The Brown analysis seminar typically meets on Mondays at 4pm in Kassar 105.
The Brown analysis seminar typically meets on Mondays at 4pm in Kassar 105.
Schedule of talks, Fall 2026
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Past talks, Spring 2026
Title: Endpoint estimates for Fourier multipliers with Zygmund singularities
Abstract: The Hilbert transform maps L¹ functions into weak-L¹ ones. In fact, this estimate holds true for any operator T(m) defined by a bounded Fourier multiplier m with singularity only in the origin. Tao and Wright identified the space replacing L¹ in the endpoint estimate for T(m) when m has singularities in a lacunary set of frequencies, in the sense of the Hörmander-Mihlin condition.
In this talk we will quantify how the endpoint estimate for T(m) for any arbitrary m is characterized by the lack of additivity of its set of singularities . This property of the set of singularities of m is expressed in terms of a Zygmund-type inequality. The main ingredient in the proof of the estimate is a multi-frequency projection lemma based on Gabor expansion playing the role of Calderón-Zygmund decomposition.
The talk is based on joint work with Bakas, Ciccone, Di Plinio, Parissis, and Vitturi.
Title: Characterizing rectifiability via metric-valued biLipschitz decompositions
Abstract: Kirchheim proved a Sard-like theorem for metric-valued Lipschitz maps on Euclidean spaces. Namely, given a Lipschitz map $f : \mathbb{R}^n \to Y$, one can give a biLipschitz decomposition of $\R^n$ into Borel pieces $N \cup E_1 \cup E_2 \cup...$ where $f|_{E_i}$ are all biLipschitz and the "critical values" $f(N)$ is null with respect to the Hausdorff $n$-measure. An easy generalization of this gives that all metric-valued Lipschitz maps on rectifiable spaces admit biLipschitz decompositions.
We will show for every $n$-dimensional purely unrectifiable metric space $X$, there is another metric space $Y$ and a Lipschitz function $f : X \to Y$ that has positive $n$-measure image but no biLipschitz pieces. This characterizes rectifiable metric spaces as those spaces for which metric-valued Lipschitz maps admit biLipschitz decompositions, a result new even for Euclidean sets.
This is joint work with Raanan Schul.
Title: Cancellative sparse domination
Abstract: We will discuss a recent improvement on the principle of sparse domination which preserves the cancellative structure of the domain. One of the novelties of the proof is that it avoids the typical weak-type (1,1) approach, which is not strong enough for our purposes. This is a joint work with Emiel Lorist and Guillermo Rey.
Title: On the Philosophy of Geometric Harmonic Analysis
Abstract: A fundamental theme in the theory of partial differential equations, which has profound and intriguing connections with many other subareas of analysis, is the well-posedness of boundary value problems for various classes of elliptic systems, categories of domains, and spaces of boundary data. In the Geometric Harmonic Analysis monograph series a new philosophy has emerged, identifying key interrelationships between these entities which guarantee (Fredholm) solvability. The goal of this talk is to elaborate on this new paradigm. Specifically, I will explain the relevance of the class of elliptic systems possessing distinguished coefficient tensors, touch on the issue of quantifying flatness, and explain what makes a Generalized Banach Function Space friendly from the point of view of harmonic analysis. This is joint work with Dorina Mitrea and Marius Mitrea from Baylor University, USA.
Title: Quantitative versions of Carleson's $\varepsilon^2$-conjecture
Abstract: Since the recent resolution of Carleson's $\varepsilon^2$-conjecture in the plane by Jaye, Tolsa, and Villa, and in higher dimensions by Fleschler, Tolsa, and Villa, it is known that the tangent points of certain domains are characterized via qualitative control on a spectral spherical function. In this talk, we will discuss how quantitative control on this same spectral spherical function characterizes higher-regularity domains. This talk is based on joint works with Xavier Tolsa and Michele Villa, and with Max Engelstein and Tatiana Toro.
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Abstract: TBA
Title: Analysis powers algebraic statistics
Abstract: A long-standing question in algebra is to determine characteristics of a random polynomial. We'll briefly survey one exciting line of inquiry, aimed at non-experts, and show how harmonic analysis played a major role in solving an almost 100 year old conjecture.
Title: Optimal Sparse Bounds and Commutator Characterizations Without Doubling
Abstract: We examine dyadic paraproducts and commutators in the non-homogeneous setting, where the underlying Borel measure $\mu$ is not assumed to be doubling. We first establish a pointwise sparse domination for dyadic paraproducts and related operators with symbols $b \in \mathrm{BMO}(\mu)$, improving on an earlier result of Lacey, where the symbol $b$ was assumed to satisfy a stronger Carleson-type condition that coincides with $\mathrm{BMO}(\mu)$ only in the doubling setting. This allows us to obtain sharpened weighted inequalities for the commutator of a dyadic Hilbert transform $H$ previously studied by T. Borges, J. Conde Alonso, J. Pipher, and N. A. Wagner. We also characterize the symbols for which the commutator $[H, b]$ is bounded on $L^p(\mu)$ for $1 < p < \infty$, showing that this class of symbols strictly depends on $p$ and is nested between symbols satisfying the $p$-Carleson packing condition and symbols in $\mathrm{BMO}$. This is joint work with F. D'Emilio, Y. Lin, and N. A. Wagner.
Title: The Dirichlet problem as the boundary of the Poisson problem.
Abstract. We will describe a novel approximation result about how solutions to the Dirichlet problem for second-order real elliptic PDEs with boundary data in Lp on rough domains may be globally approximated, up to the boundary in a precise sense, by a family of solutions to certain corresponding inhomogeneous Poisson problems with null boundary data. This result is new even for the Laplacian and on the unit ball, but is shown in high geometric generality, as well as with minimal assumptions on the coefficients.
This approximation result was inspired by our second main result: we fully characterize the dual space to the space of functions whose Kenig-Pipher modified non-tangential maximal function lies in Lp, answering a question of Hytonen and Rosen. We show that the dual space consists of exactly two components, and each of them correspond to the Banach spaces in which the Dirichlet and Poisson problems are solved with control of the Lp norm of the non-tangential maximal function. Moreover, one component is the weak-* boundary of the other component. This relationship at the level of the data in fact "lifts" to the level of solution spaces to the boundary value problems, at least partially, allowing us to make the following statement precise: the Dirichlet problem lies on the boundary of the Poisson problem, in a certain topology.
These results may be thought of as deeper instances of the robust relationship between (singular) boundary value problems and the inhomogeneous problems recently investigated in the literature. We start our talk with a discussion of these results. This talk contains joint works with Mihalis Mourgoglou, Xavier Tolsa, and Martin Ulmer.
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If you have questions or comment, please contact the organizers:, Jill Pipher. Sergei Treil, or Martin Ulmer