Cubic fourfolds have been classically studied up to birational equivalence, with an eye towards rationality problems. On the other hand, the Fano variety of lines F(X) on a cubic fourfold X is a hyperkahler manifold, and the rationality/irrationality of X is conjecturely reflected in the geometry of the Fano variety of lines.  We give examples of conjecturally irrational cubic fourfolds with birationally equivalent Fano varieties of lines. Two of our examples, which are special families in C_12, provide new examples of pairs of cubic fourfolds with equivalent Kuznetsov components. Further, we show the cubic fourfolds themselves are birational. Our examples were discovered by studying the group of birational transformations of the Fano varieties of lines of these cubic fourfolds. This is joint work with Corey Brooke and Sarah Frei, building on our previous work with Xuqiang Qin. 

We show that for any distinct n-bit strings x and y, there is a deterministic finite automaton on n^{1/3} states that accepts x but not y. The methods involve complex analysis and elementary number theory.

Lattice models play a pivotal role in the investigation of microscopic multi-particle systems, with their continuum limits forming the foundation of macroscopic effective theory. These models have found wide-ranging applications in condensed matter physics, numerical analysis, and analysis of PDEs. In this talk, I will present our recent work on the continuum limits of some lattice models to the corresponding nonlinear dispersive equations. Using the integrable Ablowitz–Ladik system as a prototype, we establish that solutions of this discrete model converge to solutions of either the cubic nonlinear Schrödinger equation (NLS) or the modified Korteweg–de Vries equation (mKdV) in certain limiting regimes. Notably, we consider white-noise-like initial data which excites Fourier modes throughout the circle, and demonstrate convergence to a system of NLS/mKdV. This result suggests that a sole continuum equation may not suffice to encapsulate the lattice dynamics in such a low-regularity setting akin to thermal equilibrium. I will also outline the framework of our proof and discuss its broader implications, including its extension to more general lattice approximations of dispersive PDEs. In particular, our approach provides new insights into constructing dynamics for the Landau–Lifshitz spin model in its Gibbs state.

Most statistical models for networks focus on pairwise interactions between nodes. However, many real-world networks involve higher-order interactions among multiple nodes, such as co-authors collaborating on a paper. Hypergraphs provide a natural representation for these networks, with each hyperedge representing a set of nodes. The majority of existing hypergraph models assume uniform hyperedges (i.e., edges of the same size) or rely on diversity among nodes. In this work, we propose a new hypergraph model based on non-symmetric determinantal point processes. The proposed model naturally accommodates non-uniform hyperedges, has tractable probability mass functions, and accounts for both node similarity and diversity in hyperedges. For model estimation, we maximize the likelihood function under constraints using a computationally efficient projected adaptive gradient descent algorithm. We establish the consistency and asymptotic normality of the estimator. Simulation studies confirm the efficacy of the proposed model, and its utility is further demonstrated through edge predictions on several real-world datasets.

We study the cone of pseudoeffective divisors on moduli spaces of K3 surfaces. We give numerical criteria for when (the irreducible components of) a Noether-Lefschetz divisor on these moduli spaces is an extremal ray of the pseudoeffective cone and use this to exhibit many new extremal divisors. We also discuss the question of whether the pseudoeffective cone is generated by Noether-Lefschetz divisors.

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In regression models fitted to data from complex survey designs, sampling weights often incorporate non-essential variation, inflating variance estimates. Stabilised weights mitigate this issue by adjusting sampling weights to account for variation explained by covariates. We evaluate the performance of optimal stabilised weights and propose combining the stabilised weights estimator with generalised raking, a class of efficient design-based estimators. This combination improves efficiency by reducing unnecessary weight variation and leveraging information from auxiliary variables. We show this combination can be implemented using the standard statistical package that handles two-phase samples and generalised raking. Simulation studies demonstrate that the proposed estimator enhances precision under realistic two-phase designs, though efficiency gains may be limited in highly informative designs.

Given an family of polarized abelian varieties, let $\mathcal{L}$ be a relatively ample line bundle, consider the family effective divisors on the fibers or equivalently the space of theta-functions induced by $\mathcal{L}$. Mumford-Welters work endowed this family with a flat projective connection which is realized via the heat equations. Hitchin's paper on {\em Flat connections and geometric quantizations, 1990} generalizes the above construction. Namely, for a complex simply connected Lie group $G$ and a surface $\Sigma$, the Hitchin connection is a flat projective connection on a vector bundle over $\mathcal{M}_g$, whose fiber over a compact Riemann surface $C$ ($\Sigma$ and a complex structure) is the space of holomorphic sections of natural line bundle $\mathcal{L}$ on the moduli space of principal $G$ bundles on $C$. Hitchin's work can be realized as an initial step in the direction of Witten's proposal of quantization of Chern-Simons theory via the method of geometric quantization. In this talk, we will discuss the construction of a Hitchin-type connection for the moduli of parabolic bundles. We will start with a more general framework of constructing connections from heat operators with a given symbol map due to van Geeman-de Jong. We will also identify our connection with the Wess-Zumino-Witten/Tsuchiya-Ueno-Yamada connection in conformal field theory generalizing a result of Laszlo. If time permits, we will make further comments on the abelian case. This is a joint work with Indranil Biswas and Richard Wentworth.

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Let $X$ be a K3 surface. Consider a finitely supported probability measure $\mu$ on $\operatorname{Aut}(X)$ such that $\Gamma_{\mu} = \langle \operatorname{Supp}(\mu)\rangle < Aut(X)$ is non-elementary. We do not assume that $\Gamma_{\mu}$ contains any parabolic elements. We study and classify hyperbolic ergodic $\mu$-stationary probability measures on $X$.

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Hamiltonian systems have long been a cornerstone of classical mechanics, providing a powerful framework to describe and analyze the motion of physical systems. But what happens when these mathematical structures take flight? In this talk, we will explore how the geometric principles of Hamiltonian mechanics play a crucial role in the modeling and control of modern aerial vehicles, including drones. From symplectic structures and variational principles to optimal control and real-world applications, we will uncover the elegant mathematical tools that bridge fundamental physics with cutting-edge drone technology. Whether you're interested in geometry, control theory, or just fascinated by the math behind autonomous systems, this talk will offer a compelling journey from theory to application in the skies.

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