Math 296: Graduate Student Colloquium
Winter 2019

Coordinator: Brendon Rhoades
Email: bprhoades (at) math.ucsd.edu
Office: 7250 APM
Colloquium Location: 6402 APM
Colloquium Time: Wednesdays, 1:00-1:50 pm

Description: This is a weekly colloquium meant to give graduate students an introduction to the research areas of faculty in the department.

Schedule

1/9/2019: Brendon Rhoades
Title: Spanning configurations
Abstract: A finite sequence (V_1, ... , V_n) of subspaces of a fixed vector space V is called a *spanning configuration* if V_1 + ... + V_n = V as vector spaces. We discuss some (sometimes conjectural) combinatorial and geometric properties of these objects.

1/16/2019: Steven Sam
Title: Diagram algebras and invariant theory
Abstract: Diagrams algebras serve many roles in combinatorial representaiton theory and invariant theory. I will discuss Brauer algebras in particular, and their categorical upgrade and how it connects with representation theory.

1/23/2019: No Seminar

1/30/2019: Wen-Xin Zhou
Title: Challenging the least squares: A new perspective on robustness
Abstract: Heavy-tailed distribution is a viable model for data contaminated by outliers that are typically encountered in applications. In this talk, we first revisit the classical Huber’s M-estimator, in which the robustification parameter balancing efficiency and robustness plays a critical role. It turns out that the validity of classical robust regression relies on the symmetry of the noise distribution and skewness gives rise to non-negligible bias. To better deal with heavy-tailed and/or asymmetric data, we reinvent robust regression, typified by Huber’s method, from a nonasymptotic viewpoint. Guided by the nonasymptotic deviation analysis, we propose a new data-driven tuning scheme to choose the robustification parameter for Huber-type sub-Gaussian estimators in three fundamental problems: mean estimation, covariance estimation and (sparse) linear regression. We illustrate its promising performance with extensive numerical experiments.

2/6/2019: Andrew Suk
Title: Geometrically separating the clique and chromatic number
Abstract: In this talk, I will sketch a construction of a family of segments in the plane, no three of which are pairwise crossing, such that no matter how you color the segments with k colors, there is a pair of segments with the same color that are crossing. This construction is due to Pawek et. al. (2012), which answered a question of Erdos from the 1970s.

2/13/2019: TBA
Title: TBA
Abstract: TBA

2/20/2019: Jon Novak
Title: Group Integrals 101
Abstract: TBA

2/27/2019: Ila Varma
Title: Some open problems in arithmetic statistics
Abstract: The most fundamental objects in number theory are number fields, field extensions of the rational numbers that are finite dimensional as vector spaces over \mathbb{Q}. Their arithmetic is governed heavily by certain invariants such as the discriminant, Artin conductors, and the class group; for example, the ring of integers inside a number field has unique prime factorization if and only if its class group is trivial. The behavior of these invariants is truly mysterious: it is not known how many number fields there are having a given discriminant or conductor, and it is an open conjecture dating back to Gauss as to how many quadratic fields have trivial class group. Nonetheless, one may hope for statistical information regarding these invariants of number fields, the most basic such question being “How are such invariants distributed amongst number fields of degree $d$ with fixed Galois group?” There are many foundational conjectures that predict the statistical behavior of these invariants in such families; however, only a handful of unconditional results are known. In this talk, I will describe a combination of algebraic, analytic, and geometric methods used to prove many new instances of these conjectures.

3/6/2019: Alex Cloninger
Title: Spectral Theory, Laplacians, Two Sample Statistics, and Data Science
Abstract: This talk introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The techniques and theory touch on spectral theory of Laplacians and heat kernels, optimization, and linear algebra. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

3/13/2019: TBA
Title: TBA
Abstract: TBA