(A. Olevskii): Let X be a Banach function space on R. Does there exist a function f in X and a uniformly dicrete sequence S in R such that the family of translates {f(t-s)} for s in S spans the whole space X? I will present a survey on the subject and discuss the last results joint with Alexander Ulanovskii.
(A. Chertock): Chemotaxis is a movement of micro-organisms or cells towards the areas of high concentration of a certain chemical, which attracts the cells and may be either produced or consumed by them. In its simplest form, the chemotaxis model is described by a system of nonlinear PDEs: a convection-diffusion equation for the cell density coupled with a reaction- diffusion equation for the chemoattractant concentration. It is well-known that solutions of such systems may develop spiky structures or even blow up in finite time provided the total number of cells exceeds a certain threshold. This makes development of numerical methods for chemotaxis systems extremely delicate and challenging task. In this talk, I will present a family of high-order numerical methods for the Keller-Segel chemotaxis system and several related models. Applications of the proposed methods to to multi-scale and coupled chemotaxis–fluid system and will also be discussed.
(J. Lewis): 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.
(M. Csörnyei): We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set. This is a joint work with Alan Chang.
(R. Thomas): Many results in extremal graph theory can be formulated as inequalities on graph densities. While many inequalities are known,many more are conjectured. A standard tool to establish an inequality is to write the expression whose nonnegativity needs to be certified, as a sum of squares. This technique has had many successes but also limitations. In this talk I will describe new restrictions that show that several simple inequalities cannot be certified by sums of squares. These results extend to the powerful frameworks of flag algebras by Razborov and graph algebras by Lovasz and Szegedy. Joint work with Greg Blekherman, Annie Raymond, and Mohit Singh.
(B. Tsygan): I will discuss how to construct a category starting with a symplectic manifold using microlocal methods, following works of Tamarkin, Nadler-Zaslow, and myself. The talk will be very introductory and concentrate mainly on examples of the plane, cylinder, and 2-torus.
(A. Greenleaf): Electrical impedance tomography (EIT) is an imaging technique that has been proposed for imaging and nondestructive testing of oil fields, manufactured parts and human bodies. Over the last 40 years it has led to much beautiful mathematics, but the very features that make the mathematical foundation of EIT, the Calderon inverse problem, so interesting mathematically also makes the images EIT produces very blurry. I will describe some of the underlying analysis, and work in progress on using some old ideas from PDE to try to apply EIT to stroke diagnosis.
(L. Pick): Many important tasks in mathematics and its applications turn into investigation of the action of operators on various algebraic structures endowed with appropriate analytic properties. A principal instance of such an investigation involves various classes of functions and sequences, usually labelled in general as function spaces. In connection with various applications in mathematical physics or the theory of partial differential equations, enormous effort has been spent by many authors in order to improve classical results in the sense of nailing down more and more precise function spaces on which the operators act. However, little seems to be known about sharpness or optimality of such results. We shall survey a new approach to the investigation of action of operators on function spaces, which we developed roughly during the last two decades, and whose principal aim is to push the borders of the knowledge as far as one can and to show that further improvement is impossible. The key innovation is nailing down the actual optimal function spaces for given operators. Techniques involved in this approach vary from symmetrization and interpolation to new types of inequalities or to iteration processes.
(B. McReynolds): The profinite completion of a finitely generated, residually finite group is the universal group for the finite representation theory of the group. One of the basic questions one can ask is whether or not the group is determined up to isomorphism by its profinite completion. Said alternatively, when is a finitely generated, residually finite group determined by its finite quotients. In this talk, I will discuss some positive and negative results to this question with the main focus on lattices in Lie groups. This is based on recent work with Martin Bridson, Alan Reid, and Ryan Spitler.
(A. Lauda): In this talk we will explain how Lie theory leads to interesting families of invariants for knots and links that can all be defined in an elementary diagrammatic fashion. The Reshetikhin-Turaev construction associated knot invariants to the data of a simple Lie algebra and a choice of irreducible representation. The Jones polynomial is the most famous example coming from the Lie algebra sl(2) and its two-dimensional representation. In this talk we will explain Cautis-Kamnitzer-Morrison's novel new approach to studying RT invariants associated to the Lie algebra sl(n). Rather than delving into a morass of representation theory, we will show how two relatively simple Lie theoretic ingredients can be combined with a powerful duality (Howe duality) to give an elementary and diagrammatic construction of these invariants. We will explain how this new framework solved an important open problem in representation theory, proves the q-holonomic conjecture in knot theory (joint with Garoufalidis and Lê), and leads to a new elementary approach to 'categorifying' these knots invariants to link homology theories.
(T. Panov): A combinatorial 3-dimensional polytope P can be realized in Lobachevsky 3-space with right dihedral angles if and only if it is simple, flag and does not have 4-belts of facets. This criterion was proved in the works of A.Pogorelov and E.Andreev of the 1960s. We refer to combinatorial 3-polytopes admitting a right-angled realization in Lobachevsky 3-space as Pogorelov polytopes. The Pogorelov class contains all fullerenes, i.e. simple 3-polytopes with only 5-gonal and 6-gonal facets. There are two families of smooth manifolds associated with Pogorelov polytopes. The first family consists of 3-dimensional small covers (in the sense of M.Davis and T.Januszkiewicz) of Pogorelov polytopes P, also known as hyperbolic 3-manifolds of Loebell type. These are aspherical 3-manifolds whose fundamental groups are certain extensions of abelian 2-groups by hyperbolic right-angled reflection groups in the facets of P. The second family consists of 6-dimensional quasitoric manifolds over Pogorelov polytopes. These are simply connected 6-manifolds with a 3-dimensional torus action and orbit space $P$. Our main result is that both families are cohomologically rigid, i.e. two manifolds M and M' from either family are diffeomorphic if and only if their cohomology rings are isomorphic. We also prove that a cohomology ring isomorphism implies an equivalence of characteristic pairs; in particular, the corresponding polytopes P and P' are combinatorially equivalent. This leads to a positive solution of a problem of A.Vesnin (1991) on hyperbolic Loebell manifolds, and implies their full classification. Our results are intertwined with classical subjects of geometry and topology such as combinatorics of 3-polytopes, the Four Color Theorem, aspherical manifolds, a diffeomorphism classification of 6-manifolds and invariance of Pontryagin classes. The proofs use techniques of toric topology. This is a joint work with V. Buchstaber, N. Erokhovets, M. Masuda and S. Park.
(E. Palsson): Finding and understanding patterns in data sets is of significant importance in many applications. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk, I will give an introduction to both the discrete and continuous questions, report on recent progress, and share some exciting open questions.
2018-2019
2017-2018
2016-2017