Date/Time : 12 Aug 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Matias Delgadino, University of Texas at Austin

We prove sharp estimates for the mean-field limit of weakly interacting diffusions with repulsive logarithmic interaction in arbitrary dimension. More precisely, we show that the associated partition function is uniformly bounded in the number of particles N for an arbitrary bounded base measure. Combined with the modulated free energy method, this amounts to a logarithmic improvement in N of the current best available closeness estimates in the literature. Our arguments are inspired by and borrow ideas from Nelson’s classical construction of the \Phi^4_2 Euclidean quantum field theory.

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Date/Time : 08 Aug 2025 10:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Timo Sprekeler, Texas A&M University

We begin by discussing the finite element approximation of stationary Fokker–Planck–Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings, based on the finite element scheme for stationary FPK problems developed in the first part. This talk includes joint work with Endre Süli (University of Oxford) and Zhiwen Zhang (The University of Hong Kong).

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Date/Time : 30 Jul 2025 14:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Sang-hyun Kim, Korea Institute for Advanced Study

Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that for two real numbers r,s ≥ 1, and for two smooth closed manifolds M and N, the C^r diffeomorphism group of M is elementarily equivalent to the C^s diffeomorphism group of N if and only if r=s and M is diffeomorphic to N. In the case of r=s=0, we can even weaken the hypothesis so that the manifolds are only assumed to be compact, dropping the smoothability or closedness hypotheses. This strengthens (1) Whittaker’s theorem (1963) on the C^0 case, and (2) Takens–Filipkiewicz theorems (1982) on C^p case with an integer p. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS)

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Date/Time : 29 Jul 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Vilas Sreenivasan Winstein, University of California, Berkeley

Independent sets have long been studied as a model for gases in statistical mechanics, and Sapozhenko’s famous graph container method was developed to understand their quantity in the hypercube, where they correspond to error-detecting codes. When removing edges at random (i.e. percolating the hypercube), the number of independent sets in the resulting graph becomes the partition function of a disordered system, akin to popular models like spin glasses. I will discuss a new probabilistic framework which establishes sharp distributional characterizations of this partition function and the underlying disordered system, settling several open problems and explaining various phase transitions observed in this model. This is joint work with Mriganka Basu Roy Chowdhury and Shirshendu Ganguly.

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Date/Time : 28 Jul 2025 16:00
Venue : S17 #04-04 (SR3)
Speaker : Vilas Winstein , University of California, Berkeley

This is part 2 of a two-talk series. We will discuss some recent results on the limiting Gaussian nature of fluctuations of various observables under exponential random graph models; in particular, we obtain quantitative error bounds via Stein’s method. A key novelty is that our results apply in the low-temperature parameter regime which has previously been unexplored and requires new methods. These include the application of novel concentration inequalities derived from carefully analyzing the mixing behavior of relevant Markov chains.

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Date/Time : 25 Jul 2025 16:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Vilas Winstein , University of California, Berkeley

This is part 1 of a two-talk series. We will introduce exponential random graph models, which are exponential tilts of Erdős–Rényi models by functions which increase or decrease the number of certain small subgraphs (e.g. triangles). We will also discuss relevant tools for studying these modelThis is part 1 of a two-talk series. We will introduce exponential random graph models, which are exponential tilts of Erdős–Rényi models by functions which increase or decrease the number of certain small subgraphs (e.g. triangles). We will also discuss relevant tools for studying these models, such as large deviations principles, mixing results, and concentration inequalities.s, such as large deviations principles, mixing results, and concentration inequalities.

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Date/Time : 16 Jul 2025 15:00
Venue : S17 #04-04 (SR3)
Speaker : Naihuan Jing, North Carolina State University

I will discuss some polynomials parameterized by Young diagrams, whose coefficients are central elements of the quantized enveloping algebra of the general linear algebra. These polynomials are q-immanants and are shown to provide the higher quantum Capelli identities and generalize Drinfeld-Reshetikhin’s construction of central elements in Z(U_q(g)) as well as q-analogues of Okounkov’s quantum immanants for U(gl(n)). The Harish-Chandra image of the these central elements are certain factorial Schur polynomials. The main results are joint works with Ming Liu and Alex Molev.

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Date/Time : 10 Jul 2025 10:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Nimish Shah, The Ohio State University

For the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real-analytic submanifold under a diagonal flow. This extends the earlier result of Shah in the case of non-degenerate submanifolds.

We apply the above dynamical result to show that if the affine span of a real-analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet-improvable.

This is a joint work with Pengyu Yang

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Date/Time : 10 Jul 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Si-Jong Kwak, Korea Advanced Institute of Science & Technology

The gonality conjecture of a smooth projective curve raised by Green and Lazarsfeld. has been one of the leading problems in geometry and syzygy. The conjecture says that the gonality of a curve can be encoded in the Betti table if a given curve is embedded by the line bundle of sufficiently large degree. It was proved by Ein-Lazarsfeld in 2016.

In this talk, we generalize this conjecture to the secant varieties of a curve. The gonality sequence d_1, d_2, . . . , d_g are encoded in syzygies of the secant varieties of a curve. This is a joint work with J. Choe and J. Park.

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