# Mathematics Colloquium Winter 2017

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

**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

**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 Universit****y**

**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.

**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**