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Schedule of talks:

Abstracts

(A. Yavicoli): Let E be a measurable set of positive Lebesgue measure on the real line. A simple argument using a density point shows that E contains a translated and rescaled copy of every finite subset of the real line. What happens if we replace a finite set by an infinite one? Erdös conjectured that the answer changes completely: for every infinite set A, there should be a measurable set E of positive measure that contains no translated and rescaled copy of A. I will introduce this conjecture, explain some of the ideas surrounding it, and discuss recent progress toward its resolution.

(S. Fenley): Anosov flows are flows that admit invariant stable and unstable bundles, which are respectively contracted or expanded when flowing forward. The Verjosky conjecture states that every codimension one Anosov flow in dimensions 4 or higher is orbitally equivalent to a suspension Anosov flow. Codimension one means that either the weak stable or the weak unstable foliation of the flow has codimension one. This conjecture is 50 years old. In joint work with K. Mann and R. Potrie, we construct infinitely many counterexamples to the conjecture in 4-manifolds. To achieve that, we need a closed hyperbolic 3-manifold M, a faithful minimal representation of \pi_1(M) into Homeo+(S^1) (the circle), and a group equivariant Cannon-Thurston map f from S^1 to the sphere at infinity of hyperbolic 3-space. With this data we can construct a topological Anosov flow in dimension 4. If the set of non injective points of the map f has image in the sphere at infinity which has measure zero, we show how to perturb the topological Anosov flow to obtain a (smooth) Anosov flow, which is orbitally equivalent to it. We then show that there are infinitely many examples satisfying the data, producing counterexamples to the Verjovsky conjecture.

(Y. Yuan): We survey some old and new and the most recent, positive and negative results on a priori estimate, regularity, rigidity, and constant rank results for special Lagrangian and quadratic Hessian equations. These equations originate in calibrated, convex, conformal, and complex geometries among other fields. Unlike all order Hessian equations such as the linear (Laplace) and top order (Monge-Ampere) equations, and the most recently settled quadratic Hessian equations in all dimensions, the regularity/irregularity issue for special Lagrangian equations still remains.

(A. Barvinok): A matching in a graph is a collection of edges such that every vertex is incident to at most one edge of the collection. Let v be the number of vertices of the graph and let d be the maximum degree of a vertex. It turns out that the number of all matchings in the graph, up to a relative error 0 < epsilon < 1, is determined by the number of matchings with k edges for k=O( sqrt{d} ln (v/epsilon)). We will discuss what the Koebe function has to do with it, and how this is related to the absence of a phase transition in monomer-dimer systems.

(A. Gelb): Recovering signals, images, and physical states from noisy or incomplete data is typically framed as a regularization problem: enforce that some transform Lx of the unknown x is sparse. Classical choices such as total variation and its higher-order variants fix a single assumed order of smoothness, which forces a trade-off between staircasing and ringing whenever the true variability changes across the domain-exactly the situation for piecewise smooth signals and for physical states governed by hyperbolic conservation laws, whose solutions develop shocks even from smooth initial data. We introduce the residual transform operator R = T-S, built from two distinct operators, T = S, that nevertheless satisfy Tx \simeq Sx to the same order of approximation in the transform domain. Because T and S are matched in this way regardless of the underlying signal's local smoothness, R acts as an annihilating operator: it is small wherever the underlying signal is smooth and large only at genuine discontinuities, without ever needing to know the signal's variability in advance. Placed inside a generalized sparse Bayesian learning framework, R yields not only a robust point estimate but full uncertainty quantification, with the regularization strength learned automatically from the data rather than hand-tuned. We then carry R into a new setting: variational data assimilation (3D-Var) for hyperbolic conservation laws, where the usual Gaussian background term is ill-suited to a shock's genuinely multimodal position uncertainty. Replacing this with residual-based regularization yields analyses that remain accurate even when the background-error covariance is misspecified, with the advantage over standard baselines growing precisely as observations become sparser-the regime where this structure-preserving approach earns its keep. This research is done in collaboration with Yao Xiao and Adityvikram Viswanathan, and is partially supported by supported by DOD (ONR MURI) #N00014-20-1-2595 and DOE ASCR #DE-SC0025555.

W. Schlag: (Virtual only) The speaker will share some data on the current mathematical landscape that is changing rapidly. Over the past two months we have witnessed enormous advances both in the power of LLMs, a steep increase in the number of submissions to arxiv, as well as to many journals. Editorial boards are searching for strategies to cope with these challenges. The talk will present some of the thoughts that have come up in this process. You will be asked to participate by sharing your ideas and concerns.

(P. Hajlasz): Advanced calculus as we usually know it is formulated in Euclidean spaces and, more generally, on smooth manifolds. In this talk, I will discuss a perhaps surprising fact: advanced calculus can be developed in arbitrary metric spaces. More precisely, I will explain how notions such as differentiability, the implicit function theorem, the change of variables formula, and the area and coarea formulas extend to Lipschitz mappings with values in general metric spaces. This theory was initiated by Bernd Kirchheim in 1994 and has since become a cornerstone of contemporary geometric measure theory. The talk will be elementary and accessible to anyone with a basic background in advanced calculus, metric spaces, and measure theory.

Past Ohio State University Mathematics Department Colloquia

• 2025-2026

• 2024-2025

• 2023-2024

• 2022-2023

• 2020-2021

• 2019-2020

• 2018-2019

• 2017-2018

• 2016-2017

(V. Kobzar): We establish a direct connection between combinatorial discrepancy minimization problems and curvature flows in R^m. This is done by determining the long-time asymptotics of an online version of the classic vector balancing problem, known as the Komlós conjecture, and showing it is exactly determined by the extinction time of mean curvature flow on the m-dimensional cube. Our proof builds upon Kohn and Serfaty's work on deterministic games and mean curvature flow, and Banaszczyk's Euclidean analogue of the Beck-Fiala theorem. As a consequence of this geometric characterization, we show that the leading order term of the value of this game grows as \Theta (\sqrt { T \log m}) as the time horizon T gets large. This is joint work with Nestor Guillen at Courant available here. A geometric PDE-inspired perspective is also fruitful in the more classic, offline Komlós setting, and I will briefly preview our most recent results in this direction at the end of the talk.

(R. Fang): Reliable prediction of fluid flows is challenging because small errors can grow rapidly. Incorporating observational data can improve predictions; however, observations are often incomplete, and mathematical models are imperfect. This talk will focus on nudging-based data assimilation, which uses observational feedback to improve simulated flows. I will present adaptive algorithms for selecting the feedback strength, a modular framework that separates forecasting from observational correction, and local nudging methods that recover global flows from observations confined to part of the domain. I will also discuss data assimilation with model errors, including omitted rotational effects and the use of incompressible models for slightly compressible flows. Through rigorous analysis and numerical experiments, I will show how these methods address parameter selection, extensibility, limited observations, and model mismatch, with long-time accuracy guarantees under suitable assumptions. I will conclude with a brief overview of complementary research on simplified turbulence models, efficient ensemble algorithms, adaptive penalty methods, and optimization-based structure-preserving methods for nonlinear dispersive wave equations, highlighting a broader research program in reliable and efficient scientific computing.

(A. Luna): We give an overview of how various tools from dynamical systems and dimension theory have been used in the study of the spectral theory of certain ergodic Schr\"odinger operator models. In particular, we describe how tools from smooth hyperbolic dynamics have played a role in understanding the topological properties of the spectra of such operators with Sturmian potentials. We also discuss ideas concerning sums of Cantor sets and their connections to questions about the spectral properties of certain multidimensional models.

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