Date/Time : 23 Oct 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Roberto Imbuzeiro Oliveira , Instituto Nacional de Matemática Pura e Aplicada (IMPA)

Estimating the mean of a random vector from i.i.d. data has received considerable attention. When the data take values in more general metric spaces, an appropriate extension of the notion of the mean is the Fréchet mean. While asymptotic properties of the most natural Fréchet mean estimator (the empirical Fréchet mean) have been
thoroughly researched, non-asymptotic performance bounds have only been studied recently.

This talk considers the performance of estimators of the Fréchet mean in general metric spaces under possibly heavy-tailed and contaminated data. In such cases, the empirical Fréchet mean is a poor estimator. We propose a general estimator based on high-dimensional extensions of trimmed means and prove general performance bounds. Unlike all previously established bounds, ours generalize the optimal bounds known for Euclidean data. Much like in the Euclidean case, the optimal accuracy is governed by two “variance” terms: a “global variance” term that is independent of the prescribed confidence, and a potentially much smaller, confidence-dependent “local variance” term. We apply our results for metric spaces with curvature bounded from below, such as Wasserstein spaces, and for uniformly convex Banach spaces.

The talk is based on arXiv:2509.13606, which is joint work with Daniel Bartl (NUS), Gabor Lugosi (ICREA/UPF), and Zoraida Rico (Bocconi).

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Date/Time : 22 Oct 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Richard Canary, University of Michigan

Margulis proved asymptotic counting results for the number of closed geodesics less than a fixed length in the quotient of a uniform lattice in a rank one Lie group. Patterson and Sullivan developed a technology to generalize his results to convex cocompact and geometrically finite subgroups of rank one Lie groups.

An element of a higher rank Lie group has a vector-valued translation length function, which gives rise to more than one reasonable notion of the length of an element. Anosov groups of higher rank semi-simple Lie groups are the natural analogue of convex cocompactness in rank one. Sambarino proved counting results for all well-behaved length functions in this setting.

Relatively Anosov subgroups of higher rank Lie groups are one generalization of geometrically finite subgroups of rank one Lie groups. In this talk we obtain counting results in this setting. The crucial new difficulty is that the relevant flow spaces are non-compact, whereas they were compact in the Anosov setting. These results are based on work with Zhang and Zimmer on Poincare series of relatively Anosov groups and a rather general formulation of Patterson-Sullivan theory which was developed with Blayac, Zhu and Zimmer.

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Date/Time : 22 Oct 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Roberto Imbuzeiro Oliveira , Instituto de Matemática Pura e Aplicada

The problem of reliably estimating the expectation (or mean) of a random variable from a random sample is arguably the most fundamental problem in statistics. It also seemed to be a basically settled problem: what could be better than taking the sample mean? Contrary to this received wisdom, Catoni showed 15 years ago that there was a way
to come up with higher-confidence, higher-precision estimators for expectations in one dimension.

The goal of this talk is to introduce this line of work, its many ramifications and some of its open problems to a broad audience. We will discuss the estimation of expectations in higher dimensions and nonlinear spaces, as well as some consequences of these results for statistical tasks. One of our goals will be to show that the classical
notion of trimmed mean can be put to use and lead to good rates even when the random sample is corrupted. The talk will discuss results by many researchers, including our own work with Daniel Bartl, Gábor Lugosi, Lucas Resende, Zoraida Rico and Philip Thompson.

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Date/Time : 17 Oct 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Yibo Gao, BICMR, Peking University

A partially ordered set is called Sperner if the size of its largest antichain does not exceed the size of its largest rank. In this talk, we show that the weak Bruhat order of the symmetric group is strongly Sperner via sl2 representations, resolving an old conjecture by Björner. We will also discuss its relations with Schubert polynomials, which are polynomial representations of the Schubert classes in the cohomology ring of the flag variety.

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Date/Time : 15 Oct 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Milind Hegde , Nanyang Technological University

A recurring theme in probability theory is that of universality: when extremely different looking systems have the same large scale statistical behavior. In the last few decades, an important new universality class has been discovered, called the Kardar-Parisi-Zhang (KPZ) universality class. However, the universality is only putative as only a handful of “metric” type models have been shown to lie in it in the strongest sense.

