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Abstract: The HRT conjecture predicts that finite Gabor systems generated by time-frequency shifts of a nonzero function are linearly independent. I will discuss a recent result by Oussa (Trichotomy for the HRT Conjecture for mixed integer configuration (August 2025)) proving the conjecture for mixed integer configurations - that is, sets of lattice points together with one off-lattice point - for Schwartz class functions.

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Abstract: Recall that two countable groups are measure equivalent, if they admit commuting, measure preserving, free actions on the same measure space with finite measure fundamental domains. This notion was first proposed by Gromov as a measure-theoretical version of quasi-isometry, and it is closely related to the notion of orbit equivalence of groups acting on probability measure spaces. Given a countable group G, one may asks what are other groups which are measure equivalent to G. This question has generated many interesting mathematics, and there were rather strong rigidity results proved in the case when G is a higher rank lattice or the mapping class group of a surface. I will talk about recently work with Camille Horbez, on measure equivalence invariants of certain countable groups acting on combinatorial complexes with certain form of negative curvature or non-positive curvature. This has application to a number of rigidity results in Artin groups, Coxeter groups and Higman groups.

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Abstract: The complex unit disc is conformally equivalent to the upper half-plane.  In 2002, Faltings proved a p-adic version:  the p-adic unit disc is isomorphic to Drinfeld’s upper half-plane, up to the action of some profinite groups.  We present a family of Faltings-style isomorphisms, occurring entirely in characteristic p, which relates a moduli space of formal groups to a "mod p period domain".  The motivation is entirely from chromatic homotopy theory, but along the way there is much interesting p-adic geometry and mod p representation theory of p-adic groups.  This is joint work with many people.

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Abstract: Generative artificial intelligence models—including variational autoencoders, normalizing flows, generative adversarial networks, and diffusion models—have dramatically advanced the realism and quality of generated images, text, and audio. Beyond these tasks, generative models hold great promise as powerful tools for probability density estimation and high-dimensional sampling, which are central to uncertainty quantification (UQ) tasks such as amortized Bayesian inference and data assimilation. However, while research on image synthesis emphasizes producing high-quality individual samples, UQ applications require accurate approximation of statistical quantities of interest rather than visually realistic samples. As a result, direct application of existing generative models to UQ problems can lead to biased approximations or unstable training. In this talk, we will introduce several new generative approaches tailored to UQ. These include training-free diffusion models for density estimation, a score-based nonlinear filter for data assimilation, and training-free conditional diffusion models for amortized Bayesian inference. We will demonstrate their effectiveness across a range of tasks, including density estimation for unimodal and multimodal distributions, learning stochastic dynamical systems, parameter estimation via amortized inference, and scalable data assimilation for atmospheric models.

Bio: Dr. Guannan Zhang is a Distinguished Staff Scientist in Computer Science and Mathematics Division at Oak Ridge National Laboratory (ORNL). He earned my Ph.D. in applied mathematics at Florida State University in 2012. He joined ORNL in 2012 as the Householder fellow. He received the DOE Early Career Award in 2022. He has been holding a joint faculty appointment with the Department of Mathematics and Statistics at Auburn University since 2014, and a joint faculty appointment with Department of Mathematics at University of Tennessee since 2022. Guannan's research interests include high-dimensional approximation, uncertainty quantification, machine learning and artificial intelligence, stochastic methods for scientific inverse problems.

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Abstract: The systole of a 3-manifold is the length of the shortest closed geodesic. Given a closed 3-manifold M and link L such that M\L is hyperbolic, the essential systole of M\L is the length of the shortest closed geodesic which is not nullhomotopic in M. In this talk, we will discuss and motivate the study of essential systoles of hyperbolic link complements, including their application towards answering a question of Freedman and Krushkal about the existence of "filling links" in closed 3-manifolds. This is joint work with Chris Leininger.

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Abstract: The ocean’s abyssal overturning circulation provides the long timescale memory of the climate system, with a round trip between the surface and the deep interior taking around 1000 years. Maintaining this slow overturning requires turbulent mixing at depth, providing a pathway for dense waters to return to the surface. The canonical picture of how this mixing occurs involves remotely generated internal waves reflecting, and breaking, along the bottom topography. However, recent work by our group suggests that low-frequency flows along topography may also provide novel pathways to mixing, both through their role in wave-mean-flow interactions and via the generation of small-scale hydrodynamic instabilities. Large-eddy simulations of flows interacting with bottom topography indicate the presence of symmetric instability (a 2D mixed gravitational-inertial instability of rotating flows), which is shown to facilitate a transition to turbulence both along the bottom and in topographic wakes behind seamounts and headlands. Similar conditions are also amenable to parametric subharmonic instability, a wave triad instability. Linear stability analysis, and nonlinear simulations, show that this also provides a pathway for the breakdown of internal waves into turbulence, a novel source of mixing along the bottom. These results will be summarized in the context of their potential unresolved effects in our understanding and modeling of the ocean circulation, and several related open questions (including available PhD projects) will be discussed.

