Tue, Feb 11 2025
  • 2:00 pm
    Sara Billey - University of Washington
    Enumerating Quilts of Alternating Sign Matrices and Generalized Rank Functions

    Math 269: Combinatorics Seminar

    APM 7321

    We present new objects called quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a quilt with respect to two Boolean lattices. When the two posets are chains, a quilt is equivalent to an alternating sign matrix and its corresponding corner sum matrix. Such rank functions are used in the definition of Schubert varieties in both the Grassmannian and the complete flag manifold. Quilts also generalize the monotone Boolean functions counted by the Dedekind numbers, which is known to be a #P-complete problem. Quilts form a distributive lattice with many beautiful properties and contain many classical and well known sublattices, such as the lattice of matroids of a given rank and ground set. While enumerating quilts is hard in general, we prove two major enumerative results, when one of the posets is an antichain and when one of them is a chain. We also give some bounds for the number of quilts when one poset is the Boolean lattice. Several open problems will be given for future development. This talk is based on joint work with Matjaz Konvalinka in arxiv:2412.03236.

Wed, Feb 12 2025
  • 4:00 pm
    Santiago Arango-Piñeros - Emory University
    Counting 5-isogenies of elliptic curves over the rationals

    Math 209: Number Theory Seminar

    APM 7321 and online (see https://www.math.ucsd.edu/~nts/)

    In collaboration with Han, Padurariu, and Park, we show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$ action of weights $(4, 4, 2)$.

    Pretalk: I will explain how to count isomorphism classes of elliptic curves over the rationals. On the way, I will introduce some basic stacky notions: torsors, quotient stacks, weighted projective stacks, and canonical rings.

    [pre-talk at 3:00PM]

Thu, Feb 13 2025
  • 10:00 am
    Siyuan Tang - Beijing International Center for Mathematical Research
    Effective density of surfaces near Teichmüller curves

    Math 211B: Group Actions Seminar

    Zoom (Link)

    The study of orbit dynamics for the upper triangular subgroup $P \subset \mathrm{SL}(2, \mathbb R)$ holds fundamental significance in homogeneous and Teichmüller dynamics. In this talk, we shall discuss the quantitative density properties of $P$-orbits for translation surfaces near Teichmüller curves. In particular, we discuss the Teichmüller space $H(2)$ of genus two Riemann surfaces with a single zero of order two, and its corresponding absolute period coordinates, and examine the asymptotic dynamics of $P$-orbits in these spaces.

  • 11:00 am
    Kunal Chawla - Princeton University
    The Poisson boundary of hyperbolic groups without moment conditions

    Math 288: Probability & Statistics

    APM 6402

    Given a random walk on a countable group, the Poisson boundary is a measure space which captures all asymptotic events of the markov chain. The Poisson boundary can sometimes be identified with a concrete geometric "boundary at infinity", but almost all previous results relied strongly on moment conditions of the random walk. I will discuss a technique which allows us to identify the Poisson boundary on any group with hyperbolic properties without moment conditions, new even in the free group case, making progress on a question of Kaimanovich and Vershik. This is joint work with Behrang Forghani, Joshua Frisch, and Giulio Tiozzo.

  • 2:00 pm
    Harish Kannan - UCSD
    Emergent features and pattern formation in dense microbial colonies

    Math 218: Seminar in Mathematics of Complex Biological Systems

    APM 7321

    Growth of bacterial colonies on solid surfaces is commonplace; yet, what occurs inside a growing colony is complex even in the simplest cases. Robust colony expansion kinetics featuring linear radial growth and saturating vertical growth in diverse bacteria indicates a common developmental program, which will be elucidated in this talk using a combination of findings based on modeling and experiments.  Agent-based simulations reveal the crucial role of emergent mechanical constraints and spatiotemporal dynamics of nutrient gradients which govern observed expansion kinetics. The consequences of such emergent features will also be examined in the context of pattern formation in multi-species bacterial communities. Future directions and opportunities in theoretical modeling of such pattern formation systems will be discussed.

