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10:00 am
Professor Nicolas Monod - École Polytechnique Fédérale de Lausanne
The Furstenberg boundary of Gelfand pairs
Math 211B - Group Actions Seminar
APM 7321
AbstractMany classical locally compact groups $G$ admit a very large compact subgroup $K$, where "very large" has been formalized by Gelfand in 1950. Examples include $G=\mathrm{SL}_n(\mathbb{R})$ with $K=\mathrm{SO}(n)$, or $G=\mathrm{SL}_n(\mathbb{Q}_p)$ with $K=\mathrm{SL}_n(\mathbb{Z}_p)$. More generally, all semi-simple algebraic groups and some tree automorphism groups.
In these explicit examples, there is also an "Iwasawa decomposition" which formalizes the fact that $G$ has a homogeneous Frustenberg boundary, even homogeneous under $K$. This is a very strong restriction for general groups.
Using no structure theory whatsoever, we prove that this homogeneity (and Iwasawa decomposition) holds for all Gelfand Pairs. This implies, in some geometric cases, a classification of Gelfand pairs. (This is related to a small part of my 2021 zoom colloquium at UCSD).
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11:00 am
Prof. Elliot Paquette - McGill University
Kahane’s coverage question and the image of Gaussian analytic function
Math 288 - Probability & Statistics
APM 6402
AbstractWe consider the range of Gaussian analytic functions (GAF) with finite radius of convergence. We show that any unbounded GAF has dense image in the plane. We moreover show that if in addition the coefficients have sufficiently regular variances, then the image is the whole complex plane. We do this by exploiting an approximate connection between the coverage problem and spatial branching processes, analogous to the branching structure that appears in the log-correlated GAF and circular beta ensembles. This answers a long-standing open question of J.-P. Kahane, with sufficient regularity.
Joint work with Alon Nishry.
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2:00 pm
Baiming Qiao - UCSD (baqiao@ucsd.edu)
Nonabelian Chabauty method
Final Defense
AbstractApplication of non-abelian Chabauty method to diophantine equations
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2:00 pm
Professor Joachim Dzubiella - University of Freiburg, Germany
Modeling responsive microgel particles: from soft colloids to artificial cells
Joint Mathematical Biology and Mathematical Modeling/Applied Analysis Seminar
APM 7321
AbstractMicrometer-sized particles made from responsive polymer networks (that is, responsive microgel colloids) are of high potential for the design of functional soft materials due to their adaptive compressibility and stimuli-triggered volume transition. In this talk, I will discuss models and theoretical approaches, such as Langevin simulations and classical (dynamic) density functional theory (DFT), to describe the structural and dynamical behavior of dispersions of these responsive colloids in and out of equilibrium. Moreover, I will argue that chemical fueling and the inclusion of chemomechanical feedback loops may lead to excitable and oscillatory dynamics of the active colloids, establishing first steps to a well-controlled design of artificial cells and their emergent behavior.
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3:00 pm
Dr. Lihan Wang - California State University Long Beach
How rare are simple Steklov eigenvalues
Math 248: Real Analysis
APM 6218
AbstractSteklov eigenvalues are eigenvalues of the Dirichlet-to-Neumann operator which are introduced by Steklov in 1902 motivated by physics. And there is a deep connection between the extremal Steklov eigenvalue problems and the free boundary minimal surface theory in the unit Euclidean ball as revealed by Fraser and Schoen in 2016. In the talk, we will discuss the question of how rare simple Steklov eigenvalues are on manifolds and its applications in nodal sets and critical points of eigenfunctions.
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3:00 pm
Amit Ophir - UCSD
Pseudo-representations
Postdoc Seminar
APM 7218
AbstractBy a pseudo-representation, I mean an umbrella term for several abstractions/generalizations of finite dimensional representations. In my talk, I will discuss two types of pseudo-representations, explain their relevance to number theory, and highlight the relationship between the two. I will conclude by mentioning a few open questions.
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12:00 pm
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2:00 pm
Elise Alvarez-Salazar - UCSD
The De Rham Theorem, Stokes’ Theorem, and Hodge Theory
Food for Thought
APM 7321
AbstractDifferent constructions of cohomology allow us to measure different properties of manifolds. To construct either the singular cohomology or the de Rham cohomology groups, the representatives of each are distinct things: cochains and closed differential forms. Yet the de Rham theorem states there is a correspondence between them. And we can see this directly via examples displaying how integration on a manifold is greatly influenced by the manifold’s topology. In this talk, I will cover highlights of the proof of the De Rham theorem and show how Stokes’ theorem lends itself to defining an explicit homomorphism between the two cohomology groups: de Rham and singular. I will also discuss a further direction that this correspondence allows which leads us into Hodge theory.
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4:00 pm
Shubham Saha - UCSD
The Chow ring of the universal moduli space of (semi)stable bundles over smooth curves.
Math 208: Seminar in Algebraic Geometry
APM 7321
AbstractWe will discuss some ongoing work on the subject, specifically in the rank $2$, genus $2$ case. The talk will start with a quick review of existing literature on $M_2$ and some of its étale covers, along with results and constructions involving moduli of rank $2$ bundles. We will go over their generalizations to the universal setting and outline the current plan on using these tools for computing the Chow ring.
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9:00 am
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4:00 pm
Dr. Alfonso Castro - Harvey Mudd College (castro@g.hmc.edu)
Critical point theory and the existence of seven solutions for a semilinear elliptic boundary value problem
Math 248: Real Analysis Seminar
APM 7321
AbstractAiming to understand the solvability of semilinear elliptic boundary values problems in bounded domains, we will review the best known techniques for establishing the existence of critical points of functionals whose critical points are solutions to such problems. The mountain pass lemma, the Nehari manifold,
the Morse index, and bifurcation analysis will be discussed to conclude the existence of seven solutions for an asymptotically linear elliptic problem.
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3:00 pm
Dr. Harold Jimenez Polo - UC Irvine
A Goldbach Theorem for Polynomial Semirings
Math 211A: Seminar in Algebra
APM 7321
AbstractWe discuss an analogue of the Goldbach conjecture for polynomials with coefficients in semidomains (i.e., subsemirings of an integral domain).