Week of February 11, 2018

Mon Feb 12, 2018
4:00pm to 5:00pm - RH306 - Applied and Computational Mathematics
Xuehai Huang - (Wenzhou University)
Decoupling of Mixed Methods Based on General Helmholtz Decompositions

A framework to systematically decouple high order elliptic equations into combination of Poisson-type and Stokes-type equations is developed using the tools of differential complexes and Helmholtz decompositions. The key step is to systematically construct the underling commutative diagrams involving the complexes and Helmholtz decompositions in a general way.

Discretizing the decoupled formulation leads to a natural superconvergence between the Galerkin projection and the decoupled approximation. Examples include but not limit to: the primal formulations and mixed formulation of biharmonic equation, fourth order curl equation, and triharmonic equation etc. As a by-product, Helmholtz decompositions for many dual spaces are obtained.

4:00pm to 5:30pm - RH 440R - Logic Set Theory
Zach Norwood - (UCLA)
Coding along trees and remarkable cardinals

A major project in set theory aims to explore the connection between large cardinals and so-called generic absoluteness principles, which assert that forcing notions from a certain class cannot change the truth value of (projective, for instance) statements about the real numbers. For example, in the 80s Kunen showed that absoluteness to ccc forcing extensions is equiconsistent with a weakly compact cardinal. More recently, Schindler showed that absoluteness to proper forcing extensions is equiconsistent with a remarkable cardinal. (Remarkable cardinals will be defined in the talk.) Schindler's proof does not resemble Kunen's, however, using almost-disjoint coding instead of Kunen's innovative method of coding along branchless trees. We show how to reconcile these two proofs, giving a new proof of Schindler's theorem that generalizes Kunen's methods and suggests further investigation of non-thin trees.

4:00pm to 5:00pm - RH 340P - Geometry and Topology
Qiongling Li - (Caltech)
Hodge metric of nilpotent Higgs bundles

On a complex manifold, a Higgs bundle is a pair containing a holomorphic vector bundle E and a holomorphic End(E)-valued 1-form. In this talk, we focus on nilpotent Higgs bundles, for example, the ones arising from variations of Hodge structures for a deformation family of Kaehler manifolds. We first give an optimal upper bound of the curvature of Hodge metric of the deformation space of Calabi-Yau manifolds. Secondly, we prove a rigidity theorem of the holonomy of polystable nilpotent Higgs bundles via the non-abelian Hodge theory when the base manifold is a Riemann surface. This is joint work with Song Dai.

Tue Feb 13, 2018
3:00pm to 4:00pm - RH 306 - Nonlinear PDEs
Jeremy LeCrone - (University of Richmond)
Mean Value Theorems for Riemannian Manifolds via the Obstacle Problem

In this talk, I will discuss recent results produced with co-authors Ivan Blank (KSU) and Brian Benson (UCR) regarding a formulation of the Mean Value Theorem for the Laplace-Beltrami operator on smooth Riemannian manifolds. We define the sets upon which mean values of (sub)-harmonic functions are computed via a particular obstacle problem in geodesic balls. I will thus begin by discussing the classical obstacle problem and then an intrinsic formulation on manifolds developed in our recent paper. After demonstrating how the theory of obstacle problems is leveraged to produce our Mean Value Theorem, I will discuss local and global theory for our family of mean value sets and potential connections between the properties of these sets and the geometry of the underlying manifold.

4:00pm to 5:00pm - RH 306 - Differential Geometry
Dan Knopf - (UT Austin)
Non-Kahler Ricci flow singularities that converge to Kahler-Ricci solitons

We describe Riemannian (non-Kahler) Ricci flow solutions that develop finite-time Type-I singularities whose parabolic dilations converge to a shrinking Kahler–Ricci soliton singularity model. More specifically, the singularity model for these solutions is the “blowdown soliton” discovered in 2003 by Feldman, Ilmanen, and the speaker. Our results support the conjecture that the blowdown soliton is stable under Ricci flow. This work also provides the first set of rigorous examples of non-Kahler solutions of Ricci flow that become asymptotically Kahler, in suitable space-time neighborhoods of developing singularities, at rates that break scaling invariance. These results support the conjectured stability of the subspace of Kahler metrics under Ricci flow.

Thu Feb 15, 2018
2:00pm - RH 340P - Mathematical Physics
Christoph Marx - (Oberlin)
Dependence of the density of states on the probability distribution for discrete random Schrödinger operators

We prove the Hölder-continuity of the density of states measure (DOSm) and the integrated density of states (IDS) for discrete random Schrödinger operators with finite-range potentials with respect to the probability measure. In particular, our result implies that the DOSm and the IDS for smooth approximations of the Bernoulli distribution converge to the corresponding quantities for the Bernoulli-Anderson model. Other applications of the technique are given to the dependency of the DOSm and IDS on the disorder, and the continuity of the Lyapunov exponent in the weak-disorder regime for dimension one. The talk is based on joint work with Peter Hislop (Univ. of Kentucky) 

3:00pm to 4:00pm - RH 306 - Number Theory
Sean Howe - (Stanford University)
Sideways Katz-Sarnak and motivic random variables

A fundamental observation in Katz-Sarnak's study of the zero spacing of L-functions is that Frobenius conjugacy classes in suitable families of varieties over finite fields approximate infinite random matrix statistics. For example, the normalized Frobenius conjugacy classes of smooth plane curves of degree d over F_q approach the Gaussian symplectic ensemble as we take first q to infinity, then d to infinity. In this talk, we explain a sideways version of this result where the limits in d and q are exchanged, and give a Hodge theoretic analog in characteristic zero. 

4:00pm to 6:00pm - NS II 1201 - Colloquium
Barry Simon - (Caltech)
More tales of our fathers

This is not a mathematics talk but it is a talk for mathematicians. Too often, we
think of historical mathematicians as only names assigned to theorems. With
vignettes and anecdotes, I'll convince you they were also human beings and that, as
the Chinese say, "May you live in interesting times" really is a curse. More tales
following up on the talk I gave at Irvine in May, 2014. It is not assumed listeners
heard that earlier talk.

Fri Feb 16, 2018
11:00am to 12:00pm - 340P - Applied and Computational Mathematics
Zhaosong Lu - (Simon Fraser University)
Algorithmic Development for Computing B-stationary Points of a Class of Nonsmooth DC Programs

In the first part of this talk, we study a convex-constrained nonsmooth DC program
in which the concave summand of the objective is an infimum of possibly infinitely many smooth
concave functions. We propose some algorithms by using nonmonotone linear search and extrapolation
techniques for possible acceleration for this problem, and analyze their global convergence, sequence
convergence and also iteration complexity. We also propose randomized counterparts for them
and discuss their convergence.

In the second part we consider a class of DC constrained nonsmooth DC programs. We propose penalty and
augmented Lagrangian methods for solving them and show that they converge to a B-stationary
point under much weaker assumptions than those imposed in the literature.

This is joint work with Zhe Sun and Zirui Zhou.

4:00pm - MSTB 120 - Graduate Seminar
Song-Ying Li - (UC Irvine)
Characterizations of the unit ball in Euclidean spaces.

In this talk, I will give you many ways to characterize  the unit ball

in $R^n$ or in $C^n$. It involves, differential equations, first eigenvalue of Laplace-Beltrami 

operator, etc.