Abstract: A major goal of the Langlands program is to describe the multiplicity of an irreducible discrete automorphic representation of a connected reductive group G in its discrete L2-spectrum. The first goal of this talk is to explain work from the last few years which gives the first conjectural formula for this multiplicity for general G over a global field, as envisioned by Kottwitz in 1984. We then discuss recent work which we hope lays the foundations for proving cases of these formulas using the geometric framework of Fargues and Scholze.
Abstract: My research develops mathematically grounded methods for inference in high-dimensional biomedical imaging data, spanning functional brain dynamics (functional neuroimaging) and viral heterogeneity (cryo-EM). In neuroimaging, I model time-varying interactions among brain regions, moving beyond static and correlational connectivity to infer directed, spatiotemporally evolving network organization. These approaches support mechanistic interpretation and yield innovative biomarkers relevant to conditions such as post-concussive vestibular syndrome (PCVD). In structural virology, I study 3D reconstruction of virus particles from cryo-EM images. I develop symmetry-aware methods that preserve particle-specific asymmetry while enforcing global symmetry constraints across the population, improving reconstruction of virus(-like) particles such as bacteriophage HK97. The unifying theme is to exploit dynamics, constraints, and invariances for reliable inference under noise and heterogeneity.
Abstract: Shrinking gradient Kähler-Ricci solitons (Kähler-Ricci shrinkers) are fundamental objects in the study of the Kähler-Ricci flow, characterizing much of the behavior of finite-time singularities. Recently, Sun--Zhang have developed an algebraic theory for Kähler-Ricci shrinkers, which in particular implies that such spaces are naturally quasiprojective varieties. Moreover, they propose a YTD correspondence between the existence of such a metric and an algebro-geometric notion of K-stability, analogous to and in fact extending the well-known situations for Fano manifolds and Kähler cones. In this talk, I will discuss the proof of one direction of the correspondence, namely that the existence of a Kähler-Ricci shrinker metric implies K-polystability, in the case that the Ricci curvature decays at infinity. This is joint work with Carlos Esparza.
Abstract: Many contemporary public health challenges involve substantial heterogeneity in individual characteristics, contact structures, comorbidities, and social determinants of health. Addressing these interacting layers requires modeling frameworks capable of representing individuals, diseases, and interventions within a unified and dynamically evolving system.
In this talk, I will introduce the Model of Inter-Generational Health, Transmission, and Interventions (MIGHTI), a modular multi-disease agent-based simulation platform designed to jointly model infectious diseases, non-communicable diseases, and social determinants of health within a single computational framework.
Using Eswatini as a case study, I will demonstrate how MIGHTI enables evaluation of life expectancy gains, cause-specific mortality attribution, and policy-relevant intervention scenarios. The goal is to illustrate how an integrated, multi-disease modeling platform can support quantitative analysis of complex, multi-layered population health systems.
Abstract: We prove that the theory of the Farey graph is pseudofinite by constructing a sequence of finite structures that satisfy increasingly large subsets of its first-order axiomatization. The Farey graph was recently axiomatized by Tent and Mohammadi. We show that while no finite planar graph can satisfy these axioms for sufficiently large substructures, they can be satisfied by triangulations densely embedded on orientable surfaces of higher genus. By applying a result of Archdeacon, Hartsfield, and Little on the existence of triangulations with representativity and connectedness, we establish that every finite subset of the theory of the Farey graph has a finite model.
Abstract: Mining industries use apparatuses that separate materials immersed in fluids based on physical properties such as density and size.
The typical setup is a spiral slide apparatus in which a particle-fluid mixture is introduced at the top and flows down an inclined spiral channel as a thin film. This produces a stable configuration in which particles accumulate toward the central axis of the spiral, while clear fluid appears toward the outer edge of the channel. Notably, contrary to what one might expect from devices such as centrifuges, the denser particles accumulate toward the channel axis. This behavior is observed both experimentally and in our analysis of conservation law models.
