrary field, and let k be the dimension of the anisotropic part of q over the function field of the quadric p=0. We then conjecture that the dimension of q lies within k of an integer multiple of 2s+1, where 2s+1 is the least power of 2 bounding the dimension of p from above. This generalizes the so-called ``separation theorem'' of D. Hoffmann, bridging the gap between it and some other classical results on Witt kernels of function fields of quadrics. The statement holds trivially if k≥2s−1. In this talk, I will discuss recent work that confirms its validity in the case where k≤2s−1+2s−2 (among other cases).

uk: вхід без авторизації

name: On an extension of the separation theorem for quadratic forms over fields


uk: Математика, природничі науки

description: Toric varieties defined over the complex numbers provide an important testing ground for computing various algebro-geometric invariants (e.g., the coherent derived category associated to a variety), as many computations of interest may be phrased entirely in terms of combinatorial data such as fans, cones, polytopes. Over general fields, we consider twisted forms of such objects called "arithmetic toric varieties", whose analysis is naturally Galois-theoretic. In this talk, we will present results on the structure of derived categories of arithmetic toric varieties via exceptional collections. In particular, we will focus on some highly symmetric classes of such objects, including centrally symmetric toric Fano varieties and toric varieties associated to root systems of type A. The latter class yields examples of varieties which are arithmetically interesting but whose derived categories are well understood. A conjecture of Orlov posits that the structure of the derived category influences the rationality type of a variety. We will discuss how this plays out in the setting of toric varieties. This is joint work with Matthew Ballard, Alexander Duncan, and Alicia Lamarche.

uk: вхід без авторизації

name: Twisted forms of toric varieties, their derived categories, and their rationality


From flips to functors

uk: Математика, природничі науки

description: A healthy body of evidence says that birational geometry and derived categories are intimately bound. Even so, many basic questions are still open. One of the most central questions is the conjecture of Bondal and Orlov (later extended by Kawamata) that says two smooth projective varieties related by a flop are actually derived equivalent. The first step in resolving this question is understanding how to produce functors from rational maps. In work with Diemer and Favero, we provided a method to construct an integral kernel associated to any D-flip of normal varieties with Q-Cartier D. Conjecturally, this can be used to answer Bondal and Orlov's question. In this talk, we will discuss the construction and natural extensions of it. In particular, we will highlight work with Chidambaram, Favero, McFaddin, and Vandermolen relating to the what has been termed a Grassmann flop.

uk: вхід без авторизації


Quadratic forms and Brauer groups

uk: Математика, природничі науки

description: The Clifford algebras provide a connection between the study of quadratic forms and the Brauer group. We explain how a bound for the u-invariant of a field can be given purely in terms of certain period-index bounds for the Brauer group. We survey some progress in finding such bounds for function fields of p-adic curves. There are open questions in this direction for function fields of curves over number fields.

uk: вхід без авторизації


Numerical dimension revisited

uk: Математика, природничі науки

description: The Iitaka dimension of a line bundle D on a projective variety X is the dimension of the image of the rational map given by |mD| for large and divisible m. The Iitaka dimension is not a numerical invariant of D, and there are several approaches to constructing a "numerical dimension", which should be an analogous invariant depending only on the numerical class of D. I will discuss some divisors of dynamical origin that have unexpected behavior with respect to these invariants.

uk: вхід без авторизації


uk: Математика, природничі науки

description: The theory of flasque resolutions gives a complete solution to the problem of determining whether an algebraic torus is stably- or retract-rational. More generally, flasque resolutions of more general connected linear algebraic groups are a powerful tool for studying R-equivalence, weak approximation, and the Hasse principle of such groups. There is a dual notion of coflasque for tori, which can also be extended to more general linear algebraic groups.  A common technique for classifying objects over non-closed fields is to associate a set of simple algebras to each object. The most famous example is the deep connection between Severi-Brauer varieties and central simple algebras. However, it may be that there is no way to distinguish non-isomorphic objects through any such association if it is functorial in the base field. Using coflasque resolutions of general algebraic groups, we can precisely describe the set of such indistinguishable objects.  I will recall flasque and coflasque resolutions, discuss how they can be used to describe these indistinguishable sets, and comment on the implications for detecting arithmetic properties using Brauer groups, K-theory, and derived categories. This is based on joint work with Matthew Ballard, Alicia Lamarche, and Patrick McFaddin.

