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