Difference between revisions of "Fall 2017 and Spring 2018 Analysis Seminars"

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'''Analysis Seminar
 +
'''
 +
 +
The seminar will  meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
 +
 +
If you wish to invite a speaker please  contact  Brian at street(at)math
 +
 +
===[[Previous Analysis seminars]]===
 +
 +
= Analysis Seminar Schedule =
 +
{| cellpadding="8"
 +
!align="left" | date 
 +
!align="left" | speaker
 +
|align="left" | '''institution'''
 +
!align="left" | title
 +
!align="left" | host(s)
 +
|-
 +
|Sept 11
 +
| Simon Marshall
 +
| UW Madison
 +
|[[#Simon Marshall  | Integrals of eigenfunctions on hyperbolic manifolds ]]
 +
|
 +
|-
 +
|'''Wednesday, Sept 12'''
 +
| Gunther Uhlmann 
 +
| University of Washington
 +
| Distinguished Lecture Series
 +
| See colloquium website for location
 +
|-
 +
|'''Friday, Sept 14'''
 +
| Gunther Uhlmann 
 +
| University of Washington
 +
| Distinguished Lecture Series
 +
| See colloquium website for location
 +
|-
 +
|Sept 18
 +
| Grad Student Seminar
 +
|
 +
|
 +
|
 +
|-
 +
|Sept 25
 +
| Grad Student Seminar
 +
|
 +
|
 +
|
 +
|-
 +
|Oct 9
 +
| Hong Wang
 +
| MIT
 +
|[[#Hong Wang  |  About Falconer distance problem in the plane ]]
 +
| Ruixiang
 +
|-
 +
|Oct 16
 +
| Polona Durcik
 +
| Caltech
 +
|[[#Polona Durcik  |  Singular Brascamp-Lieb inequalities and extended boxes in R^n ]]
 +
| Joris
 +
|-
 +
|Oct 23
 +
| Song-Ying Li
 +
| UC Irvine
 +
|[[#Song-Ying Li  |  Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold ]]
 +
| Xianghong
 +
|-
 +
|Oct 30
 +
|Grad student seminar
 +
|
 +
|
 +
|
 +
|-
 +
|Nov 6
 +
| Hanlong Fang
 +
| UW Madison
 +
|[[#Hanlong Fang  |  A generalization of the theorem of Weil and Kodaira on prescribing residues ]]
 +
| Brian
 +
|-
 +
||'''Monday, Nov. 12, B139'''
 +
| Kyle Hambrook
 +
| San Jose State University
 +
|[[#Kyle Hambrook  |  Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]]
 +
| Andreas
 +
|-
 +
|Nov 13
 +
| Laurent Stolovitch
 +
| Université de Nice - Sophia Antipolis
 +
|[[#Laurent Stolovitch  |  Equivalence of Cauchy-Riemann manifolds and multisummability theory ]]
 +
|Xianghong
 +
|-
 +
|Nov 20
 +
| Grad Student Seminar
 +
|
 +
|[[#linktoabstract  |    ]]
 +
|
 +
|-
 +
|Nov 27
 +
| No Seminar
 +
|
 +
|[[#linktoabstract  |    ]]
 +
|
 +
|-
 +
|Dec 4
 +
| No Seminar
 +
|[[#linktoabstract  |    ]]
 +
|
 +
|-
 +
|Jan 22
 +
| Brian Cook
 +
| Kent
 +
|[[#Brian Cook  |  Equidistribution results for integral points on affine homogenous algebraic varieties ]]
 +
| Street
 +
|-
 +
|Jan 29
 +
| No Seminar
 +
|
 +
|[[#linktoabstract  |    ]]
 +
|
 +
|-
 +
|Feb 5, '''B239'''
 +
| Alexei Poltoratski
 +
| Texas A&M
 +
|[[#Alexei Poltoratski  |  Completeness of exponentials: Beurling-Malliavin and type problems ]]
 +
| Denisov
 +
|-
 +
|'''Friday, Feb 8'''
 +
| Aaron Naber
 +
| Northwestern University
 +
|[[#linktoabstract  |  A structure theory for spaces with lower Ricci curvature bounds ]]
 +
| See colloquium website for location
 +
|-
 +
|Feb 12
 +
| Shaoming Guo
 +
| UW Madison
 +
|[[#Shaoming Guo | Polynomial Roth theorems in Salem sets  ]]
 +
|
 +
|-
 +
|'''Wed, Feb 13, B239'''
 +
| Dean Baskin
 +
| TAMU
 +
|[[# Dean Baskin  |  Radiation fields for wave  equations ]]
 +
|  Colloquium
 +
|-
 +
|'''Friday, Feb 15'''
 +
| Lillian Pierce
 +
| Duke
 +
|[[#Lillian Pierce  |  Short character sums ]]
 +
|  Colloquium
 +
|-
 +
|'''Monday,  Feb 18, 3:30 p.m, B239.'''
