Seminars and Colloquia by Series

Essentially Coercive Forms and asympotically compact semigroups

Series
Analysis Seminar
Time
Wednesday, March 11, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Isabelle Chalendar Université Paris-Est - Marne-la-Vallée

Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that $\dist(S(t),{\mathcal K}(H))\to 0$ as $t\to\infty$, where ${\mathcal K}(H)$ denotes the space of all compact operators on the underlying Hilbert space. 

Stable phase retrieval for infinite dimensional subspaces of L_2(R)

Series
Analysis Seminar
Time
Wednesday, March 4, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel FreemanSt. Louis University

 The problem of phase retrieval for a set of functions $H$ can be thought of as being able to identify a function $f\in H$ or $-f\in H$ from the absolute value $|f|$.  Phase retrieval for a set of functions is called stable if when $|f|$ and $|g|$ are close then $f$ is proportionally close to $g$ or $-g$.  That is, we say that a set $H\subseteq L_2({\mathbb R})$ does stable phase retrieval if there exists a constant $C>0$ so that
$$\min\big(\big\|f-g\big\|_{L_2({\mathbb R})},\big\|f+g\big\|_{L_2({\mathbb R})}\big)\leq C \big\| |f|-|g| \big\|_{L_2({\mathbb R})} \qquad\textrm{ for all }f,g\in H.
$$
 It is known that phase retrieval for finite dimensional spaces is always stable.  On the other hand, phase retrieval for infinite dimensional spaces using a frame or a continuous frame is always unstable.  We prove that there exist infinite dimensional subspaces of $L_2({\mathbb R})$ which do stable phase retrieval.  This is joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman.

Geometric averaging operators and points configurations

Series
Analysis Seminar
Time
Wednesday, February 26, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eyvindur Ari PalssonVirginia Tech

Two classic questions -- the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem -- both focus on the distance, which is a simple two point configuration. When studying the Falconer distance problem, a geometric averaging operator, namely the spherical averaging operator, arises naturally. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will give a brief introduction to the motivating point configuration questions and then report on some novel geometric averaging operators and their mapping properties.

Optimal measures for three-point energies and semidefinite programming

Series
Analysis Seminar
Time
Wednesday, February 12, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Josiah ParkGeorgia Tech

Given a potential function of three vector arguments, $f(x,y,z)$, which is $O(n)$-invariant, $f(Qx,Qy,Qz)=f(x,y,z)$ for all $Q$ orthogonal, we use semidefinite programming bounds to determine optimizing probability measures for interaction energies of the form $\int\int\int f(x,y,z) d\mu(x)d\mu(y)d\mu(z)$ over the sphere. This approach builds on previous use of such bounds in the discrete setting by Bachoc-Vallentin, Cohn-Woo, and Musin, and is successful for kernels which can be shown to have expansions in a particular basis, for instance certain symmetric polynomials in inner products $u=\langle x,y \rangle$, $v=\langle y,z\rangle$, and $t=\langle z, x \rangle$. For other kernels we pose conjectures on the behavior of optimizers, partially inferred through numerical studies.

A Szemeredi-type theorem for subsets of the unit cube.

Series
Analysis Seminar
Time
Wednesday, January 29, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Vjeko KovacGeorgia Tech

  We are interested in arithmetic progressions in positive measure subsets of [0,1]^d. After a counterexample by Bourgain, it seemed as if nothing could be said about the longest interval formed by sizes of their gaps. However, Cook, Magyar, and Pramanik gave a positive result for 3-term progressions if their gaps are measured in the l^p-norm for p other than 1, 2, and infinity, and the dimension d is large enough. We establish an appropriate generalization of their result to longer progressions. The main difficulty lies in handling a class of multilinear singular integrals associated with arithmetic progressions that includes the well-known multilinear Hilbert transforms, bounds for which still constitute an open problem. As a substitute, we use the previous work with Durcik and Thiele on power-type cancellation of those transforms, which was, in turn, motivated by a desire to quantify the results of Tao and Zorin-Kranich. This is joint work with Polona Durcik (Caltech).

On the Log-Brunn-Minkowski conjecture and other questions

Series
Analysis Seminar
Time
Wednesday, January 22, 2020 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Galyna LivshytsGeorgia Tech

We will discuss certain isoperimetric-type problems for convex sets, such as the Log-Brunn-Minkowski conjecture for Lebesgue measure, and will explain the approach to this type of problems via local versions of inequalities and why it arises naturally. We consider a weaker form of the conjecture and prove it in several cases, with elementary geometric methods.  We shall also consider several illustrative ``hands on’’ examples. If time permits, we will discuss Bochner’s method approach to the question and formulate some new results in this regard. The second (optional!) part of this talk will be at the High-dimensional seminar right after, and will involve a discussion of more involved methods. Partially based on a joint work with Hosle and Kolesnikov.

Small deviation estimates for norms of Gaussian vectors

Series
Analysis Seminar
Time
Wednesday, November 13, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Konstantin TikhomirovGeorgia Tech
Let |.| be a norm in R^n, and let G be the standard Gaussian vector.
We are interested in estimating from above the probabilities
P{|G|<(1-t)E|G|} in terms of t. For 1-unconditional norms
in the L-position, we prove small deviation estimates which match those for the
ell-infinity norm: in a sense, among all 1-unconditional norms in the L-position,
the left tail of |G| is the heaviest for ell-infinity. Results for general norms are also obtained.
The proof is based on an application of the hypercontractivity property combined with
certain transformations of the original norm.
Joint work with G.Paouris and P.Valettas.

Singular Brascamp-Lieb inequalities

Series
Analysis Seminar
Time
Wednesday, November 6, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Polona DurcikCaltech

Brascamp-Lieb inequalities are estimates for certain multilinear forms on functions on Euclidean spaces. They generalize several classical inequalities, such as Hoelder's inequality or Young's convolution inequality. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions in the Brascamp-Lieb inequality is replaced by a singular integral kernel. Examples include multilinear singular integral forms such as paraproducts or the multilinear Hilbert transform. We survey some results in the area. 

 

Quantum graphs, convex bodies, and a century-old problem of Minkowski

Series
Analysis Seminar
Time
Wednesday, October 30, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Yair ShenfeldPrinceton University

That the ball minimizes surface area among all sets of fixed volume, was known since antiquity; this is equivalent to the fact that the ball is the unique set which yields equality in the isoperimetric inequality. But the isoperimetric inequality is only a very special case of quadratic inequalities about mixed volumes of convex bodies, whose equality cases were unknown since the time of Minkowski. This talk is about these quadratic inequalities and their unusual equality cases which we resolved using degenerate diffusions on the sphere. No background in geometry will be assumed. Joint work with Ramon van Handel.

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