Departmental Colloquium 2018-2019

Thursdays, 4:00 PM, JWB 335

Fall 2018

September 27:
Speaker: Raphaël Rouquier, UCLA
Title: Higher representation theory
Abstract: Representation theory is the study of vector spaces with prescribed symmetries. Replacing vector spaces by more complicated structures (of categorical, topological or geometrical origin) leads to new directions in algebra. This sheds a new light on geometric representation theory. More general moduli spaces constructions and enumerative invariants should fit in that framework. An additional motivation is the construction of invariants of low-dimensional manifolds from quantum field theories.

November 8:
Speaker: Mark Walker, University of Nebraska
Title:The Total and Toral Rank Conjectures
Abstract: Assume X is a nice space (a compact CW complex) that admits a fixed-point free action by a d-dimensional torus T. For example, X could be T acting on itself in the canonical way. The Toral Rank Conjecture, due to Halperin, predicts that the sum of the (topological) Betti numbers of X must be at least 2^d. Put more crudely, this conjecture predicts that it takes at least 2^d cells to build X by gluing. Now suppose M is a module over the polynomial ring k[x_1, \dots, x_d] that is finite dimensional as a k-vector space. The Total Rank Conjecture, due to Avramov, predicts that the sum of algebraic Betti numbers of M must be at least 2^d. Here, the algebraic Betti numbers refer to the ranks of the free modules occuring in the minimal free resolution of M. The Total Rank Conjecture is a weak form of the well-known Buchsbaum-Eisenbud-Horrocks Conjecture. In this talk I will discuss the relationship between these conjectures, some related conjectures, and recent progress toward settling them.

Special Colloquium: Tuesday November 27, 4-5pm, JWB 335:
Speaker: Eric Cator, Radboud University
Title: From WW II bombs to robust statistics.
Abstract: In this talk I will highlight some of the statistical problems that I have been working on over the years.
The first project is about unexploded bombs from World War II still in the ground in Amsterdam. There is a lot of historical information on bombardment the Allies carried out on German industry in Amsterdam, and the city would like to use this information to determine the risk of starting new building projects in these areas. We developed a new model for this, and were able to implement a Bayesian statistics approach through the Metropolis-Hastings algorithm. Quantative statements about the risk of specific areas can now be given, and this method is about to be implemented across the Netherlands.
The second project is on the Minimum Covariance Determinant (MCD) estimator that is frequently used in robust statistics. Consider a sample from some multidimensional distribution. In robust statistics, we try to estimate properties of the underlying distribution using only a fraction of the data, so that if part of the data is corrupted, this would not influence our estimators too much. The MCD tries to find that fixed fraction of the dat that has a covariance matrix with minimal determinant. It was known, for some special multivariate distributions, that this estimator has better convergence properties than the more intuitive Minimum Volume Estimator, but the asymptotic normality and even consistency for general distributions remained out of reach for more than 10 years. By extending the definition of MCD from a sample to a continuous distribution in a proper way, we were able to prove consistency and asymptotic normality, with explicit covariance structure, under mild conditions on the underlying distribution.

Special Colloquium: Monday December 3, 3-4pm, JWB 335: (Note special time)
Speaker: Nivedita Bhaskhar, UCLA
Title: On rational points, zero cycles and norm principles
Abstract: From the epitaphs of Diophantus to the margins of Fermat, the study of solutions to systems of polynomial equations has left its mark everywhere in history. In broad terms, given nice systems of polynomial equations over a field k which define varieties with tractable structure, one would like to ascertain whether the set of solutions over k is empty or not, i.e. whether the variety in question has a k-rational point.
A zero cycle on a k-variety X is any element of the free abelian group of closed points of X and its degree is the sum of its coefficients, weighted by the degrees of the residue fields. Any k-rational point of X is a zero cycle of degree one. In this talk, we discuss Serre’s injectivity question which asks whether the converse is true for torsors X under connected linear algebraic groups, i.e. whether such an X admitting a zero cycle of degree one in fact has a rational point. This naturally brings into the picture the so-called norm principles, which examine the behaviour of the images of group morphisms over field extensions from a linear algebraic group into a commutative one with respect to the norm map.

Special Colloquium: Thursday December 6, 4-5pm, JWB 335:
Speaker: Elina Robeva, MIT
Title: Orthogonal Tensor Decomposition
Abstract: Tensor decomposition has many applications. However, it is often a hard problem. In this talk I will discuss a family of tensors, called orthogonally decomposable tensors, which retain some of the properties of matrices that general tensors don't. A symmetric tensor is orthogonally decomposable if it can be written as a linear combination of tensor powers of n orthonormal vectors. Such tensors are interesting because their decomposition can be found efficiently. We study their spectral properties and give a formula for all of their eigenvectors. We also give equations defining all real symmetric orthogonally decomposable tensors. Analogously, we study nonsymmetric orthogonally decomposable tensors, describing their singular vector tuples and giving polynomial equations that define them. In an attempt to extend the definition to a larger set of tensors, we define tight-frame decomposable tensors and study their properties. Finally, I will conclude with some open questions and future research directions.

