### TBA by Vitali Vougalter

- Series
- Analysis Seminar
- Time
- Wednesday, February 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
- Location
- ONLINE
- Speaker
- Vitali Vougalter – University of Toronto

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- Series
- Analysis Seminar
- Time
- Wednesday, February 24, 2021 - 14:00 for 1 hour (actually 50 minutes)
- Location
- ONLINE
- Speaker
- Vitali Vougalter – University of Toronto

- Series
- Analysis Seminar
- Time
- Wednesday, February 10, 2021 - 14:00 for 1 hour (actually 50 minutes)
- Location
- https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
- Speaker
- Manasa Vempati – Washington University in St Louis

For (X,d,w) be a space of homogeneous type in the sense of Coifman and Weiss, suppose that u and v are two locally finite positive Borel measures on (X,d,w). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderon--Zygmund operator T from L^{2}(u) to L^{2}(v) in terms of the A_{2} condition and two testing conditions. The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

We also give the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type. These commutators turn out to be controlled by the sparse operators in the setting of space of homogeneous type. The lower bound of the maximal commutator is also obtained.

Zoom link:

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

- Series
- Analysis Seminar
- Time
- Wednesday, February 10, 2021 - 02:00 for 1 hour (actually 50 minutes)
- Location
- Speaker
- Nathan Wagner – Washington University, St Louis – nathanawagner@wustl.edu

- Series
- Analysis Seminar
- Time
- Wednesday, January 27, 2021 - 14:00 for 1 hour (actually 50 minutes)
- Location
- https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
- Speaker
- Michael Lacey – Georgia Tech

An initial result of Bourgain and Chang has lead to a number of striking advances in the understanding of polynomial extensions of Roth's Theorem.

The most striking of these is the result of Peluse and Prendiville which show that sets in [1 ,..., N] with density greater than (\log N)^{-c} contain polynomial progressions of length k (where c=c(k)). There is as of yet no corresponding result for corners, the two dimensional setting for Roth's Theorem, where one would seek progressions of the form(x,y), (x+t^2, y), (x,y+t^3) in [1 ,..., N]^2, for example.

Recently, the corners version of the result of Bourgain and Chang has been established, showing an effective bound for a three term polynomial Roth theorem in the finite field setting. We will survey this area. Joint work with Rui Han and Fan Yang.

The link for the seminar is the following

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

- Series
- Analysis Seminar
- Time
- Tuesday, November 24, 2020 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Online
- Speaker
- Carlos Cabrelli – University of Buenos Aires – cabrelli@dm.uba.ar

- Series
- Analysis Seminar
- Time
- Tuesday, November 17, 2020 - 14:00 for 1 hour (actually 50 minutes)
- Location
- https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
- Speaker
- Mariusz Mirek – Rutgers University – mm2809@math.rutgers.edu

**Please Note:** We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages. This is my recent work with Ben Krause and Terry Tao.

- Series
- Analysis Seminar
- Time
- Tuesday, November 10, 2020 - 02:00 for 1 hour (actually 50 minutes)
- Location
- https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
- Speaker
- Bobby Wilson – University of Washington

We will discuss Marstrand's classical theorem concerning the interplay between density of a measure and the Hausdorff dimension of the measure's support in the context of finite-dimensional Banach spaces. This is joint work with David Bate and Tatiana Toro.

- Series
- Analysis Seminar
- Time
- Tuesday, October 27, 2020 - 14:00 for 1 hour (actually 50 minutes)
- Location
- https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
- Speaker
- Yao Yao – Georgia Tech

For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] <= E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will discuss a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities.

I will also discuss another work with Matias Delgadino and Xukai Yan, where we constructed an interpolation curve between any two radially decreasing densities with the same mass, and show that the interaction energy is convex along this interpolation. As an application, this leads to uniqueness of steady states in aggregation-diffusion equations with any attractive interaction potential for diffusion power m>=2, where the threshold is sharp.

- Series
- Analysis Seminar
- Time
- Tuesday, October 20, 2020 - 14:00 for 1 hour (actually 50 minutes)
- Location
- https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
- Speaker
- Robin Neumayer – Northwestern University

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.

We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

- Series
- Analysis Seminar
- Time
- Tuesday, October 6, 2020 - 14:00 for 1 hour (actually 50 minutes)
- Location
- https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
- Speaker
- Joaquim Ortega-Cerda – University of Barcelona

We improve on some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function f with mean zero, then either the size of the zero set of the function or the cost of transporting the mass of the positive part of f to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the Laplacian.

This proves a conjecture of Steinerberger and provides a lower bound of the size of a nodal set of the eigenfunction. Finally, we use a similar technique to provide a measure of how well the points in a design in a manifold are equidistributed. This is a joint work with Tom Carroll and Xavier Massaneda.

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