On Maz’ya’s \Phi-inequalities.

Speaker: 

Dmitriy Stolyarov

Speaker Link: 

https://chebyshev.spbu.ru/dmitriystolyarov/

Institution: 

St. Petersburg State University

Time: 

Thursday, November 4, 2021 - 11:00am

Location: 

zoom ID: 949 5980 5461. Password: the last four digits of the zoom ID in the reverse order

I will speak about an unusual way to correct the (invalid) endpoint case of the Hardy—Littlewood—Sobolev inequality. Usually the correction is done by imposing additional linear constraints on the function we apply the Riesz potential to. Being the gradient of another function is an example of such a constraint. The inequalities obtained this way are often called Bourgain—Brezis inequalities. In 2010, Maz’ya suggested another approach: instead of constraining the right hand side we should replace the L_p norm on the left with an expression \Phi, which alongside with having the same homogeneity properties as the L_p norm, possesses additional cancellations. He conjectured that if \Phi satisfies a natural necessary condition, then the modified Hardy—Littlewood—Sobolev inequality holds true. I will try to survey the proof of Maz’ya’s conjecture. Based on https://arxiv.org/abs/2109.08014

Estimating average of function on Boolean cube without risk

Speaker: 

March Boedihardjo

Institution: 

UCI

Time: 

Thursday, October 7, 2021 - 11:00am

Location: 

zoom ID: 949 5980 5461. Password: the last four digits of the zoom ID in the reverse order

Strong law of large numbers gives a method to estimate the
average of a function on the Boolean cube so that it is accurate with
high probability. But there is still a little risk that it is
inaccurate. I will present a polynomial time method to estimate the
averages of certain functions on Boolean cube without risk of being
inaccurate.

A moment ratio bound for polynomials on the boolean cube

Speaker: 

Alex Samorodnitsky

Speaker Link: 

https://www.cs.huji.ac.il/~salex/

Institution: 

The Hebrew University of Jerusalem

Time: 

Thursday, November 18, 2021 - 11:00am

Location: 

zoom ID: 949 5980 5461. Password: the last four digits of the zoom ID in the reverse order

We prove an almost tight upper bound on the ratio ||f||_p / ||f||_2, when f is a polynomial of a given degree on the Boolean cube {0,1}^n and describe some applications.  In particular, we describe a family of hypercontractive inequalities for functions on {0,1}^n which take into account the concentration of a function.

CANCELED

Speaker: 

Oscar Dominguez [CANCELED]

Speaker Link: 

https://sites.uci.edu/analysis/

Institution: 

Universidad Complutense de Madrid, Spain

Time: 

Friday, April 10, 2020 - 3:00am

Host: 

Paata Ivanisvili

Location: 

RH 340P

Maximal functions play a central role in the study of differentiation,
singular integrals and almost everywhere convergence. With Sergey Tikhonov
(ICREA, Barcelona) we recently proved some pointwise estimates for maximal
functions in terms of smoothness and rearrangements. I plan to discuss the recent
progress on these topics and some applications. In particular, I will discuss the
Fefferman-Stein inequality for the sharp maximal function for r.i. spaces which
are close to L∞.

The multilinear restriction estimates for hypersurfaces with curvature

Speaker: 

Ioan Bejenaru

Institution: 

UCSD

Time: 

Wednesday, February 19, 2020 - 3:00pm to 3:50pm

Host: 

Hamid Hezari

Location: 

340P

Abstract: I will present an overview of the multilinear restriction theory, with an emphasis on the case when the hypersurfaces have some curvature. I will discuss a new result: the case of n-1 hypersurfaces in n dimensions where a fairly general theory is developed. 

Selberg integral and polynomial identities

Speaker: 

Fedor Petrov

Speaker Link: 

https://sites.uci.edu/analysis/

Institution: 

Steklov Institute of Mathematics at St. Petersburg

Time: 

Monday, February 10, 2020 - 3:00pm to 3:50pm

Host: 

Paata Ivanisvili

Location: 

RH 306

Selberg-type integrals that can be turned into constant term identities for Laurent polynomials arise naturally in conjunction with random matrix models in statistical mechanics. We discuss a general method based on the multi-dimensional polynomial interpolation identity related to the so called Combinatorial Nullstellensatz of Alon that is powerful enough to establish many such identities, both known before and new, in a simple manner. Based on joint works with R. Karasev, G. Karolyi, Z. Nagy and V. Volkov.

Phase transition of capacity for the uniform $G_{\delta}$-sets and another counterexample to Nevanlinna's conjecture

Speaker: 

Fernando Quintino

Institution: 

UCI

Time: 

Wednesday, January 15, 2020 - 3:00pm

Location: 

340P

We consider a family of dense $G_{\delta}$ subsets of $[0,1]$, defined as intersections of unions of small uniformly distributed intervals, and study their capacity. Changing the speed at which the lengths of generating intervals decrease, we observe a sharp phase transition from full to zero capacity. Such a $G_{\delta}$ set can be considered as a toy model for the set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products.

The same mass re-distribution construction that we use to obtain a full capacity statement, allows us to construct another counter-example to a conjecture by Nevanlinna. 

Fourier analysis and discrete structures

Speaker: 

Paata Ivanishvili

Institution: 

UCI

Time: 

Thursday, December 5, 2019 - 11:00am

Location: 

RH 306

Let a_ij be a finite collection of real numbers, and let s_i and s_j be spins (they take only two values +1 or -1). The goal is to choose the spins s_i and s_j so  that to maximize the double sum a_ij s_i s_j, where the summation runs over all indecies from 1 to n such that i does not qual to j.  If a_ij =1 then the sum is maximized if all spins have the same sign which gives the result to be of order n^2. However, if a_ij=-1 then the sum is maximized when half of the spins take value 1 and the other half -1. When a_ij are arbitrary real numbers this becomes a nontrivial problem (also known as Dean's problem). The problem is well understood  in average when a_ij are i.i.d symmetric  +1 or -1 random vairables  (the Sherrington--Kirkpatrick model).  We will speak about possible  lower bounds of  the maximum in terms of arbitrary real numbers a_ij,  its extensions to d-spin case, some conjectures in this area, and their applications in quantum computing. 

Geometry of nodal domains

Speaker: 

Alexander Logunov

Speaker Link: 

https://www.math.princeton.edu/people/aleksandr-logunov

Institution: 

Princeton University

Time: 

Thursday, December 19, 2019 - 11:00am

Location: 

RH 306

Courant's theorem states that the k-th eigenfunction of the Laplace operator on a closed Riemannian manifold has at most k nodal domains. Given a ball of radius r, we will discuss how many of nodal domains can intersect a ball (depending on r and k). Based on a joint work (in progress) with S.Chanillo and E.Malinnikova.

Volume of intersections of convex bodies with their symmetric images and efficient coverings

Speaker: 

Tomasz Tkocz

Speaker Link: 

http://www.math.cmu.edu/~ttkocz/

Institution: 

Carnegie Mellon University

Time: 

Thursday, December 12, 2019 - 11:00am

Location: 

RH 306

Let K be a convex body with volume one and barycentre at the origin.
How small is the volume of the intersection of K and -K? I shall
discuss such lower bounds and present applications to the Hadwidger
covering/illumination conjecture. Based on joint work with H. Huang,
B. Slomka and B. Vritsiou.

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