Special Colloquium: Tuesday December 11, 4-5pm, JWB 335:
Speaker: Natalie Sheils, University of Minnesota
Title: Interface Problems Using the Unified Transform Method
Abstract: Interface problems for partial differential equations are initial boundary value problems for which the solution of an equation in one domain prescribes boundary conditions for the equations in adjacent domains. These types of problems occur widely in applications including heat transfer, quantum mechanics, and mathematical biology. These problems, though linear, are often not solvable analytically using classical approaches. In this talk I present an introduction to the Unified Transform Method as well as an extension appropriate for solving interface problems.

Special Colloquium: Thursday December 13, 4-5pm, JWB 335:
Speaker: Priyam Patel, UC Santa Barbara
Title: Quantitative methods in hyperbolic geometry
Abstract: Peter Scott’s famous result states that the fundamental groups of hyperbolic surfaces are subgroup separable, which has many powerful consequences. For example, given any closed curve on such a surface, potentially with many self-intersections, there is always a finite cover to which the curve lifts to an embedding. It was shown recently that hyperbolic 3-manifold groups share this separability property, and this was a key tool in Ian Agol's resolution to the Virtual Haken and Virtual Fibering conjectures for hyperbolic 3-manifolds.
I will begin this talk by giving some background on separability properties of groups, hyperbolic manifolds, and these two conjectures. There are also a number of interesting quantitative questions that naturally arise in the context of these topics. These questions fit into a recent trend in low-dimensional topology aimed at providing concrete topological and geometric information about hyperbolic manifolds that often cannot be gathered from existence results alone. I will highlight a few of them before focusing on a quantitative question regarding the process of lifting curves on surfaces to embeddings in finite covers.

Special Colloquium: Thursday January 10, 4-5pm, JWB 335:
Speaker: Will Feldman, University of Chicago
Title: Interfaces in inhomogeneous media: pinning, hysteresis, and facets
Abstract: I will discuss some models for the shape of liquid droplets on rough solid surfaces. The framework of homogenization theory allows to study the large scale effects of small scale surface roughness, including interesting physical phenomena such as contact line pinning, hysteresis, and formation of facets.

Special Colloquium: Tuesday January 15, 4-5pm, JWB 335:
Speaker: Sean Howe, Stanford University
Title: Probabilistic structures in the topology and arithmetic of moduli spaces
Abstract: The average smooth surface in P^3 is a plane, the universal curve of genus g approaches a Poisson random variable, and the probability that a random hypersurface is smooth is given by a special value of a zeta function! In this talk, we explain how to make sense of these claims using the language of motivic random variables to adapt basic concepts from probability, like moment generating functions and independence, to the study of the topology and arithmetic of moduli spaces. This framework leads to new results on the topology of moduli spaces of hypersurface sections, a new perspective on representation stability for configuration spaces of algebraic varieties, and the computation of higher order terms in Katz-Sarnak style statistics for the zero-spacings of zeta functions; we will explain some of these applications, and speculate about future directions.

Special Colloquium: Thursday January 17, 4-5pm, JWB 335:
Speaker: Ian Tobasco, University of Michigan
Title: The cost of crushing: curvature-driven wrinkling of thin elastic shells
Abstract: How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on wrinkling patterns formed by thin floating shells, we develop a rigorous method (via Gamma-convergence) for evaluating the cost of crushing to leading order in the shell’s thickness and other small parameters. The observed patterns involve regions of well-defined wrinkling alongside totally disordered regions where no single direction of wrinkling is preferred. Our goal is to explain the appearance of such “wrinkling domains”. Our analysis proves that energetically optimal patterns maximize their projected planar area subject to a shortness constraint. This purely geometric variational problem turns out to be explicitly solvable in many cases of interest, and a straightforward scheme for predicting wrinkle patterns results. We demonstrate our methods with concrete examples and offer comparisons with simulation and experiment.
This talk will be mathematically self-contained, not assuming prior background in elasticity or calculus of variations.

Special Colloquium: Tuesday February 12, 4-5pm, JWB 335:
Speaker: Andreas Malmendier, Utah State University
Title: TBA
Abstract: TBA

Special Colloquium: Thursday February 14, 4-5pm, JWB 335:
Speaker: Tim Jacobbe, University of Florida
Title: TBA
Abstract: TBA

April 4:
Speaker: Philip Kutzko, University of Iowa