To investigate size-based segregation, we consider a simpler apparatus consisting of a straight, inclined plane. Experiments reveal an unexpected behavior: smaller particles settle rapidly, sinking to the bottom upstream, while larger particles continue to flow and roll, accumulating at the front of the particle mass. In contrast, while our existing models for density-based particle segregation on an inclined plane are consistent with experimental observations, the corresponding size-based models predict the opposite trend, with smaller particles accumulating at the front of the flow. This discrepancy highlights the need for future work to better understand complex particle interactions using techniques such as HPC and PIV.
Abstract: I will first give a review of the well-studied relationship between Loop groups, vector bundles on complex P^1 and Langlands duality for reductive groups. Then I will discuss recent progress on establishing a similar relationship in the setting of loop spaces of symmetric spaces (or spherical varieties), vector bundles on twistor P^1 (or real projective line RP^1) and Langlands duality for real groups (or Relative Langlands duality). The key ingredients include a Matsuki duality for loop groups and a version of derived Satake equivalence for ramified groups. I will discuss applications of such connections to Ben-Zvi-Nadler's conjecture on Betti Geometric Langlands for real groups and Ben-Zvi-Sakellaridis-Venkatesh's conjecture on relative derived Satake equivalence for symmetric spaces. If time permits, I will mention a hope / speculation in the mixed characteristic setting.
Abstract: The Tracy-Widom (TW) distributions, indexed by a positive parameter b, arise as limiting distributions in many different probabilistic models including random matrices and the longest increasing subsequences of random permutations, thus exemplifying a universality phenomenon in probability. However, they are rather mysterious, with no explicit form for their densities or characteristic functions, and little has been proved analytically about their properties. Although simulations suggest that the TW distributions are log-concave, the only (partial) result in this direction is a theorem of Percy Deift that the TW distribution with parameter 2 is log-concave on the positive real line. We settle this as well as several related questions: Not only are all TW distributions shown to be log-concave, we do this by establishing log-concavity of certain pre-limit distributions, including the largest eigenvalues of Gaussian beta-ensembles and the Poissonized Plancherel measure on Young diagrams that arises in the representation theory of the symmetric group. In particular, one consequence of our results is that a Poissonized version of a 2008 conjecture of W.Y.C.Chen— asserting that for any fixed natural number N, the number of permutations in the symmetric group S(N) that have a longest increasing subsequence of length K, is a log-concave sequence in K — is true. The talk is based on joint work with Jnaneshwar Baslingker and Manjunath Krishnapur.
Abstract: Conformal prediction provides a distribution-free framework for uncertainty quantification via prediction sets with exact finite-sample coverage. In low dimensions these sets are easy to interpret, but in high-dimensional or structured output spaces they are difficult to represent and use, which can limit their ability to integrate with downstream tasks such as sampling and probabilistic forecasting. We show that any differentiable nonconformity score induces a deterministic flow on the output space whose trajectories converge to the boundary of the corresponding conformal prediction set. This leads to a computationally efficient, training-free method for sampling conformal boundaries in arbitrary dimensions. Boundary samples can be reconformalized to form pointwise prediction sets with controlled risk, and mixing across confidence levels yields conformal predictive distributions whose quantile regions coincide exactly with conformal prediction sets. We evaluate the approach on PDE inverse problems, precipitation downscaling, climate model debiasing, and hurricane trajectory forecasting.
Abstract: In this talk, I will present a quantitative relative entropy framework for the diffusion limit of kinetic equations with Riesz-type interactions and Fokker-Planck relaxation. The focus will be on the derivation of a drift-diffusion equation from the Vlasov-Fokker-Planck equation. By combining entropy dissipation, Fisher-information bounds, and modulated interaction energies, the method yields stability estimates that provide the quantitative convergence rates in this regime. I will also discuss how the framework distinguishes between well-prepared and mildly prepared initial data, leading respectively to strong and weak convergence results.
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