uk: вхід без авторизації

name: Indistinguishability and simple algebras


uk: Математика, природничі науки

description: Let X be a projective algebraic surface. We recall the K-group K_{1,\ind}^{(2)}(X) of indecomposables and provide evidence that membrane integrals are sufficient to detect these indecomposable classes.

uk: вхід без авторизації

name: Indecomposable K_1 classes on a Surface and Membrane Integrals


uk: Математика, природничі науки

description: Donaldson-Thomas invariants are virtual counts of stable sheaves with fixed numerical invariants, on a Calabi-Yau threefold. I will survey old and new results with emphasis on the peculiarities arising when substituting the CY with an abelian threefold.

uk: вхід без авторизації

name: Donaldson-Thomas theory for abelian threefolds


uk: Математика, природничі науки

description: The concept of birational divisor arises in different contexts; for example in work of V. V. Shokurov and, independently, P. Vojta. The aim of this talk is to survey related recent developments. Some emphasis will be placed on results that deal with Brauer groups of function fields of algebraic varieties (e.g., joint work with C. Ingalls). Time permitting, I hope also (to at least briefly) mention results which deal with complexity and distribution of rational points (e.g., extensions to the recent work of Ru and Vojta).

uk: вхід без авторизації

name: Birational divisors and consequences for noncommutative algebra and arithmetic


VGIT for CDGAs

uk: Математика, природничі науки

description: By an observation of Reid, any birational morphism between algebraic varieties can be realized through variation of geometric invariant theory quotients (VGIT). As a consequence, one way of comparing the derived categories of birational algebraic varieties is through VGIT. It turns out that this can be done even in singular cases by, in some sense, "deriving" the VGIT procedure. I will discuss how this can be done very concretely by studying VGIT for commutative differential graded algebras. This talk is about joint work with Chidambaram motivated by work with Ballard and Diemer.

uk: вхід без авторизації


uk: Математика, природничі науки

description: In this talk we study the behavior of special classes of fibrations onto normal projective varieties that admit a finite morphism to an abelian variety under derived equivalence of smooth projective complex varieties. Our first result is that any derived equivalence of such varieties induces a base preserving correspondence between their sets of isomorphism classes of fibrations onto smooth projective curves of genus greater or equal to two. The proof of this result involves earlier results, obtained in collaboration with M. Popa, regarding the derived invariance of the non-vanishing loci attached to the canonical bundle, and generic vanishing theory. Concerning fibrations onto higher-dimensional bases, we show how the problem of the derived invariance of fibrations is related to the conjectural derived invariance of Hodge numbers. I will report on some progress in this direction.

uk: вхід без авторизації

name: Fibrations of algebraic varieties and derived equivalence


uk: Математика, природничі науки

description: Cohomological invariants play an important role in the classification of G-torsors, where G is an algebraic group over a field. However these invariants are hard to compute. In case of a Weyl group G they have been recently computed (with some restrictions on the base field). A crucial role in this computation is played by a splitting principle, which roughly says that an invariant of a Weyl group is determined by its restriction to elementary abelian 2-subgroups generated by reflections. In the talk I will discuss the generalization of this principle to orthogonal reflection groups. (joint work with Christian Hirsch)

uk: вхід без авторизації

name: A splitting principle for cohomological invariants of reflection groups


uk: Математика, природничі науки

description: We study non-commutative projective varieties in the sense of Artin-Zhang, which are given by non-commutative homogeneous coordinate rings, which are finite over their centre. We construct moduli spaces of stable modules for these, and construct a symmetric obstruction theory in the CY3-case. This gives deformation invariants of Donaldson-Thomas type. The simplest example is the Fermat quintic in quantum projective space, where the coordinates commute only up to carefully chosen 5th roots of unity. We explore the moduli theory of finite length modules, which mixes features both of the Hilbert scheme of commutative 3-folds, and the representation theory of quivers with potential. This is mostly a report on the work of Yu-Hsiang Liu, with contributions by myself and Atsushi Kanazawa.

