Fri, Apr 18 2025
  • 11:00 am
    Raphael Meyer - Caltech
    Optimal Trace Estimation, and the Strangeness of the Kronecker Trace Estimation

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

    APM 6402

    A fundamental task in linear algebra is that of trace estimation: Suppose we have a PSD matrix A that can be accessed only by matrix-vector products. Then, with as few matrix-vector products as possible, estimate the trace of A to relative error with high probability. This is an essential subroutine in all sorts of applications, for instance in efficiently estimating the log-determinant of a matrix.

    In the first part of the talk, I'll rigorously introduce this problem, the prior state-of-the-art algorithm (the Girard-Hutchinson Estimator), and our improvement upon it (the Hutch++ Estimator), which we show to have asymptotically optimal matrix-vector complexity. In the second part of the talk, I'll introduce a Kronecker-structured variant of this problem with applications for tensorized data, alongside the only known algorithm that solves this problem.

    However, we'll see that this algorithm converges very slowly. We will show this is a result of this Kronecker-structured computational model, which elicits strange computational properties. We will see that good design decisions in the non-Kronecker case can cause catastrophic failure in the Kronecker case, that using complex random variables leads to exponential speedups over reals, and that subgaussianity does not suffice to understand the performance of randomized algorithms here.

    Joint work with Haim Avron, David Woodruff, and William Swartworth.

  • 2:00 pm
    Professor Feng Xu
    Rigorous results about entropies in QFT

    Math 243: Functional Analysis Seminar

    APM 7218

    I will discuss some recent results about relative entropies in QFT, with particular emphasis on the singular limits of such entropies.

  • 4:00 pm
    Dr. Weihong Xu - California Institute of Technology
    Quantum K-theory of IG(2,2n)

    Math 208: Seminar in Algebraic Geometry

    APM 7321

    We give an explicit geometric computation of the quantum K rings of symplectic Grassmannians of lines, which are deformations of their Grothendieck rings of vector bundles and refinements of their quantum cohomology rings. We prove that their Schubert structure constants have signs that alternate with codimension (just like in the Grothendieck ring) and vanish for degrees at least 3. We also give closed formulas that characterize the multiplicative structure of these rings. This is based on joint work with V. Benedetti and N. Perrin.

Mon, Apr 21 2025
  • 1:00 pm
    Shubhankar Sahai - UCSD
    Some arithmetic problems related to p-adic K-theory

    Advancement to Candidacy

    APM 7218

    We survey some recent observations and ongoing work motivated by a hope to better understand p-adic K-theory. More specifically, we discuss arithmetic problems—and potential approaches—related to syntomic cohomology in positive and mixed characteristics. At the level of the structure sheaf, syntomic cohomology is an 'intelligent version' of p-adic étale Tate twists at the characteristic and (among other things) provides a motivic filtration on p-adic étale K-theory via the theory of trace invariants.

  • 3:00 pm
    Dr. Sankhaneel Bisui - Arizona State University
    Algebraic Properties of Invariant Ideals

    Math 211A: Algebra Seminar

    APM 7321

    Let R be a polynomial ring with m x n many indeterminate over the complex numbers. We can think of the indeterminates as a matrix X of size m x n.  

    Consider the group G = Gl(m) x Gl(n). Then G acts on R via the group action (A,B)X =AXB^{-1}. In 1980, DeConcini, Eisenbud, and Procesi introduced the ideals that are invariant under this group action.

    In the same paper, they described various properties of those ideals, e.g., associated primes, primary decomposition, and integral closures.  In recent work with Sudipta Das, Tài Huy Hà, and Jonathan Montaño, we described their rational powers and proved that they satisfy the binomial summation formula. In an ongoing work, Alexandra Seceleanu and I are formulating symbolic properties of these ideals. In this talk, I will describe these ideals and the properties we are interested in. I will also showcase some results from my collaborations.

