Sat, Apr 13 2024
  • 11:00 am
    Southern California Geometric Analysis Seminar - April 13-14, 2024

    Southern California Geometric Analysis Seminar

    Natural Science Building Auditorium (04/13)
    and Center Hall 105 (04/14)

    The 29th SCGAS will be held at the Department of Mathematics of University of California at San Diego on Saturday, April 13, 2024 and Sunday, April 14, 2024. The lectures will be held in Natural Science Building Auditorium (04/13) and Center Hall 105 (04/14) due to the campus event of Triton Day. For directions on how to get to Natural Science Building see map; For the Center Hall, here is a map.

    And here are directions from the BW-Del Mar to the UCSD campus.

    Registration starts at 10am Saturday morning. The first talk will be at 11:00am and the last talk will finish at 12:30pm on Sunday, to allow for travel.

    Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to join our annual seminar. Partial financial support is available.

    The Seminar is supported by the NSF and by the School of Physical Sciences at UC San Diego.

    Invited Speakers: Guido De Philippis (CIMS), Bruce Kleiner (CIMS), Yi Lai (Stanford), Bill Minicozzi (MIT), Song Sun (Berkeley/Zhejiang Univ.), Guofang Wei (UCSB), Xin Zhou (Cornell)

    Registration: Participants are asked to register online: the electronic registration form is now available. 

Tue, Apr 16 2024
  • 11:00 am
    Dr. Ian Charlesworth & Dr. David Jekel - Cardiff University/Fields Institute for Research in Mathematical Sciences
    Algebraic soficity and graph products

    Math 243, Functional Analysis

    APM 7218 and Zoom (meeting ID:  94246284235)

     We show that a graph product of tracial von Neumann algebras is strongly $1$-bounded if the first $\ell^2$-Betti number vanishes for an associated dense $*$-subalgebra.  Graph products of tracial von Neumann algebras were studied by Caspers and Fima, and generalize Green's graph product of groups.  Given groups $G_v$ for each vertex of a graph $\Gamma$, the graph product is the free product modulo the relations that $G_v$ and $G_w$ commute when $v \sim w$; for von Neumann algebras, graph products are described by a certain moment relation.  In our paper, the crux of the argument is a generalization to tracial von Neumann algebras of the statement that soficity of groups is preserved by graph products.  We replace soficity for groups with a more general notion of algebraic soficity for a $*$-algebra $A$, which means the existence of certain approximations for the generators of $A$ by matrices with algebraic integer entries and approximately constant diagonal.  We show algebraic soficity is preserved under graph products through a random permutation construction, inspired by previous work of Charlesworth and Collins as well as Au-C{\'e}bron-Dahlqvist-Gabriel-Male.  In particular, this gives a new probabilistic proof of Ciobanu-Holt-Rees's result that soficity of groups is preserved by graph products.

    This is based on joint work with Rolando de Santiago, Ben Hayes, Srivatsav Kunnawalkam Elayavalli, Brent Nelson.

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, 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)