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 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. We will discuss more cases of the simple Kim-forking conjecture proven in joint work with John Baldwin and James Freitag, including a finite-variable global variant of the simple Kim-forking conjecture 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. Time permitting, we will then discuss examples of finite F_Mb and the definable Morley property, which measure dependence within the indiscernible sequences in a type in quantitative and qualitative ways.

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.

I will start by discussing joint work with P. Bandi and D. Kleinbock on ergodic theorems for dilates of sub-manifolds in mixing $\mathbb{R}^d$-actions – a generalization of spherical averages in ergodic theory. Such theorems have interesting applications to Diophantine approximation: in particular, based on these results, one can show that almost every vector in $\mathbb{R}^d$ is Dirichlet improvable in the multiplicative sense. In the second part of this talk (and time permitting), I will present a further joint work with Jiajun Cheng and Beinuo Guo on the validity of pointwise ergodic theorems as above in connection to the regularity of the test functions.

In this talk, I will give the idea to prove that $DCF_0$ has $2^\lambda$ non-isomorphic models for any cardinal $\lambda$.

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.

TBA

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.

Precision medicine is the future of drug development, and subgroup identification plays a critical role in achieving the goal. In this presentation, we propose a powerful end-to-end solution squant (available on CRAN) that explores a sequence of quantitative objectives. The method converts the original study to an artificial 1:1 randomized trial, and features a flexible objective function, a stable signature with good interpretability, and an embedded false discovery rate (FDR) control. We demonstrate its performance through simulation and provide a real data example.

In many applications of scientific and engineering interest, the accurate modeling of linear waves scattered by quasiperiodic media plays a crucial role. The ability to numerically simulate such configurations robustly and rapidly is of overwhelming importance in photonics applications. In this talk we will discuss the specific problem of electromagnetic radiation interacting with a two-dimensional multiply layered diffraction grating with quasiperiodic interfaces. We describe how the classical boundary perturbation method of Field Expansions can be extended to this two-dimensional problem, and with specific numerical experiments we will show the remarkable efficiency, fidelity, and high-order accuracy one can achieve with an implementation of this algorithm.

In network settings, interference between units makes causal inference more challenging as outcomes may depend on the treatments received by others in the network. Typical estimands in network settings focus on treatment effects aggregated across individuals in the population. We propose a framework for estimating node-wise counterfactual means, allowing for more granular insights into the impact of network structure on treatment effect heterogeneity. We develop a doubly robust and non-parametric estimation procedure, KECENI (Kernel Estimator of Causal Effect under Network Interference), which offers consistency and asymptotic normality under network dependence. The utility of this method is demonstrated through an application to microfinance data, revealing the node-wise impact of network characteristics on treatment effects.

Many geometric spaces come equipped with a natural collection of special submanifolds that reflect their internal symmetries. Examples include abelian varieties with their sub-abelian varieties, locally symmetric spaces with totally geodesic subspaces, period domains with sub–period domains, and strata of abelian differentials with affine invariant submanifolds. In recent years, significant progress has been made in understanding such structures through the lens of unlikely intersections and functional transcendence. I will outline the general framework of variations of Hodge structures and period domains, and explain how the so-called completed Zilber–Pink philosophy provides a unifying way to describe the qualitative behaviour of these special loci. This perspective reveals deep connections between arithmetic geometry, Hodge theory, and dynamical systems.

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.

TBA

TBA