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Abstract: Using a reaction-diffusion model with free boundaries in one space dimension for a single population species with density $u(t,x)$ and population range $[g(t), h(t)]$, we demonstrate that the Allee effects can be eliminated if the species maintains its population density at a suitable level at the range boundary by advancing or retreating the fronts. It is proved that with such a strategy at the range edge the species can invade the environment successfully with all admissible initial populations, exhibiting the dynamics of super invaders. Numerical simulations are used to help understand what happens if the population density level at the range boundary is maintained at other levels. If the invading cane toads in Australia used this strategy at the range boundary to become a super invader, then our results may explain why toads near the invading front evolve to have longer legs and run faster.

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Abstract: https://drive.google.com/file/d/1Nkb68KO7CLI_vfxQGJsnPMYqh3ZuPAdE/view?usp=sharing

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Abstract: The question of finding and classifying complete Calabi--Yau metrics on smooth affine varieties of Euclidean volume growth goes back to Tian--Yau, who constructed such metrics on X given by the complement of a Kähler--Einstein divisor in a Fano variety. Recent classification results suggest that such metrics on smooth affine varieties come from prescribing the asymptotic geometry using a negative valuation. In this talk, I will revisit Tian--Yau's example, in which case the Kähler--Einstein divisor defines a K-semistable valuation which does not admit a center on X. Generalizing this leads to a valuative criterion for K-semistable valuations at infinity on a given affine variety. Time permitting, I will also explain that these valuations in fact come from Fano type compactifications generalizing the Tian--Yau case. This is based on joint work in progress with Mattias Jonsson.

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Abstract: In this talk, I discuss how the study of Ramsey structures and generalized indiscernibles allows us to define new versions of well-known model-theoretic tree properties. Focusing attention on the colored linear order as an index structure, I prove that the so-called colored tree property gives a characterization of stability in terms of positive formulas, while colored versions of TP1 and TP2 give equivalent versions of TP1 and the Independence Property, respectively.

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Abstract: We consider Chorin's projection method combined with a finite element spatial discretization for the time-dependent incompressible Navier-Stokes equations. The projection method advances the solution in two steps: A prediction step which computes an intermediate velocity field that is generally not divergence-free, and a projection step which enforces (approximate) incompressibility by projecting this velocity onto the (approximately) divergence-free subspace. We establish convergence, up to a subsequence, of the numerical approximations generated by the projection method with finite element spatial discretization to a Leray-Hopf solution of the incompressible Navier-Stokes equations, without any additional regularity assumptions beyond square-integrable initial data and square-integrable forcing. A discrete energy inequality yields a priori estimates, which we combine with a Aubin-Lions type compactness result to prove precompactness of the approximations in $L^2([0,T]\times\dom)$, where $[0,T]$ is the time interval and $\dom$ is the spatial domain. Passing to the limit as the discretization parameters vanish, we obtain a weak solution of the Navier–Stokes equations. A central difficulty is that different a priori bounds are available for the intermediate and projected velocity fields; our compactness argument carefully integrates these estimates to complete the convergence proof. If time permits, I will also discuss how the proof can be adapted to prove convergence of a second-order in time method.

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Abstract: The congruent number problem is one of the oldest problems in number theory. We'll talk a little about its (partial) resolution as well as its relationship to arithmetic statistics.

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Abstract: Serre's classical GAGA theorem states that coherent sheaves on a proper complex algebraic variety are the same as those of its analytification. In particular, one finds that the algebraic and analytic Picard groups agree for proper varieties. The Picard group being the degree-two weight-one motivic cohomology group, one may wonder to what extend motivic cohomology does satisfy some GAGA comparison in general. In this talk, I want to discuss some answers to the non-archimedean analogue of this question. In particular, I will explain how to use recent progress in motivic cohomology to define a good notion of "analytic cycles" on rigid-analytic varieties, and report on a GAGA theorem between algebraic and analytic zero-cycles. This is based on joint work in progress with Elden Elmanto, Brian Shin, and Mahdi Rafiei.

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Abstract: The density of a plasma is governed by the Landau equation, a nonlocal kinetic equation with quadratic nonlinearity that is connected to the Euler equations via a hydrodynamic limit.  A longstanding open problem about the Landau equation is whether global-in-time solutions exist.  We show that (classical) solutions that may be continued for as long as they remain bounded in L^\infty.  At first this may seem unassuming.  However, it is widely believed that blow-up, if it were to occur, would happen by "tail fattening," not L^\infty unboundedness, because tail fattening corresponds to the type of blow-up seen in the Euler equations.  Our theorem rules this out.  Additionally, our work suggests that one cannot construct blow-up for the Landau equation by using singularity formation in the Euler equations. This is a joint work with Will Golding.

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Abstract: In this article, we discuss a Bayesian Empirical Likelihood (BayesEL) based method for complex survey data, leading to applications in non-probability sampling. Bayesian formulation of complex survey data presents several computational, methodological, as well as philosophical problems. Since the observations are sampled with unequal probability, the distributions of the observations in the population and the sample are different. Thus, when it comes to constructing the posterior, a practitioner has a choice of using one of the above distributions. Associated to this, we also need to consider if the posterior is constructed based on the sample or on the whole of the finite population. This question is also related to the method used to incorporate the information in the sampling weights in the procedure. We will show that the BayesEL provides an orderly way to construct a posterior for complex survey data by addressing all the above questions. Some properties of the posterior, e.g., asymptotic validity, objective prior construction, etc, will be discussed. Finally, an application to the non-probability sampling problem will be presented.