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Mathematics Colloquium Winter 2017

Tuesdays - 4:00 p.m. McHenry Room 4130 Refreshments served at 3:30 in the Tea Room (4161)
For further information please contact Professor Junecue Suh or call the Mathematics Department at 459-2969

Tuesday, January 10, 2017

NO COLLOQUIUM

Tuesday, January 17,2017

Chenyang Xu, Beijing International Center of Mathematics Research

Title: Dual Complex of a Singular Pair

Abstract: The topology of an algebraic variety is a central subject in algebraic geometry. Instead of a variety, we consider the topology of a pair (X,D) which is a variety X with a divisor D, but in the coarsest level. More precisely, we study the dual complex defined as the combinatorial datum characterizing how the components of D intersect with each other. We will discuss how to use the minimal model program (MMP) to investigate it. As one concrete application, we will explain how close the dual complex of a log Calabi-Yau pair (X,D) is to a finite quotient of a sphere.

Tuesday, January 24, 2017

Jenny Wilson, Stanford University

Title: Stability in the homology of configuration spaces

Abstract: This talk will illustrate some topological properties of the space F_k(M) of ordered k-tuples of distinct points in a manifold M. For a fixed manifold M, as k increases, we might expect the topology of the configuration spaces F_k(M) to become increasingly complicated. Church and others showed, however, that when M is a connected and open, there is a representation-theoretic sense in which these configuration spaces stabilize. In this talk I will explain these stability patterns, and how they generalize classical notions of homological stability proved by McDuff and Segal in the 1970s. I will describe higher-order "secondary stability" phenomena established in recent work joint with Jeremy Miller.

Tuesday, January 31, 2017

Tuesday, February 7, 2017

Beren Sanders, University of Copenhagen

Title: An Introduction to Tensor Triangular Geometry

Abstract: Tensor triangulated categories arise in a truly diverse range of mathematical disciplines, from algebraic geometry and modular representation theory to stable homotopy theory, symplectic topology, and beyond. Tensor triangular geometry is a recent theory --- initiated and developed by Paul Balmer and his collaborators --- which studies tensor triangulated categories geometrically via methods motivated by algebraic geometry. Recent successes of the theory include applications to equivariant stable homotopy theory and the introduction of descent methods to modular representation theory. A key tool for these applications has been a tensor triangular analogue of the étale topology, and the surprising fact that in equivariant contexts, restriction to a subgroup can be regarded as an étale extension. In this talk, I will give an introduction to this area of mathematics, with an emphasis on the big picture.

Thursday, February 9, 2017

Benjamin Sambale, TU Kaiserslautern, Germany

Title: The Number of Characters in a Block of a Finite Group

Abstract: After a brief introduction to the character theory of finite groups we present some progress on open problems of Richard Brauer. These problems concern the global-local relationship of blocks. In the last part we give an overview of the tools used in the proofs.

Tuesday, February 14, 2017

You Qi, Yale University

Title: Categorification at a Prime Root of Unity

Abstract: Topological quantum field theory, in the sense of Atiyah and Segal, is an excellent organizational principle in understanding different kinds of manifold invariants. We outline a program aimed at categorically lifting the 3-dimensional Witten-Reshetikhin-Turaev topological quantum field theory into a 4-dimensional theory. This would eventually give rise to a combinatorial construction of 3- and 4-manifold invariants, previously only obtainable only through gauge theoretical methods. This is based on previous joint work and work in progress with B. Elias, M. Khovanov and J. Sussan.

Thursday, February 16, 2017

Oleksandr Tsymbaliuk, Stony Brook University

Title: Shifted Yangians and Shifted Quantum Affine Algebras

Abstract: In this talk, I will speak about the shifted versions of Yangians and quantum affine algebras as well as their incarnations through geometry of parabolic Laumon spaces, additive/multiplicative slices, and Todda lattice.

I will start by reminding the notion of the shifted Yangian (originally introduced by Brundan-Kleshchev in the gl(n) case with a dominant shift and later generalized by Kamnitzer-Webster-Weekes-Yacobi to any simple Lie algebra with an arbitrary shift) as well as the recent work relating these algebras to the Coulomb branches.

In the second half, I will discuss the multiplicative analogue of that story (which is a joint project with Michael Finkelberg). On the algebraic side this leads to the notion of shifted quantum affine algebras, while on the geometric side we replace cohomology by K-theory and additive slices are replaced by multiplicative slices.

Tuesday, February 21, 2017

Ricardo Sanfelice, University of California, Santa Cruz

Title: Structural Properties and Tools for Robustness in Hybrid Systems: Flows, Jumps, Zeno, and other Misbehaviors

Abstract: Hybrid systems have become prevalent when describing control systems that mix continuous and impulsive dynamics. Continuous dynamics usually govern the evolution of the physical variables in a system, while impulsive (or discrete) behavior is typically due to events in the control algorithm or abrupt changes in the dynamics. A mathematical framework comprised of differential and difference equations/inclusions with constraints will be introduced to model, analyze, and design such systems. An appropriate notion of solution and basic properties on the system data guaranteeing sequential compactness of solutions will be introduced. Tools for the analysis and synthesis of robust hybrid feedback control systems will be presented. The focus will be on asymptotic stability, invariance of sets, and robustness. The tools will be exercised in examples throughout the talk. Relevant applications in science and engineering will be highlighted.

Tuesday, February 28, 2017

Andras Vasy, Stanford University

Tuesday, March 7, 2017

Tomoyuki Arakawa, Research Institute for Mathematical Sciences Kyoto University

Title: Vertex Operator Algebras and Symplectic Varieties

Abstract: In physics, for each 4d N=2 superconformal field theory (SCFT) one associates a hyperkähler manifold called the Higgs branch. Remarkably, it has been recently turned out that the Higgs branch of 4d N=2 SCFTs can be (at least conjecturally) obtained from vertex operator algebras (VOAs) (which are purely algebraic objects) via the 4d-2d correspondence discovered by Beem, Lemos, Liendo, Peelaers, Rastelli and van Rees.

In my talk, I will start with explaining the Moore-Tachikawa conjecture on symplectic varieties, which is a mathematical conjecture that describes the Higgs branch of some interesting class of 4d N=2 SCFTs called “the theory of class S”. And then I will explain the above mentioned background. Finally I will introduce/prove the VOA analogue of the Moore-Tachikawa conjecture due to Rastelli et al, and reduce the original Moore-Tachikawa conjecture to a conjecture for VOAs.

Thursday, March 16, 2017

*SPECIAL Mathematics Colloquium*

Yitang Zhang, University of California, Santa Barbara

Title: Methods of Undetermined Quantities in Number Theory

Abstract: We present two typical examples to describe the application of methods of undetermined coefficients and functions to problems in analytic number theory. The first one is to reduce the problem of bounded gaps between primes to determining the coefficients of an arithmetic sum; the second one is to reduce the estimate of simple zeros of the Riemann zeta function, on assuming the Riemann Hypothesis, to determining a function that is called a mollifier of the Riemann zeta function.

Tuesday, March 21, 2017

NO COLLOQUIUM