Tue, Apr 22 2025
  • 11:00 am
    Dr. Changying Ding - UCLA
    Relative solidity in measure equivalence and applications

    Math 243: Functional Analysis Seminar

    APM 7218

    In his seminal paper, Ozawa demonstrated the solidity property for ${\rm II}_1$ factor arising from Biexact groups. In this talk, I will discuss a relative version of the solidity property for biexact groups in the setting of measure equivalence and its applications to measure equivalence rigidity. This is a joint work with Daniel Drimbe.

  • 11:00 am
    Ray Zirui Zhang - UC Irvine
    BiLO: Bilevel Local Operator Learning for PDE inverse problems with uncertainty quantification

    Center for Computational Mathematics Seminar & MINDS Seminar

    APM 2402 and Zoom ID 946 4079 7326

    We introduce BiLO (Bilevel Local Operator Learning), a novel neural network-based approach for solving inverse problems in partial differential equations (PDEs). BiLO formulates the PDE inverse problem as a bilevel optimization problem: at the upper level, we optimize PDE parameters by minimizing data loss, while at the lower level, we train a neural network to locally approximate the PDE solution operator near given PDE parameters. This localized approximation enables accurate descent direction estimation for the upper-level optimization. We apply gradient descent simultaneously on both the upper and lower level optimization problems, leading to an effective and fast algorithm. Additionally, BiLO can infer unknown functions within PDEs by introducing an auxiliary variable. Extensive experiments across various PDE systems demonstrate that BiLO enforces strong PDE constraints, is robust to sparse and noisy data, and eliminates the need for manually balancing residual and data loss, a common challenge in soft PDE constraints. We also discuss how to apply the BILO for uncertainty quantification in a Bayesian framework.

  • 2:00 pm
    Stephan Pfannerer - University of Waterloo
    Rotation-invariant web bases from hourglass plabic graphs

    Math 269: Combinatorics Seminar

    APM 5829

    In 1995, Kuperberg introduced a remarkable collection of trivalent web bases which encode tensor invariants of $U_q(\mathfrak{sl}_3)$. Extending these bases to general $\mathfrak{sl}_r$ has been an open problem ever since. We present a solution to the $r=4$ case by introducing hourglass plabic graphs - a new generalization of Postnikov's plabic graphs. Joint work with Christian Gaetz, Oliver Pechenik, Jessica Striker and Joshua Swanson.

  • 3:00 pm
    Prof. Kiran Kedlaya - UC San Diego
    Counting curves (and their rational points) over finite fields

    Math 296: Graduate Student Colloquium

    APM 6402

    For q a prime power, let F_q be the finite field of order q. There are a finite number of isomorphism classes of (smooth, projective, geometrically irreducible) curves of genus g over F_q. Can one give a closed form expression for this number? We discuss how to correctly interpret this question; how to generalize it by also counting marked points; what is known for small g; and what information can be gained by making complete tables of curves of a given genus.

Wed, Apr 23 2025
  • 4:00 pm
    Prof. Anna Ma - UC Irvine
    Tensor Iterative Methods for Large-Scale Linear Systems

    Math 278C: Optimization and Data Science

    APM 6402 and Zoom (Meeting ID: 941 4642 0185 / Password: 278C2025)

    Solving linear systems is a crucial subroutine and challenge in the large-scale data setting. In this presentation, we introduce an iterative method for approximating the solution of large-scale multi-linear systems, represented in the form A*X=B under the tensor t-product. Unlike previously proposed randomized iterative strategies, such as the tensor randomized Kaczmarz method (row slice sketching) or the tensor Gauss-Seidel method (column slice sketching), which are natural extensions of their matrix counterparts, our approach delves into a distinct scenario utilizing frontal slice sketching. In particular, we explore a context where frontal slices, such as video frames, arrive sequentially over time, and access to only one frontal slice at any given moment is available. This talk will present our novel approach, shedding light on its applicability and potential benefits in approximating solutions to large-scale multi-linear systems.
     

