Thu, Apr 18 2024
  • 11:30 am
    Prof. Keaton Hamm - University of Texas at Arlington
    Tensor decompositions by mode subsampling

    Math 278B - Mathematics of Information, Data, and Signals

    APM 2402
     

    We will overview variants of CUR decompositions for tensors. These are low-rank tensor approximations in which the constituent tensors or factor matrices are subtensors of the original data tensors. We will discuss variants of Tucker decompositions and those based on t-products in this framework. Characterizations of exact decompositions are given, and approximation bounds are shown for data tensors contaminated with Gaussian noise via perturbation arguments.  Experiments are shown for image compression and time permitting we will discuss applications to robust PCA.

  • 2:00 pm
    Professor Ruth J. Williams - UCSD
    Stochastic Analysis of Chemical Reaction Networks with Applications to Epigenetic Cell Memory

    Math 218: Seminar on Mathematics for Complex Biological Systems

    AP&M 2402

    Epigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms. Simulation studies have shown how stochastic dynamics and time-scale differences between establishment and erasure processes in chromatin modifications (such as histone modifications and DNA methylation) can have a critical effect on epigenetic cell memory. 

    In this talk, we describe a mathematical framework to rigorously validate and extend beyond these computational findings. Viewing our stochastic model of a chromatin modification circuit as a singularly perturbed, finite state, continuous time Markov chain, we extend beyond existing theory in order to characterize the leading coefficients in the series expansions of stationary distributions and mean first passage times. In particular, we characterize the limiting stationary distribution in terms of a reduced Markov chain, provide an algorithm to determine the orders of the poles of mean first passage times, and describe a comparison theorem which can be used to explore how changing erasure rates affects system behavior. These theoretical tools not only allow us to set a rigorous mathematical basis for the computational findings of prior work, highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains especially those associated with chemical reaction networks.

    Based on joint work with Simone Bruno, Felipe Campos, Yi Fu and Domitilla Del Vecchio.

  • 3:00 pm
    David Jekel - Fields Institute for Research in Mathematical Sciences
    Infinite-dimensional, non-commutative probability spaces and their symmetries

    Postdoc Seminar

    APM 5829

    There is a deep analogy between, on the one hand, matrices and their trace, and on the other hand, random variables and their expectation.  This idea motivates "quantum" or non-commutative probability theory. Tracial von Neumann algebras are infinite-dimensional analogs of matrix algebras and the normalized trace, and there are several ways to construct von Neumann algebras that represent suitable "limits" of matrix algebras, either through inductive limits, random matrix models, or ultraproducts.  I will give an introduction to this topic and discuss the ultraproduct of matrix algebras and its automorphisms or symmetries. This study incorporates ideas from model theory as well as probability and optimal transport theory.

Fri, Apr 19 2024
  • 1:00 pm
    Hugo Jenkins - UCSD
    No Prerequisites Cayley-Bacharach

    Food for Thought

    AP&M 6402

    The Cayley-Bacharach theorem says that if two plane cubics intersect in exactly 9 points, then any third cubic passing through eight of these must pass through the ninth. We'll give a weird, elementary but cute proof which shows something a tiny bit stronger. The prerequisites will be not nil but nilpotent, limited to Bezout's theorem which I'll state carefully in the form I need. This proof came from Math 262A, which apparently got it from Terence Tao's blog.

Tue, Apr 23 2024
  • 11:00 am
    Jesse Peterson - Vanderbilt University
    Biexact von Neumann algebras

    Functional Analysis Seminar (Math 243)

    APM 7218

    The notion of biexactness for groups was introduced by Ozawa in 2004 and has since become a major tool used for studying solidity of von Neumann algebras. In joint work with Changying Ding, we extended this notion from the group theory setting to the setting of von Neumann algebras, thereby giving a unified setting for proving solidity type results. We will discuss biexactness and solidity and give examples of solid von Neumann algebras that are not biexact.  

  • 11:00 am
    Haoyu Zhang - UCSD
    An interacting particle consensus method for constrained global optimization

    Math 278A - Center for Computational Mathematics Seminar

    APM 2402 and Zoom ID 982 8500 1195

    This talk presents a particle-based optimization method designed for addressing minimization problems with equality constraints, particularly in cases where the loss function exhibits non-differentiability or non-convexity. A rigorous mean-field limit of the particle system is derived, and the convergence of the mean-field limit to the constrained minimizer is established.