In this talk we will discuss a recent proof of membership in the KPZ class of the first non-metric type of model, namely the colored stochastic six-vertex model. The model arises naturally in probability theory and has connections to many areas of statistical physics and quantum integrable systems; e.g., the first form of the six-vertex model was introduced by Pauling in 1935 to model the crystal structure of ice. The Yang-Baxter equation and line ensembles (collections of random non-intersecting curves) will play fundamental roles in our discussion, but no prior background will be assumed. This is based on joint work with Amol Aggarwal and Ivan Corwin.

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Date/Time : 15 Oct 2025 17:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Luca San Mauro, University of Bari

Borel reducibility has been a decisive tool in assessing whether familiar classes of countable structures can be neatly classified. However, when the corresponding isomorphism problems admit only countably many isomorphism types, they all share the same Borel complexity. To study such cases, we propose a finer framework, inspired by algorithmic learning theory. In this setting, a countable family of countable structures is learnable exactly when its isomorphism problem continuously reduces to E0 (the relation of eventual agreement on reals). Replacing E0 with other equivalence relations then yields a natural hierarchy for ranking isomorphism problems that escape the Borel analysis. Within this framework, we investigate both classical benchmark relations and those arising from established learning criteria. We also revisit the Friedman-Stanley tower from a learning-theoretic viewpoint. This talk is based on a series of joint works with N. Bazhenov, V. Cipriani, E. Fokina, S. Jain, A. Marcone, and F. Stephan.

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Date/Time : 15 Oct 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Kai Dong, National University of Singapore

In this talk, we give a new proof of the past nonlinear stability of Kasner solutions for $3+1$-dimensional Einstein vacuum equations with polarized $U(1)$-symmetry. Our method, following Beyer, Oliynyk, and Arturo Olvera-Santamaria \cite{beyer2021fuchsian}, relies on energy estimates for a class of first order hyperbolic PDEs, namely Fuchsian systems.

We show that the perturbed solutions are asymptotically Kasner, geodesically incomplete and crushing at the big bang singularity. This is achieved by reducing the $3+1$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of propagation, we obtain global existence and precise asymptotic behavior of the solutions up to the singularities.

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Date/Time : 15 Oct 2025 15:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Liang Zhao, National University of Singapore

In this talk, we will consider the simplification from the multi-dimensional (M-D) bipolar Euler–Poisson (2EP) equations to the unipolar Euler–Poisson (1EP) equations of electrons, which are both important models in plasma physics and semiconductors. Based on some rigorous convergence analysis of certain physical parameters, we are able to justify these simplifications for two classes of global-in-time solutions of the M-D 2EP system. The first is the smooth solution of the damped M-D 2EP system near the constant equilibrium state, and the second is the regular solution to the Cauchy problem for the undamped M-D 2EP system with far-field vacuum. The global-in-time error estimates between solutions to 2EP and 1EP are also established.

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Date/Time : 15 Oct 2025 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Vladimir Spokoiny, Weierstrass Institute for Applied Analysis and Stochastics (WIAS)

The recent paper Gao et al. (2023) on estimation and inference for top-ranking problem in Bradley-Terry-Luce (BTL) model presented a surprising result: component-wise estimation and inference can be done under much weaker conditions on the number of comparison then it is required for the full dimensional estimation. The present paper revisits this finding from completely different viewpoint. Namely, we show how a theoretical study of estimation in sup-norm can be reduced to the analysis of plug-in semiparametric estimation. For the latter, we adopt and extend the general approach from Spokoiny (2025b) to high-dimensional estimation and inference. The main tool of the analysis is a theory of perturbed marginal optimization when an objective function depends on a low-dimensional target parameter along with a high-dimensional nuisance parameter. A particular focus of the study is the critical dimension condition. Full-dimensional estimation requires in general the condition n ≫ p between the effective parameter dimension p and the effective sample size corresponding to the smallest eigenvalue of the Fisher information matrix . Inference on the estimated parameter is even more demanding: the condition n ≫ p^2 cannot be generally avoided; see Spokoiny (2025b). However, for the sup-norm estimation, the critical dimension condition can be reduced to n ≥ C log p.

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