Thu, Apr 24 2025
  • 10:00 am
    Pratyush Sarkar - UCSD
    Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications

    Math 211B - Group Actions Seminar

    APM 7321

    A celebrated theorem of Eskin–Mozes–Shah gives an asymptotic counting formula for the number of integral (n x n)-matrices with a prescribed irreducible (over the integers/rationals) integral characteristic polynomial. We obtain a power saving error term for the counting problem for (3 x 3)-matrices. We do this by using the connection to homogeneous dynamics and proving effective equidistribution of translates of tori in SL_3(R)/SL_3(Z). A key tool is that the limiting Lie algebra corresponding to the translates of tori is a certain nilpotent Lie algebra. This allows us to use the recent breakthrough work of Lindenstrauss–Mohammadi–Wang–Yang on effective versions of Shah's/Ratner's theorems. We actually study the phenomenon more generally for any semisimple Lie group which we may discuss if time permits.

     

  • 11:00 am
    Jonas Luhrmann - Texas A&M University
    Asymptotic stability of the sine-Gordon kink outside symmetry

    Math 248: Real Analysis Seminar

    APM 7321

    We consider scalar field theories on the line with Ginzburg-Landau  (double-well) self-interaction potentials. Prime examples include the  $\phi^4$ model and the sine-Gordon model. These models feature simple  examples of topological solitons called kinks. The study of their = asymptotic stability leads to a rich class of problems owing to the  combination of weak dispersion in one space dimension, low power  nonlinearities, and intriguing spectral features of the linearized  operators such as threshold resonances or internal modes.

    We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

    The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known $\phi^4$ model.

    This is joint work with Gong Chen (GeorgiaTech).

  • 2:00 pm
    Professor Mykhailo Potomkin - UC Riverside
    Computational analysis of microscopic motility: Individual and collective scales in two case studies

    Math 218: Seminar on Mathematics for Complex Biological Systems

    APM 7321

    In this talk, I will present two recent pieces of research that are connected by the common theme of multiscale models for motile microorganisms. 

    In the first part, I will discuss the orientational dynamics of microscopic organisms, such as bacteria, swimming in biofluids with properties that differ from those of isotropic Newtonian fluids, instead exhibiting characteristics of liquid crystals. These environments have a preferred direction, which forces the swimmers to align with it. However, certain types of bacteria can overcome this external torque and swim across the preferred direction. I will present a nonlinear PDE system that couples liquid crystal hydrodynamics with a model of a prototypical microswimmer. This model identifies the conditions for non-trivial reorientation dynamics and allows for deriving the homogenized limit, effectively describing the dynamics of the microswimmer colony. This is the joint work with I. Aronson (PSU), L. Berlyand (PSU), H. Chi (PSU), A. Yip (Purdue U.), and L. Zhang (SJTU). 

    In the second part of the talk, I will focus on a computational model that describes how motile cancer cells interact with the extracellular matrix (ECM) during the initial invasion phase, including ECM degradation and mechanical remodeling. The model highlights the role of elastic interactions in the dynamics of cell clusters, including their shapes, sizes, and orientations. These results are joint work with O. Kim (Virginia Tech), Y. Klymenko (Indiana U.), M. Alber (UCR), and I. Aranson (PSU).

  • 4:00 pm
    Prof. Andrew Snowden - University of Michigan, Ann Arbor
    Oligomorphic groups and tensor categories

    Mathematics Colloquium

    APM 6402

    If G is a finite group then the collection of all finite dimensional complex representations of G carries two important operations: direct sum and tensor product. A tensor category is an abstraction of this situation. Finding new examples of tensor categories is a very difficult problem. In recent work with Harman, we gave a general construction of tensor categories based on oligomorphic groups, a class of infinite permutation groups best known in model theory. I will give an overview of our work.

