Spring 2026

2/17/2026, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Fabio Isaza (Case Western Reserve University)
Title: Depth, Compositional Structure, and the Curse of Dimensionality in ReLU Networks
Abstract: Classical approximation theory shows that generic smooth functions on [0, 1]^d require a number of parameters that grows exponentially with dimension, reflecting the so-called “curse of dimensionality”. I will discuss recent results of Poggio and collaborators showing that deep ReLU neural networks can avoid this phenomenon for hierarchically compositional function classes, achieving approximation rates governed by intrinsic structural dimension rather than ambient dimension. These results clarify that the expressive advantage of depth in neural networks arises from its ability to exploit compositional structure, rather than from universality alone.

2/10/2026, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Bartłomiej Zawalski (Case Western Reserve University)
Title: On the homothety conjecture for the body of flotation and the body of buoyancy on a plane
Abstract: Let K be a convex body in R^n, i.e., a compact convex set with non-empty interior. For any $\delta \in (0, vol_n(K))$, the convex body of flotation of K is the set of points that remain above the water level when a solid body of shape K and uniform density $\delta / vol_n(K)$ floats in any orientation.

For each orientation, the centroid of the underwater part is known as the center of buoyancy. The hypersurface of buoyancy is defined as the geometric locus of all the centers of buoyancy. The hypersurface of buoyancy encloses a strongly convex body $B_\delta(K)$, called the body of buoyancy. We will investigate questions related to the homothety conjecture [3] for convex bodies on a plane, which asks whether an ellipse is the only convex body K that is homothetic to the body of flotation $F_\delta(K)$ for some $\delta \in (0, vol_n(K))$. We systematically derive all the fundamental properties of bodies of flotation and bodies of buoyancy and show that if the body of flotation is homothetic to the body of buoyancy, and if every chord of flotation cuts off from the boundary exactly 1/3 of its total affine arc length, then K is an ellipse. As a result, we get an affine counterpart of Zindler carousels introduced by J. Bracho, L. Montejano, and D. Oliveros [2]. Our result is also related to the classical floating body problem of Ulam [4, 19. Problem: Ulam] from the Scottish Book, and the work of H. Auerbach [1].

2/3/2026, Tue. 2:30-3:30 pm in Rockefeller 303
Speaker: Mengchun Cai (Case Western Reserve University)
Title: Convergence Rate for the Least Eigenvalue of the Laguerre Unitary Ensemble
Abstract: In this talk, we will derive a rate of convergence regarding the least eigenvalue of the Laguerre Unitary Ensemble (LUE) by discussing its corresponding point process. We will show that with a rate of order $N^\frac{2}{3}$, the scaled least eigenvalue of an LUE converges toward the Tracy-Widom law in distribution. To this end, we will establish necessary large $N$ expansions for the Laguerre polynomial by the approximation of the Whittaker function. Some tools will be introduced in this talk to facilitate the analysis of the Whittaker equation.