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2:00 pm
Sara Billey - University of Washington
Enumerating Quilts of Alternating Sign Matrices and Generalized Rank Functions
Math 269: Combinatorics Seminar
APM 7321
AbstractWe 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.
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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
/) AbstractIn 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]
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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)
AbstractThe 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.
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11:00 am
Kunal Chawla - Princeton University
The Poisson boundary of hyperbolic groups without moment conditions
Math 288: Probability & Statistics
APM 6402
AbstractGiven 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.
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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
AbstractGrowth 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.
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3:00 pm
Dr. Haoren Xiong - UCLA
Semiclassical asymptotics for Bergman projections
Math 248: Real Analysis Seminar
APM 6218
AbstractIn 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.
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4:00 pm
Martin Dindos - University of Edinburgh
The $L^p$ regularity problem for parabolic operators
Mathematics Colloquium
APM 6402
AbstractIn 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.
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2:00 pm
Arseniy Kryazhev - UCSD
Chess
Food for Thought
APM 7321
AbstractIn 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.
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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
AbstractFor 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.
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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
AbstractThis 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.
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9:00 am
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4:00 pm
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10:00 am
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2:00 pm
Mark Bowick
TBA
Math 218: Seminars on Mathematics for Complex Biological Systems
APM 7321
AbstractTBA
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4:00 pm
Professor Soeren Bartels - University of Freiburg, Germany
Babuska's Paradox in Linear and Nonlinear Bending Theories
Mathematics Colloquium
APM 6402
AbstractThe 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.