Fri, Apr 25 2025
  • 4:00 pm
    Dr. Miguel Moreira - Massachusetts Institute of Technology
    The Chern filtration on the cohomology of moduli spaces of (parabolic) bundles

    Math 208: Seminar in Algebraic Geometry

    APM 7321

    The Chern filtration is a natural filtration that can be defined on the cohomology of moduli spaces of sheaves. Its definition was originally made for the moduli of Higgs bundles, motivated by a comparison with the perverse and weight filtrations, but it also makes sense for the very classical moduli spaces of bundles on curves. A vanishing result conjectured by Newstead and proved by Earl-Kirwan in the 90s is secretly a statement about the Chern filtration. I will explain a new approach to this vanishing which is based on parabolic bundles: it turns out that enriching the problem with a parabolic structure gives access to powerful tools, such as wall-crossing, Hecke transforms and Weyl symmetry — together, these give a new proof of the Newstead-Jefrey-Kirwan vanishing and a related "d independence" statement. Part of the talk is based on work with W. Lim and W. Pi.
     

Thu, May 1 2025
  • 11:00 am
    Professor Zhen-Qing Chen - University of Washington
    Boundary trace of symmetric reflected diffusions

    2025 Ronald Getoor Lecture

    APM 6402

    Starting  with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$ (for example, absorbing Brownian motion in a snowflake domain), one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the theory of  reflected Dirichlet spaces. The boundary trace process $\check X$ of $X$ on the boundary $\partial D:=D^*\setminus D$ is the reflected diffusion process $\bar X$ time-changed by a smooth measure $\nu$ having full quasi-support on $\partial D$. The Dirichlet form of the trace process $\check X$ is called the trace Dirichlet form. In this talk, I will address the following two fundamental questions:

    1) How to characterize the boundary trace Dirichlet space in a concrete way?

    2) How does the boundary trace process behave? 

    Based on a joint work with Shiping Cao.

Wed, May 7 2025
  • 2:00 pm
    Professor Claire Tomlin - James and Katherine Lau Professor in the College of Engineering; Chair, Department of Electrical Engineering and Computer Sciences (University of California, Berkeley)
    Safe Learning in Autonomy

    Murray and Adylin Rosenblatt Lecture in Applied Mathematics

    Kavli Auditorium, Tata Hall, UC San Diego

Thu, May 8 2025
  • 11:00 am
    Haixiao Wang - UC San Diego
    Critical sparse random rectangular matrices: emergence of spectra outliers

    Math 288 - Probability & Statistics

    APM 6402

    Consider the random bipartite Erdos-Renyi graph $G(n, m, p)$, where each edge with one vertex in $V_{1}=[n]$ and the other vertex in $V_{2}=[m]$ is connected with probability $p$ with $n \geq m$. For the centered and normalized adjacency matrix $H$, it is well known that the empirical spectral measure will converge to the Marchenko-Pastur (MP) distribution. However, this does not necessarily imply that the largest (resp. smallest) singular values will converge to the right (resp. left) edge when $p = o(1)$, due to the sparsity assumption. In Dumitriu and Zhu 2024, it was proved that almost surely there are no outliers outside the compact support of the MP law when $np = \omega(\log(n))$. In this paper, we consider the critical sparsity regime with $np =O(\log(n))$, where we denote $p = b\log(n)/\sqrt{mn}$, $\gamma = n/m$ for some positive constants $b$ and $\gamma$. For the first time in the literature, we quantitatively characterize the emergence of outlier singular values. When $b > b_{\star}$, there is no outlier outside the bulk; when $b^{\star}< b < b_{\star}$, outlier singular values only appear outside the right edge of the MP law; when $b < b^{\star}$, outliers appear on both sides. Meanwhile, the locations of those outliers are precisely characterized by some function depending on the largest and smallest degrees of the sampled random graph. The thresholds $b^{\star}$ and $b_{\star}$ purely depend on $\gamma$. Our results can be extended to sparse random rectangular matrices with bounded entries.