(Z. Wang): In topological quantum computing, information is encoded in "knotted" quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of 10^{-10}, and harnessed to stabilize quantum memory. In this survey, we discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.
(E. Peters): Planar algebras are a formalization of "proof by pictures" techniques that are used in the study of knots, subfactors, and tensor categories, among other places. When one tries to directly construct a planar algebra, by generators and relations, one quickly runs into the standard problems, like: how do I decide if what I have is just a dressed-up version of the trivial planar algebra? Sometimes, one can answer this type of question by giving an "evaluation algorithm" based on the relations available. In this talk, I will define planar algebras and give some of my favorite examples of planar algebras with cool evaluation algorithms. No background beyond linear algebra will be assumed. If time permits, I will touch on recent results on the extended Haagerup planar algebras, which is joint work with Grossman, Morrison, Penneys and Snyder.
(I. Farah): Coarse geometry is the study of large-scale properties of metric spaces. Roughly, two spaces are coarsely equivalent if their "large-scale structures" agree. The uniform Roe algebra C^*_u(X) is a norm-closed algebra of bounded linear operators on the Hilbert space \ell^2(X). It is the algebra of all bounded linear operators on \ell^2(X) that can be uniformly approximated by operators of "finite propagation". The uniform Roe algebra is a coarse invariant of the space X. It includes \ell^\infty(X) (as the algebra of all operators of zero propagation) and the algebra of compact operators. After introducing the basics of coarse spaces and uniform Roe algebras, we will consider the following questions: (1) If the uniform Roe algebras of X and Y are isomorphic, when can we conclude that X and Y are coarsely equivalent? (2) The uniform Roe corona is obtained by modding out the compact operators from C^*_u(X). If the uniform Roe coronas of X and Y are isomorphic, when can we conclude that C^*_u(X) and C^*_u(Y) are isomorphic, or at least have large isomorphic corners? Under some additional assumptions on X and Y (the uniform local finiteness and a weakening of property A -- A stands for "amenability"), (1) has a positive answer. The answer to question (2), even for uniformly locally finite spaces with property A, is quite surprising. These talks will be based on a joint work with B.M. Braga and A. Vignati.
(P. Hajlasz): I will discuss the following result: If n=2,3 and a C^1 map f: S^1 -> S^n is not homotopic to a constant map, then there is an open set U of \mathbb{S}^{n+1} such that rank(df) = n on U, and f(U) is dense in S^n, while for any n at least 4, there is a C^1 map f:S^{n+1} -> S^n that is not homotopic to a constant map and such that rank(df)<4 everywhere. The result in the case n is at least 4 answers a question of Larry Guth. I will also discuss an application of the result to a solution of a recent conjecture of Jacek Galeski. In particular I will show that there is a C^1 mapping in R^5 with the derivative of rank at most 3 that cannot be uniformly approximated by C^2 mappings with the derivative of rank at most 3. The methods use analysis, algebraic topology and geometric measure theory. The talk will be accessible to graduate students. The presentation is based on my two joint papers. One with P. Goldstein and one with P. Goldstein and P. Pankka.
(A. Mohammadi): Rigidity phenomena in homogeneous dynamics have been extensively studied over the past few decades with several striking results and applications. In this talk we will give an overview of the more recent activities which aim at presenting quantitative versions of some of these strong rigidity results.
(R. Kuske): Stochastic averaging has a long history for systems with multiple time scales and Gaussian forcing, but far less attention has been paid to problems where the stochastic forcing has infinite variance, such as in Levy processes or alpha-stable noise. Correlated additive and multiplicative (CAM) Gaussian noise, with infinite variance or "fat tails" in certain parameter regimes, can arise generically in many models with parametric uncertainty and has received increased attention in the context of atmosphere and ocean dynamics. These applications motivate new reduced models using stochastic averaging for systems with fast processes driven by noise with fat tails. We develop these results for the case of alpha-stable noise, giving explicit results that use the Marcus interpretation, the infinite variance analog to the Stratonovich interpretation. Then we show how reduced models for systems driven by fast linear CAM noise processes can be connected with the stochastic averaging for multiple scales systems driven by alpha-stable processes. We identify the conditions under which the approximation of a CAM noise process is valid in the averaged system, and illustrate methods using effectively equivalent fast, infinite-variance processes. These new types of approximations open the door for stochastic averaging in a wider range of stochastic systems with multiple time scales. This is joint work with Prof. Adam Monahan (U Victoria) and Dr. Will Thompson (UBC/NMi Metrology and Gaming)
(E. Lubetzky): The Potts model and its special case, the Ising model, are one of the most studied models in mathematical physics, tracing back to the 1920's with the motivation of modeling ferromagnetism. In the classical 2D setting, the model assigns one of q possible colors to the sites of the square grid according to a given probability distribution, which is a function of the number of neighboring sites whose spins agree, as well as the temperature. A focal point of the study of the model has been the critical temperature, where the phase transition in the static model is accompanied by a dynamical phase transition for the natural stochastic processes that model its evolution, as well as provide efficient methods for sampling. I will survey the recent developments on understanding this dynamical phase transition, which subtly depends on the number of colors q and the boundary conditions.
