Georgia Institute of Technology College of Sciences

We discuss the recent progress on Graham's Conjecture which states that for any subset $S \subseteq \mathbb{Z}_p \setminus \{0\}$, there exists an ordering of the elements $s_1, \cdots, s_m$ of $S$ such that the partial sums $\sum_{i = 1}^k s_i$ are all distinct. This was very recently proven for all sufficiently large primes by Pham and Sauermann, however our work focuses on the more general setting where $\mathbb{Z}_p$ is replaced by an arbitrary finite group, where the result is also conjectured to hold.

By considering the Cayley Graph, we can translate the problem into the purely graph theoretic problem of finding a rainbow path of length $d - 1$ in any $d$-regular properly edge-colored directed graph. We give an asymptotic result which builds on work by Bucić, Frederickson, Müyesser, Pokrovskiy, and Yepremyan, and shows that we can find a path of length $(1 - o(1)) d$. This corresponds to showing that for any subset $S \subseteq G$, there exists a dense subset $S' \subseteq G$ and an ordering $s'_1, \cdots, s'_m$ of the elements of $S'$ such that the partial products $\prod_{i  = 1}^k s'_i$ are all distinct.

While foundation models have gained considerable attention in core AI fields such as natural language processing (NLP) and computer vision (CV), their application to learning complex responses of physical systems from experimental measurements remains underexplored. In physical systems, learning problems are often characterized as discovering operators that map between function spaces, using only a few samples of corresponding function pairs. For instance, in the automated discovery of heterogeneous material models, the foundation model must be capable of identifying the mapping between applied loading fields and the resulting displacement fields, while also inferring the underlying microstructure that governs this mapping. While the former task can be seen as a PDE forward problem, the later task frequently constitutes a severely ill-posed PDE inverse problem.

In this talk, we will explore the attention mechanism towards a foundation model for physical systems. Specifically, we show that the attention mechanism is mathematically equivalent to a double integral operator, enabling nonlocal interactions among spatial tokens through a data-dependent kernel that characterizes the inverse mapping from data to the hidden PDE parameter field of the underlying operator. Consequently, the attention mechanism captures global prior information from training data generated by multiple systems and suggests an exploratory space in the form of a nonlinear kernel map. Based on this theoretical analysis, we introduce a novel neural operator architecture, the Nonlocal Attention Operator (NAO). By leveraging the attention mechanism, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and enhancing generalizability. To demonstrate the applicability of NAO to material modeling problems, we apply it to the development of a foundation constitutive law across multiple materials, showcasing its generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning an interpretable foundation model of physical systems, but also offers a new perspective towards understanding the attention mechanism.

Quadratic Gromov--Witten invariants allow one to count curves on varieties over a field k satisfying geometric constraints while keeping track of arithmetic information about those curves. In particular, k does not need to be the field of complex or real numbers. These invariants were developed in joint work with Kass, Levine, and Solomon in genus 0 for del Pezzo surfaces. In this talk we will compute these invariants for rational del Pezzo surfaces of degree >5. To do this, we give these invariants the structure of an unramified Witt invariant for any fixed surface and degree. We then construct a multivariable unramified Witt invariant which conjecturally contains all of these invariants for k-rational surfaces. We prove this conjecture in degree >5. To do this, we study the behavior of these Gromov–Witten invariants during an algebraic analogue of surgery on del Pezzo surfaces. We obtain a surprisingly simple formula when uncomputable terms cancel out with an identity in (twisted) binomial coefficients in the Grothendieck–Witt group. This is joint work with Erwan Brugallé and Johannes Rau.

In this talk we consider the Hartree energy functional for an ensemble of identical fermions. We prove that the minimizers satisfy an important set of semi-classical commutator estimates (SCEs), which encode the uniform regularity of the states in the semi-classical parameter. In the recent years, the SCEs have been shown to play a key role in the quantitative derivation of Hartree-Fock and Vlasov dynamics from large systems of fermions, and are typically implemented as assumptions in the initial data. Proving the validity of such estimates is, however, not an easy task; up to recently only the linear smooth case was understood. In our work, we provide the first set of examples of states satisfying the SCEs which arise from a nonlinear minimization problem. In particular,  two-body interactions up to the repulsive Coulomb potential are included.  If time allows it, I will present — as an application — the quantitative convergence of the fermonic N-body ground state, towards the minimizer of the Vlasov functional.  Based on joint work with L. Lafleche (ENS Lyon).