uk: вхід без авторизації

name: Donaldson-Thomas theory of non-commutative projective schemes


uk: Математика, природничі науки

description: Given a virtually smooth quasi-projective scheme M, and a morphism from M to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M′/M that admits a perfect obstruction theory relative to B. We study the resulting virtual cycles on the fibers of M′/B and relate them to the image of the virtual cycle [M]vir under refined Gysin homomorphisms. Our main application is when M is a moduli space of stable codimension 1 sheaves on a nonsingular projective surface or Fano threefold.

uk: вхід без авторизації

name: Counting sheaves on singular curves and surfaces


uk: Математика, природничі науки

description: In 2014 Gaiotto conjectured a Lagrangian correspondence between holomorphic Lagrangian of opers in the Dolbeault moduli space of Higgs bundles and the de Rham moduli space of holomorphic connections. The conjecture was solved in 2016 for holomorphic opers in paper with Fredrickson, Kydonakis, Mazzeo, Mulase and Neitzke.  By a similar analysis method, Collier and Wentworth, extended the correspondence for more general Lagrangians consisting of stable points.  In my talk, I will present an algebraic geometry description of the Lagrangian correspondence of Gaiotto, based on the work of Simpson.

uk: вхід без авторизації

name: Lagrangian correspondence between Hitchin and de Rham moduli spaces

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


uk: Математика, природничі науки

description: The moduli stacks of sheaves on Calabi-Yau four-folds carry -2-shifted symplectic structures (Pantev, Toen, Vezzosi, Vaquie). Viewing these stacks as objects in differential geometry, one can construct Lagrangian foliations relative to these symplectic structures, such that quotients by the foliations are perfectly obstructed derived stacks, equipped with globally defined -1-shifted potentials, whose critical loci are the original moduli stacks. This is a joint work with A.Sheshmani and S-T.Yau.

uk: вхід без авторизації

name: Moduli stacks of sheaves on Calabi-Yau four-folds as critical loci


uk: Математика, природничі науки

description: The derived symmetries associated to a 3-fold admitting an Atiyah flop may be organised into an action of the fundamental group of a sphere with three punctures, thought of as a stringy Kaehler moduli space. I extend this to general flops of irreducible curves on 3-folds in joint work with M. Wemyss. This uses certain deformation algebras associated to the curve and its multiples, with applications to Gopakumar-Vafa invariants.

uk: вхід без авторизації

name: Stringy Kaehler moduli, mutation and monodromy


uk: Математика, природничі науки

description: The Geometric Langlands Conjecture (GLC) for a curve C and a group G is a non-abelian generalization of the relation between a curve and its Jacobian. It claims the existence of Hecke eigensheaves on the moduli of G-bundles on C. The parabolic GLC is a further extension to curves with punctures. After explaining and illustrating the conjectures, I will outline an approach to proving them using non-abelian Hodge theory. A key geometric ingredient is the locus of wobbly bundles: bundles that are stable but not very stable. If time allows, I will discuss two instances where this program has been implemented recently: GLC for G=GL(2) and genus 2 curves (with T. Pantev and C. Simson), and parabolic GLC for P1 with marked points (with T. Pantev).

uk: вхід без авторизації

name: On the Geometric Langlands Conjecture and Non-Abelian Hodge Theory


uk: Математика, природничі науки

description: In this talk we reframe a collection of well-known comparison results in genus-zero Gromov-Witten theory in order to relate these to integral transforms between derived categories. This implies that various comparisons between Gromov-Witten and FJRW theory are compatible with the integral structure introduced by Iritani. We conclude with a proof that a version of the LG/CY correspondence relating quantum D-modules with Orlov's equivalence is implied by a version of the crepant transformation conjecture.

uk: вхід без авторизації

name: Integral transforms and quantum correspondences