(V. Mikyoung Hur): Water waves describe the situation where water lies below a body of air and are acted upon by gravity. Describing what we may see or feel at the beach or in a boat, they are a perfect specimen of applied mathematics. They encompass wide-ranging wave phenomena, from ripples driven by surface tension to tsunamis and to rogue waves. The interface between the water and the air is free and poses profound and subtle difficulties for rigorous analysis, numerical computation and modeling. I will discuss some recent developments in the mathematical aspects of water wave phenomena. Particularly, (1) is the solution to the Cauchy problem regular, or do singularities form after some time? (2) are there solutions spatially periodic? (3) are they dynamically stable?
(C. Berkesch): The minimal free resolution of a graded module encodes many geometric properties of the corresponding sheaf on projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. By considering free complexes that are acyclic modulo irrelevant homology, which we call virtual resolutions, one can recover a relationship between the homological algebra and geometry of sheaves over these more general varieties. This is joint work with Daniel Erman and Gregory G. Smith.
(K. Igusa): Around 1975, using algebraic K-theory of spaces, F. Waldhausen predicted, as a special case of a more general theory, that there are exotic smooth disk bundles over 4k-spheres. These are smooth disk bundles which are trivial as topological bundles but nontrivial as smooth bundles. This was a difficult result requiring the work of many mathematicians. Also in 1975, A. Hatcher proposed a simple construction of these exotic bundles using G/O. Bokstedt proved that Hatcher's idea works, i.e., that Hatcher's construction gives a rational homotopy equivalence from G/O to the stablised pseudoisotopy space of a point. Later I gave another proof using parametrized Morse theory. In 2018, T. Kragh showed that the homotopy fiber of this map, which he called the "Hatcher-Waldhausen map" is the Eliashberg-Gromov space which might be an obstruction to the validity of the "nearby Lagrangian conjecture". Kragh's construction uses parametrized Morse theory to obtain "generating functions" for Lagrangian manifolds. In this talk I will tell this story in reverse order starting with generating functions for Lagrangians, the Eliashberg-Gromov construction and Kragh's argument which, running in reverse, will reproduce Hatcher's construction. The equivariant version of Hatcher construction (joint work with Goodwillie and Ohrt) will give some interesting examples in symplectic topology (joint work with Alvarez-Gavela).
(L. Bieri): In Mathematical General Relativity (GR) the Einstein equations describe the laws of the universe. This system of hyperbolic nonlinear pde has served as a playground for all kinds of new problems and methods in pde analysis and geometry. A major goal in the study of these equations is to investigate the analytic properties and geometries of the solution spacetimes. In particular, fluctuations of the curvature of the spacetime, known as gravitational waves, have been a highly active research topic. In 2015, gravitational waves were observed for the first time by Advanced LIGO (and several times since then). These waves are produced during the mergers of black holes or neutron stars and in core-collapse supernovae. Understanding gravitational radiation is tightly interwoven with the study of the Cauchy problem in GR. I will talk about the Cauchy problem for the Einstein equations, explain geometric-analytic results on gravitational radiation and the memory effect of gravitational waves, the latter being a permanent change of the spacetime. We will connect the mathematical findings to experiments.
(V. Lunts): Grothendieck introduced the group K(Var) of algebraic varieties which is generated by isomorphism classes of varieties subject to "cut and paste" relations ("scissors congruence"). This group is naturally a commutative ring. I will discuss a few interesting "motivic measures" which are homomorphisms from K(Var) to other rings. Some applications to rationality of motivic zeta function will be discussed.
(K. Hess Bellwald): Over the past decade, and particularly over the past five years, research at the interface of topology and neuroscience has grown remarkably fast. Topology has, for example, been successfully applied to objective classification of neuron morphologies and to automatic detection of network dynamics. In this talk I will focus on the algebraic topology of brain structure and function, describing results obtained by members of my lab in collaboration with the Blue Brain Project on digitally reconstructed microcircuits of neurons in the rat cortex. I will also describe our on-going work on the topology of synaptic plasticity. The talk will include an overview of the Blue Brain Project and a brief introduction to the topological tools that we use.
(B. Kleiner): The Smale Conjecture (1961) may be stated in any of the following equivalent forms: - The space of embedded 2-spheres in R^3 is contractible. - The inclusion of the orthogonal group O(4) into the group of diffeomorphisms of the 3-sphere is a homotopy equivalence. - The space of all Riemannian metrics on S^3 with constant sectional curvature is contractible. While the analogous statement one dimension lower can be proven in many ways --- for instance using the Riemann mapping theorem --- Smale's conjecture turned out to be surprisingly difficult, and remained open until 1983, when it was proven by Hatcher using a deep combinatorial argument. Smale's Conjecture has a natural generalization to other 3-dimensional space forms: if M is a Riemannian 3-manifold of nonzero constant sectional curvature, then the inclusion of the isometry group into the diffeomorphism group is a homotopy equivalence. The lecture will explain how Ricci flow through singularites, as developed in the last few years by John Lott, Richard Bamler, and myself, can be used to address this conjecture. This is joint work with Richard Bamler.
(A. Furman): A surface of genus at least two admits a variety of hyperbolic structures - the Teichmuller space. However, in higher dimensions similar locally symmetric structures turn out to be very rigid, as was shown by Weil, Selberg, Mostow, Margulis. Perhaps surprisingly, in addition to geometry, Lie groups, and number theory, some ideas from ergodic theory - dynamics on measure spaces - play an important role in these developments. In the talk I will discuss some of these rigidity phenomena with the emphasis on the interplay between ergodic theory and algebraic groups. Based on joint works with Uri Bader.
2017-2018
2016-2017
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