I will review recent developments in the study of higher Du Bois and rational singularities in characteristic zero. Then I will discuss new results on inversion of adjunction for higher rational singularities joint with T. Kawakami.

In this talk, we will present recent developments in constructing self-similar singularities in the compressible Euler equations and the nonlinear wave equation, associated with implosion. Our approach combines ODE techniques, weighted energy estimates, compact perturbation methods, and soft functional analysis arguments.

We will have a research seminar this semester on forking, broadly construed, particularly in the setting of unstable first-order theories. Graduate students are particularly encouraged to attend. Continuing from last time (when we motivated the shift from local to global versions of the stable forking conjecture), we will discuss the simple Kim-forking conjecture in NSOP_1 theories, and prove cases of this conjecture from joint work with Baldwin and Freitag, including an infinite-variable global variant in a general NSOP_1 theory, a finite-variable global variant in the case of finite F_Mb, and the full conclusion of the simple Kim-forking conjecture, given enough indices, for forking with realizations of an isolated type with the definable Morley property. All three of these results use a strong version of Kim’s lemma in NSOP_1 theories, due to Kaplan and Ramsey, which says that all Kim-independent Morley sequences exhibit Kim-dividing.

A shower thought that anyone interested in graph theory must have had at some point in their lives is the following: `How “sparse" must a given bipartite graph be, if I know that it has no “dense” subgraphs?’. This curiosity definitely crossed the mind of Polish mathematician K. Zarankiewicz, who asked a version of this question formally in 1951. In the years that followed, many central figures in the development of extremal combinatorics contemplated this problem, giving various kinds of answers. Some of these will be surveyed in the first part of my talk. So far so good, but this is a logic seminar and the title says the words “Model Theory"… In the second part of my talk, I will discuss how the celebrated Szemerédi-Trotter theorem gave a starting point to the study of Zarankiewicz’s problem in “geometric” contexts, and how the language of model theory has been able to capture exactly what these contexts are. I will then ramble about improvements to the classical answers to Zarankiewicz’s problem, when we restrict our attention to semilinear/semibounded o-minimal structures, Presburger arithmetic, and various kinds of Hrushovski constructions. The new results that will appear in the talk have been obtained jointly with Pantelis Eleftheriou.

In 1924, Banach and Tarski proved the following amazing theorem: You can cut up a ball in Euclidean space into some finite number of pieces and reassemble these pieces in such a way that you get two copies of the original ball! We will discuss the proof of this crazy result, what this has to do with a deep notion in group theory called amenability, and why we have not solved the problem of world hunger by doubling and redoubling apples, oranges, potatoes and other spherical foodstuff. No background required except mathematical curiosity.

I will discuss two notions of entropy for the geodesic flow: the topological entropy and the measure-theoretic entropy with respect to the Liouville measure. How these dynamical invariants relate to the underlying Riemannian metric has long been of interest. For negatively curved surfaces, Katok proved that equality of the Liouville and topological entropies characterizes metrics of constant negative curvature. In this setting, Manning asked how these two entropies vary along the normalized Ricci flow; more specifically, he proved the topological entropy is monotonic. The main result of this talk, joint with Erchenko, Humbert, and Mitsutani, is that the same is true for the Liouville entropy. In addition to geometric and dynamical methods, we use microlocal analysis to differentiate the Liouville entropy with respect to a conformal perturbation of the metric.

In this talk I will introduce the notion of a beautiful pair, which generalizes Poizat’s belle paires of stable structures. I will derive some of its basic properties and explain its relation to the strict pro-definability of the space of definable types.

Finite VC-dimension, a combinatorial property of families of sets, was discovered simultaneously in the 70's by Vapnik and Chervonenkis in probabilistic learning theory, and by Shelah in model theory (where it is called NIP). It plays an important role in several areas including machine learning, combinatorics, mathematical logic, functional analysis and topological dynamics. A higher arity generalization of VC-dimension for families of sets in n-fold product spaces (i.e. a bound on the sizes of n-dimensional boxes that can be shattered) is implicit in Shelah's work on n-dependent theories in model theory. Following some preliminary work in Chernikov, Palacin, Takeuchi '14, in Chernikov, Towsner '20 we developed aspects of higher-arity VC-theory, including a generalization of Haussler's packing lemma for families of sets (and real-valued functions) of bounded VC_n-dimension. Probably Approximately Correct (PAC) learning is a classical framework for mathematical analysis of machine learning, and PAC learnability is famously characterized by finite VC dimension. Generalizing this, we demonstrate that finite VC_n dimension characterizes higher arity PAC learning (PAC_n learning) in n-fold product spaces with respect to product measures introduced by Kobayashi, Kuriyama and Takeuchi '15. Joint work with Henry Towsner.