  • 3:00 pm
    Dr. Haoren Xiong - UCLA
    Semiclassical asymptotics for Bergman projections

    Math 248: Real Analysis Seminar

    APM 6218

    In this talk, we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We will first review various approaches to the construction of asymptotic Bergman projections, for smooth weights and for real analytic weights. We shall then explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, we show that Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit. We will also introduce some microlocal analysis tools in the Gevrey setting, including Borel's lemma for symbols and complex stationary phase lemma. This talk is based on joint work with Hang Xu.

  • 4:00 pm
    Martin Dindos - University of Edinburgh
    The $L^p$ regularity problem for parabolic operators

    Mathematics Colloquium

    APM 6402

     

    In this talk, I will present a full resolution of the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + {\rm div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a very natural Carleson condition (a parabolic analog of the so-called DKP-condition).

    We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known previously even in the  "small Carleson case", that is, when the Carleson norm of coefficients is sufficiently small.

    In the elliptic case the analogous question was only fully resolved recently (2022) independently by two groups using two very different methods; one involving S. Hofmann, J. Pipher and the presenter, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges. The result is a joint work with L. Li and J. Pipher.

Fri, Feb 14 2025
  • 2:00 pm
    Arseniy Kryazhev - UCSD
    Chess

    Food for Thought

    APM 7321

    In this talk, various aspects of the game of chess will be explored.  Like true philosophers, we will make random observations, pose rhetorical questions and draw strange parallels, all without claim to the truth, while touching on various topics from the nature of randomness to ways to maximize cognitive performance. Familiarity with the game is not required but will be helpful.
     

Fri, Feb 21 2025
  • 4:00 pm
    Dr. Joe Foster - University of Oregon
    The Lefschetz standard conjectures for Kummer-type hyper-Kähler varieties

    Math 208: Algebraic Geometry

    APM 7321

    For a smooth complex projective variety, the Lefschetz standard conjectures of Grothendieck predict the existence of algebraic self-correspondences that provide inverses to the hard Lefschetz isomorphisms. These conjectures have broad implications for Hodge theory and the theory of motives. In this talk, we describe recent progress on the Lefschetz standard conjectures for hyper-Kähler varieties of generalized Kummer deformation type. 

Tue, Mar 4 2025
  • 11:00 am
    Minxin Zhang - UCLA
    Inexact Proximal Point Algorithms for Zeroth-Order Global Optimization

    Center for Computational Mathematics Seminar

    AP&M 2402 and Zoom ID 946 7260 9849

    This work concerns the zeroth-order global minimization of continuous nonconvex functions with a unique global minimizer and possibly multiple local minimizers. We formulate a theoretical framework for inexact proximal point (IPP) methods for global optimization, establishing convergence guarantees under mild assumptions when either deterministic or stochastic estimates of proximal operators are used. The quadratic regularization in the proximal operator and the scaling effect of a positive parameter create a concentrated landscape of an associated Gibbs measure that is practically effective for sampling. The convergence of the expectation under the Gibbs measure is established, providing a theoretical foundation for evaluating proximal operators inexactly using sampling-based methods such as Monte Carlo (MC) integration. In addition, we propose a new approach based on tensor train (TT) approximation. This approach employs a randomized TT cross algorithm to efficiently construct a low-rank TT approximation of a discretized function using a small number of function evaluations, and we provide an error analysis for the TT-based estimation. We then propose two practical IPP algorithms, TT-IPP and MC-IPP. The TT-IPP algorithm leverages TT estimates of the proximal operators, while the MC-IPP algorithm employs MC integration to estimate the proximal operators. Both algorithms are designed to adaptively balance efficiency and accuracy in inexact evaluations of proximal operators. The effectiveness of the two algorithms is demonstrated through experiments on diverse benchmark functions and various applications.

Wed, Mar 12 2025
Thu, Mar 13 2025
  • 4:00 pm
    Edgar Knobloch
    TBA

    Mathematics Colloquium

    APM 6402

    TBA

Fri, Mar 14 2025
Thu, Mar 27 2025
  • 2:00 pm
    Mark Bowick
    TBA

    Math 218: Seminars on Mathematics for Complex Biological Systems

    APM 7321

    TBA

Thu, Apr 10 2025
  • 4:00 pm
    Professor Soeren Bartels - University of Freiburg, Germany
    Babuska's Paradox in Linear and Nonlinear Bending Theories

    Mathematics Colloquium

    APM 6402

    The plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem. The results are relevant for the construction of curved folding devices.