Fall 2025

Marco Fraccaroli (UMass Lowell): Convergence of the inverse Fourier transform and the family of bilinear Hilbert transforms

Abstract: The Fourier transform decomposes a function into frequencies, and it is a powerful tool in analysis: it "maps" convolutions into multiplications, and derivatives into polynomials. When applied twice to a smooth and rapidly decaying function f, the Fourier transform reproduces f pointwise, up to a reflection about the origin. Therefore, it can be thought of as its own inverse. But how does the inversion converge to the initial datum for general function? How well is a signal f on the real line reconstructed as a superposition of increasingly many of its frequencies?

The problem was solved by Carleson in 1966 and it is essentially governed by the Carleson maximal operator. This operator exhibits the same type of symmetries of the family of bilinear Hilbert transforms (BHTs), which appear also in other contexts in analysis, for example studying Cauchy intergrals on Lipschitz curves. In this talk, I will give a gentle introduction to the time-frequency analysis of these operators, reviewing some classical results in the area. In particular, we will consider how the operators generalize when we increase the dimension of the domain of f.

The new results are based on joint work with Olli Saari (Universitat Politècnica de Catalunya) and Christoph Thiele (Universität Bonn).

Marco Fraccaroli's Website


Rishi Sonthalia (Boston College): Universal approximation results for solving partial differential equations (PDEs) with neural network

Abstract: There has been significant recent work on solving PDEs using neural networks on infinite dimensional spaces. In this talk we consider two examples. First, we prove that transformers can effectively approximate the mean-field dynamics of interacting particle systems exhibiting collective behavior, which are fundamental in modeling phenomena across physics, biology, and engineering. We provide theoretical bounds on the approximation error and validate the findings through numerical simulations. Second, we show that finite dimensional neural networks can be used to approximate eigenfunction for the Laplace Beltrami operator on manifolds. We provide quantitative insights into the number of neurons needed to learn spectral information and shed light on the non-convex optimization landscape of training.

Rishi Sonthalia's Website


Ziming Shi (UC Irvine): Deformation theory of complex structures on manifolds with boundary

Abstract: In the late 1970s, R. Hamilton initiated a program to extend the Kodaira-Spencer's elliptic deformation theory of complex structures to manifolds with boundary. The stable case can be stated as follows. Let D be a relatively compact domain in a complex manifold M with certain complex analytic geometry. Assume H1(D,T) = 0, where T is the holomorphic tangent bundle of M. Given a formally integrable almost complex structure X defined on the closure D, and provided that X is sufficiently close to the standard complex structure on M, does there exist a complex/holomorphic coordinate that is compatible with X? In other words, does there exist a diffeomorphism from D into M that transforms X into the complex structure on M? Locally near a point inside D, such a diffeomorphism always exists by the classical Newlander-Nirenberg theorem. Thus we also call this problem the global or boundary Newlander-Nirenberg problem. In this talk I will present some recent progress on the existence of such diffeomorphism with almost sharp regularity, on a large class of domains with finite smooth boundaries and finite smooth almost complex structure. The talk is partially based on joint work with Xianghong Gong.

Ziming Shi's Website


Xueyin Wang (Texas A&M University): Discrepancy estimates and quantum dynamics

Abstract: In this talk, we present new quantum dynamical upper bounds for one-dimensional analytic long-range operators with ergodic potentials, which improve over several previously known results. Our approach is based on novel discrepancy estimates for semi-algebraic sets. Specifically, for shift dynamics, we establish an asymptotically sharp discrepancy bound by constructing a matching lower bound. For skew-shift dynamics, we reduce the exponential upper bound to a polynomial one by Vinogradov method.

Xueyin Wang's Website


Lauren Rose (Bard College): Quads: A SET-like Game with a twist

Abstract: Quads is a SET-like card game published as EvenQuads by the AWM. The goal of the game is to find “quads”, which are sets of four cards satisfying a particular pattern. The cards can be viewed as points in the finite affine geometry AG(6,2), and a quad in the card game corresponds to a plane in AG(6,2). We are particularly interested in collections of cards that don’t contain a quad. We will describe a Quads analog to the "Cap Set problem" for SET, and discuss known results. In particular, we address the question of how many cards you must lay out to guarantee a quad.

Lauren Rose's Website


Sergi Elizalde (Dartmouth College): Descents of permutations with only even or only odd cycles

Abstract: It is known that, when n is even, the number of permutations of {1,2,...,n} all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Hegedüs and Roichman recently found a surprising refinement of this equality, showing that it still holds when restricting to permutations with a given descent set J on one side, and permutations with ascent set J on the other. Their proof is algebraic and uses heavy machinery. It also yields a version for odd n.

In this talk we give a bijective proof of their result. First, using some beautiful bijections of Gessel, Reutenauer and others, we restate it in terms of multisets of necklaces, which we interpret as words. Then, we construct a bijection between words all of whose Lyndon factors have odd length and are distinct, and words all of whose Lyndon factors have even length.

Sergi Elizalde's Website

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This colloquium is coordinated by: Daniel Glasscock, Amanda Redlich, Joris Roos, Bobbie Wu