A useful vanishing theorem for understanding characteristic zero singularities is Grauert-Riemenschneider vanishing, which asserts that if f: Y -> X is a projective birational morphism and Y is smooth, then higher pushfowards of \omega_Y vanish. A remarkable consequence of this result is that characteristic zero klt singularities are rational. As one could expect, this vanishing theorem fails in positive characteristic. In this talk, we will explain how to prove a Witt vector version of Grauert-Riemenchneider vanishing, and consequences on the Witt-rationality of certain singularities in positive characteristic.

A well-known conjecture in physical literature states that high frequency waves propagating over long distances through turbulence eventually become complex Gaussian distributed. The intensity of such wave fields then follows an exponential law, consistent with speckle formation observed in physical experiments. Though fairly well-accepted and intuitive, this conjecture is not entirely supported by any detailed mathematical derivation. In this talk, I will discuss some recent results demonstrating the Gaussian conjecture in a weak-coupling regime of the paraxial approximation. The paraxial approximation is a high frequency approximation of the Helmholtz equation, where backscattering is ignored. This takes the form of a Schrödinger equation with a random potential and is often used to model laser propagation through turbulence. In particular, I will describe a diffusive scaling where the limiting probability distribution of the wavefield is completely described by a second moment which follows an anomalous diffusion. The proof relies on the asymptotic closeness of statistical moments of the wavefield under the paraxial approximation, its white noise limit and the complex Gaussian distribution itself. An additional stochastic continuity/tightness criterion allows to show the convergence of these distributions over spaces of Hölder-continuous functions. Numerical simulations illustrate theoretical results. This is joint work with Guillaume Bal.

We show that k-ary functions giving the measure of the intersection of multi-parametric families of sets in probability spaces, e.g. $(x,y,z)\in X\times Y\times Z\mapsto \mu(P_{x,y}\cap Q_{x,z}\cap R_{y,z})$, satisfy a particularly strong form of hypergraph regularity. More generally, this applies to the (integral) averages of continuous combinations of functions of smaller arity. This result is connected to higher arity stability in (continuous) model theory. In relation to that, we demonstrate that all 3-hypergraphs embedding both into the half-simplex and into $GS(\mathbb{F}_3)$, the two known sources of failure of ternary stability, do satisfy an analogous regularity lemma -- hence, unlike classical stability, strong ternary stability cannot be characterized simply by excluded hypergraphs.

In the era of digital assessments, large-scale educational data—such as that from NAEP—offers unprecedented opportunities for insight into student learning skills. Yet, the complexity and volume of this data often outpace traditional statistical approaches, calling for a fusion of statistical rigor, data science innovation, and AI-driven modeling. This talk explores a research initiative at ETS, supported by the Gates Foundation, that helps to transform multi-source NAEP data (response, process, and behavioral) into actionable insights for educators. We will discuss how statistics and data science form the foundation for extracting meaningful patterns, visualizing complex data, and how human-centered AI enables scalable, interpretable feedback with subject-matter experts and teachers. The presentation will also reflect on the speaker’s own professional evolution—from classical statistics to data science and to AI applications in the education measurement field —highlighting the synergies between these disciplines. Faculty and graduate students interested in statistical modeling, educational measurement, and AI applications are invited to join the discussion.

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We introduce the dynamical commensurator group for a generalized odometer action, that is for minimal equicontinuous group actions on Cantor sets. We show there is a map from the pointed mapping class group of a solenoidal manifold (ie a weak solenoid) to a dynamical commensurator group, and give conditions for when this map is either surjective or an isomorphism. Odden proved that this map is an isomorphism for the mapping class of the universal hyperbolic solenoid; Bering and Studenmund proved that the mapping class group of a universal solenoid over a compact K(G,1) manifold maps onto the commensurator group of G. We extend the results of both of these papers to arbitrary solenoidal manifolds. This work is joint with Olga Lukina.

We generalize an intersection-theoretic local inequality of Fulton–Lazarsfeld to weighted blowups. Using this together with the classification of 3-dimensional divisorial contractions, we prove nonrationality of many families of terminal Fano 3-folds. This is a joint work with Igor Krylov and Takuzo Okada.

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