The Brown analysis seminar typically meets on Mondays at 3pm in Kassar 105.
The Brown analysis seminar typically meets on Mondays at 3pm in Kassar 105.
Current schedule of talks, Fall 2025
Title: Vector-Valued Concentration on the Symmetric Group
Abstract: Existing concentration inequalities for functions that take values in a general Banach space, such as the classical results of Pisier, are known only in very special settings, such as the Gaussian measure on R^n and the uniform measure on the discrete hypercube {-1,1}^n. We present a novel vector-valued concentration inequality for the uniform measure on the symmetric group which goes beyond the product setting of the prior known results. Furthermore, we discuss the implications of this result for the non-embeddability of the symmetric group into Banach spaces of nontrivial Rademacher type, which is of interest in studying the metric geometry of Banach spaces. The proof uses a variety of tools related to concentration of Markov semigroups, including discrete analogs of Ricci curvature, and methods of proof for various functional inequalities of independent interest. This talk is based on joint work with Ramon van Handel.
Title: Quantitative rectifiability and singular integrals in Heisenberg groups
Abstract: Rectifiable sets extend the class of surfaces considered in classical differential geometry; while admitting a few edges and sharp corners, they are still smooth enough to support a rich theory of local analysis. However, for certain questions of global nature the notion of rectifiability is too qualitative. In a series of influential papers around the year 1990, David and Semmes developed an extensive theory of quantitative rectifiability in Euclidean spaces. A motivation for their efforts was the significance of a geometric framework for the study of certain singular integrals and their connections to removability.
We will discuss recent results which lay the foundations for a theory of quantitative rectifiability for $1$-dimensional and $1$-codimensional subsets of Heisenberg groups. As in the Euclidean case, partial motivation stems from questions involving singular integrals and removability. We will see that, in certain aspects, the situation is very different than in Euclidean spaces and new phenomena appear.
Title: Minkowski type problems: The Gauss Image Problem
Abstract:
The Gauss Image problem is a generalization of a question originally posed by Aleksandrov, who studied the existence and uniqueness of convex bodies with the prescribed Aleksandrov integral curvature (relating to the sizes of normal cones). With convexity at its core, various techniques have been applied to different aspects of this geometric problem, ranging from Aleksandrov's topological arguments to more recent optimal mass transport reformulations. In this talk, we will explore these connections and provide an overview of recent developments on the problem. Moreover, we will also juxtapose discrete and smooth versions of the problem and explore how smooth formulation of associated Monge–Ampère–type equations transfers to natural combinatorial arguments upon discretization.
Title: Independence of Weyl-Heisenberg coherent states: the HRT Conjecture
Abstract:
In 1932, J.~Von Neumann proposed that the coherent states $$\mathcal{G}(g, 1,1)=\{g_{n,k}(t):=e^{2\pi i k t}g(t-n): k , n \in \mathbb{Z}\},$$ generated by the Gaussian $g(t)=e^{-\pi t^2}$, span a dense subspace of $L^2(\R)$. A decade later, D. Gabor independently formulated a similar idea in the context of communication theory.
These foundational observations gave rise to modern time–frequency analysis, which centers on coherent state systems of the form
$$\mathcal{G}(g, \Lambda)=\{g_{\lambda}(t):=e^{2\pi i \lambda_2 t}g(t-\lambda_1): \lambda=(\lambda_1, \lambda_2) \in \Lambda \},$$ where $g\in L^2(\mathbb{R})$ and $\Lambda\subset \mathbb{R}^2$ is a discrete set.
Within this framework, C. Heil, J. Ramanathan, and P. Topiwala (1996) conjectured that for any nonzero square-integrable function $g$
and any finite subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2$, the corresponding (finite) Weyl-Heisenberg system $$\mathcal{G}(g, \Lambda)={e^{2\pi i b_k \cdot}g(\cdot - a_k)}_{k=1}^N$$ is linearly independent. This statement, now known as the HRT Conjecture, remains a central open problem in time–frequency analysis despite sustained efforts over nearly three decades.
In this talk, I will first provide a historical overview of the conjecture, and then present recent progress on cases where the discrete set $\Lambda$ is at most five.
Title: Multiple pins in distance graphs
Abstract:
Given a compact set $E \subset \mathbb{R}^d$, the distance set is
$\Delta(E) = \{ |x-y| : x,y \in E \}$, and the pinned distance set at $y \in E$ is $\Delta^y(E) = \{ |x-y| : x \in E \}$.
It is conjectured that $\Delta^y(E)$ has positive Lebesgue measure for some $y \in E$ whenever $\dim(E) > d/2$.
More generally, one can study distance sets associated with chains, trees, triangles, necklaces, and other graphs. For a graph with $n$ vertices and $m$ edges, every $n$-tuple of points in $E$ produces a distance vector in $\R^m$ (recording the lengths of all edges). A natural question is to identify thresholds on $\dim(E)$ ensuring that the resulting set has positive $m$-dimensional Lebesgue measure. This talk, based on joint work with B. Foster, Y. Ou, E. Palsson, and R. Acosta, explores multiply pinned distance sets. We ask: What changes when pins are placed in several (non-consecutive) locations of a graph? Under which thresholds can we still guarantee positive $m$-dimensional Lebesgue measure for the multiply pinned distance set? How does this depend on the number and placement of the pins?
Title: Harmonic Analysis in the Dunkl Setting
Abstract:
In this talk, we will discuss some results in the Dunkl setting of harmonic analysis. We'll first discuss the Dunkl setting and the similarities and differences between the classical setting of harmonic analysis on Euclidean space. We'll then discuss some recent theorems and work in this area which will include the commutator theorem for a family of singular integral operators in the Dunkl setting and the T1 and Tb theorems.
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Schedule of talks, Spring 2026
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If you have questions or comment, please contact the organizers:, Jill Pipher. Sergei Treil, or Martin Ulmer