Tue, May 7 2024
  • 4:00 pm
    Víctor Rivero - Center of Research in Mathematics, Guanajuato, Mexico
    An excursion from self-similar Markov processes to Markov additive processes

    2024 Ronald K. Getoor Distinguished Lecture

    AP&M 6402

    In stochastic modeling we often need to deal with one of two apparently unrelated objects. One is self-similar processes and the other is additive functionals. Self-similar Markov processes are the class of Markovian models that arise as scaling limits of stochastic processes, that are obtained after renormalization of time and space. Additive functionals arise commonly when one considers, for instance, rewards associated to a Markovian model. 

    On the one hand, the so-called Lamperti transform ensures that any $R^d$-valued self-similar Markov process admits a polar decomposition, and the argument and the radius of the process are related to a Markov additive process via an explicit time change. On the other hand, any additive functional A of a Markov process X is such that the pair (A, X) is a Markov additive process. A Markov additive process (MAP) is a stochastic process with two components: one that is additive, and real valued, the ordinator, and a general one, the modulator, that rules the behavior of the ordinator. The ordinator has independent and stationary increments, given the modulator. This general structure emulates the structure of processes with independent and stationary increments, Levy processes, as for instance Brownian motion, Cauchy and stable processes, Gamma processes, etc. 

    In general, it is too ambitious to try to determine explicitly the whole law of a self-similar Markov process or of an additive functional. But we can aim at understanding properties of the extremes of these processes and to be ready for the best and worst scenarios. In the fluctuation theory of Markov additive processes we aim at developing tools for studying the extremes of the additive part, ordinator, of the process. This has been done in a systematic way during the last four decades under the assumption that the modulator is a constant process, and hence the ordinator is a real valued Levy process. Also, in the 1980-90 period, some foundations were laid to develop a fluctuation theory for MAPs in a general setting.   

    In this talk we aim at giving a brief overview of the fluctuation theory of Markov additive processes, to describe some recent results and to provide some applications to the theory of self-similar Markov processes. These applications are mainly related to stable processes, a class of processes that arises often in mathematical physics, potential and harmonic analysis, and in other areas of mathematics. We aim at making this overview accessible to graduate and advanced undergraduate students, with some knowledge of Markov chains and Levy processes, and to point out at some open research questions.

Tue, May 14 2024
  • 11:00 am
    Aldo Garciaguinto - Michigan State University
    Schreier's Formula for some Free Probability Invariants

    Math 243, Functional Analysis

    APM 7218 and Zoom (meeting ID:  94246284235)

    Let $G\stackrel{\alpha}{\curvearrowright}(M,\tau)$ be a trace-preserving action of a finite group $G$ on a tracial von Neumann algebra. Suppose that $A \subset M$ is a finitely generated unital $*$-subalgebra which is globally invariant under $\alpha$. We give a formula relating the von Neumann dimension of the space of derivations on $A$ valued on its coarse bimodule to the von Neumann dimension of the space of derivations on $A \rtimes^{\text{alg}}_\alpha G$ valued on its coarse bimodule, which is reminiscent of Schreier's formula for finite index subgroups of free groups. This formula induces a formula for $\dim \text{Der}_c(A,\tau)$ (defined by Shlyakhtenko) and under the assumption that $G$ is abelian we obtain the formula for $\Delta$ (defined by Connes and Shlyakhtenko). These quantities and the free entropy dimension quantities agree on a large class of examples, and so by combining these results with known inequalities, one can expand the family of examples for which the quantities agree.

Thu, May 16 2024
  • 4:00 pm
    Paul K. Newton - University of Southern California
    Control of evolutionary mean field games and tumor cell population models

    UCSD Mathematics Colloquium/MathBio Seminar

     Mean field games are played by populations of competing agents who derive their update rules by comparing their own state variable with that of the mean field. After a brief introduction to several areas where they have been used recently, we will focus on models of competing tumor cell populations based on the replicator dynamics mean field evolutionary game with prisoner’s dilemma payoff matrix. We use optimal and adaptive control theory on both deterministic and stochastic versions of these models to design multi-drug chemotherapy schedules that suppress the competitive release of resistant cell populations (to avoid chemo-resistance) by maximizing the Shannon diversity of the competing subpopulations. The models can be extended to networks where spatial connectivity can influence optimal chemotherapy scheduling. 

Thu, May 23 2024
  • 10:00 am
    Josh Bowman
    TBA

    Math 211B - Group Actions Seminar

    APM 7321 and Zoom ID 967 4109 3409
    (password: dynamics)

Thu, May 30 2024
  • 10:00 am
    Carlos Ospina
    TBA

    Math 211B - Group Actions Seminar

    APM 7321 and Zoom ID 967 4109 3409
    (password: dynamics)