Large language models (LLM) have impressive performance on hard tasks, but also exhibit brittleness in simple tasks. We describe an experiment on a basic ``sub-reasoning'' task: deciding if an element belongs to a set. The results give a comprehensive picture of the various types of errors that can occur. In the second part of the talk we give a brief overview of the mathematical challenges posed by the goal of understanding how a neural network works, including understanding what an LLM ``knows''. Joint work with Gabor Berend, Lea Hergert, Mark Jelasity and Mario Szegedy.

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 Dobrowolski, Kim and Ramsey's question of whether every NSOP_1 theory has the existence axiom, and the recent example, answering this question, of an NSOP_1 theory without the existence axiom. We will then discuss problems for further collaboration.

In this talk, first I will present a very simplified proof of descent for stably dominated types in ACVF. I will also state a more general version of descent for stably dominated types in any theory, dropping the hypothesis of the existence of invariant extensions. This first part is work with Pierre Simon. Later I will present a whole theory of residual domination for henselian valued fields of equicharacteristic zero. This is joint work with Pablo Cubides and Silvain Rideau Kikuchi. Among other things, we applied the original descent to prove a change of base statement for residual domination. We also show that in any henselian valued field (over an algebraically closed base), a global invariant type is residually dominated if and only if it is orthogonal to the value group, if and only if its reduct in ACVF is stably dominated. The results extend to valued fields with operators.

Given a closed Riemannian manifold with everywhere negative sectional curvature, there exists a unique geodesic inside of every non-trivial free homotopy class. The marked length spectrum is defined to be the function which takes a free homotopy class and returns the length of this geodesic. It was conjectured by Burns and Katok that the marked length spectrum determines a Riemannian metric up to isometry. In this talk, I’ll present the "magnetized" version of this conjecture and discuss recent progress showing that for certain magnetic flows on surfaces, the periods of closed orbits still encode the underlying geometry. This is joint work with Valerio Assenza, Jacopo de Simoi, and Ivo Terek.

Non-probability samples are prevalent in various fields, such as biomedical studies, educational research, and business investigations, owing to the escalating challenges associated with declining response rates and the cost-effectiveness and convenience of utilizing such samples. However, relying on naive estimates derived from non-probability samples, without adequate adjustments, may introduce bias into study outcomes. Addressing this concern, data integration methodologies, which amalgamate information from both probability and non-probability samples, have demonstrated effectiveness in mitigating selection bias. Nonetheless, the efficacy of these methods hinges upon the assumptions underlying the models. This paper introduces innovative and robust data integration approaches, notably a semi-parametric quantile regression-based mass imputation approach and a doubly robust approach that integrates a non- parametric estimator of the participation probability for non-probability samples. Our proposed methodologies exhibit greater robustness compared to existing parametric approaches, particularly concerning model misspecification and outliers. We consider both missing at random and not missing at random scenarios. Theoretical results are established, including variance estimators for our proposed estimators. Through comprehensive simulation studies and real- world applications, our findings demonstrate the promising performance of the proposed estimators in facilitating valid statistical inference. This research contributes to the advancement of robust methodologies for handling non-probability samples, thereby enhancing the reliability and validity of research outcomes across diverse domains.

We propose a new family of regression models for analyzing categorical responses, called multinomial link models. It consists of four classes, namely, mixed-link models that generalize existing multinomial logistic models and their extensions, two-group models that can incorporate the observations with NA or unknown responses, dichotomous conditional link models that handle longitudinal binary responses, and po-npo mixture models that are more flexible than partial proportional odds models. By characterizing the feasible parameter space, deriving necessary and sufficient conditions, and developing validated algorithms to guarantee the finding of feasible maximum likelihood estimates, we solve the infeasibility issue of existing statistical software when estimating parameters for cumulative link models.

We extend Bowen’s specification approach to thermodynamic formalism to flows on complete separable metric spaces. The key point, particularly for the existence of a finite equilibrium state, is a Strong Positive Recurrence (SPR) assumption. As one application, we establish that for a sufficiently regular potential with SPR for the geodesic flow on a geometrically finite locally CAT(-1) space, there exists a unique equilibrium state. Examples of CAT(-1) spaces range from manifolds with negative curvature bounded above by -1, and at the other extreme, graphs and trees equipped with a notion of length. This is joint work with Vaughn Climenhaga and Tianyu Wang.

Regularity and VC-dimension are two fundamental notions with many applications in combinatorics and beyond. These notions are related via the result that graphs of bounded VC-dimension have (small) partitions where most pairs of parts have density close to 0 or 1. Recent work has generalized this to hypergraphs, but the quantitative aspects of these results are still far from fully understood. I will present some new results on this problem. Joint work with Asaf Shapira and Yuval Wigderson.

It is well known that asset returns usually do not follow a normal distribution, rather, they have long and fat tails. This paper focuses on the quantile portfolio methodology, which considers the whole distribution of asset returns and employs expected loss as a risk measurement. In particular, we explore statistical properties of tau risk and propose related theories of quantile portfolio optimization. We also introduce portfolio performance terms for the quantile portfolio framework.