 +
| Daniel Tataru
 +
| UC Berkeley
 +
|[[#Daniel Tataru  |  A Morawetz inequality for water waves ]]
 +
|  PDE Seminar
 +
|-
 +
|Feb 19
 +
| Wenjia Jing
 +
|Tsinghua University
 +
|Periodic  homogenization of Dirichlet problems in perforated domains: a unified proof
 +
| PDE Seminar
 +
|-
 +
|Feb 26
 +
| No Seminar
 +
|
 +
|
 +
|-
 +
|Mar 5
 +
| Loredana Lanzani
 +
| Syracuse University
 +
|[[#Loredana Lanzani  |  On regularity and irregularity of the Cauchy-Szegő projection in several complex variables ]]
 +
| Xianghong
 +
|-
 +
|Mar 12
 +
| Trevor Leslie
 +
| UW Madison
 +
|[[#Trevor Leslie  |  Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time ]]
 +
|
 +
|-
 +
|Mar 19
 +
|Spring Break!
 +
|
 +
|
 +
|
 +
|-
 +
|Mar 26
 +
| No seminar
 +
|
 +
|[[#linktoabstract  |    ]]
 +
|
 +
|-
 +
|Apr 2
 +
| Stefan Steinerberger
 +
| Yale
 +
|[[#Stefan Steinerberger  |  Wasserstein Distance as a Tool in Analysis ]]
 +
| Shaoming, Andreas
 +
|-
 +
 +
|Apr 9
 +
| Franc Forstnerič
 +
| Unversity of Ljubljana
 +
|[[#Franc Forstnerič  |  Minimal surfaces by way of complex analysis ]]
 +
| Xianghong, Andreas
 +
|-
 +
|Apr 16
 +
| Andrew Zimmer
 +
| Louisiana State University
 +
|[[#Andrew Zimmer  |  The geometry of domains with negatively pinched Kaehler metrics ]]
 +
| Xianghong
 +
|-
 +
|Apr 23
 +
| Brian Street
 +
| University of Wisconsin-Madison
 +
|[[#Brian Street  |  Maximal Hypoellipticity ]]
 +
| Street
 +
|-
 +
|Apr 30
 +
| Zhen Zeng
 +
| UPenn
 +
|[[#Zhen Zeng  |  Decay property of multilinear oscillatory integrals ]]
 +
| Shaoming
 +
|-
 +
|*[https://www.math.wisc.edu/seeger2019/?q=node/2 Madison Lectures in Fourier Analysis]
 +
|-
 +
|Summer
 +
|-
 +
|Sept  10
 +
|Jose Madrid
 +
|UCLA
 +
|
 +
|Andreas, David
 +
|-
 +
|Oct 15
 +
|Bassam Shayya
 +
|American University of Beirut
 +
|
 +
|Andreas, Betsy
 +
 +
|}
 +
 +
=Abstracts=
 +
===Simon Marshall===
 +
 +
''Integrals of eigenfunctions on hyperbolic manifolds''
 +
 +
Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X.  I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity.  I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.
 +
 +
 +
===Hong Wang===
 +
 +
''About Falconer distance problem in the plane''
 +
 +
If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.
 +
 +
===Polona Durcik===
 +
 +
''Singular Brascamp-Lieb inequalities and extended boxes in R^n''
 +
 +
Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.
 +
 +
 +
===Song-Ying Li===
 +
 +
''Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold''
 +
 +
In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates
 +
for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold,
 +
which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the
 +
Kohn Laplacian on strictly pseudoconvex hypersurfaces.
 +
 +
 +
===Hanlong Fan===
 +
 +
''A generalization of the theorem of Weil and Kodaira on prescribing residues''
 +
 +
An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.
 +
 +
===Kyle Hambrook===
 +
 +
''Fourier Decay and Fourier Restriction for Fractal Measures on Curves''
 +
 +
I will discuss my recent work on some problems concerning
 +
Fourier decay and Fourier restriction for fractal measures on curves.
 +
 +
===Laurent Stolovitch===
 +
 +
''Equivalence of Cauchy-Riemann manifolds and multisummability theory''
 +
 +
We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$  are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.
 +
 +
 +
===Brian Cook===
 +
 +
''Equidistribution results for integral points on affine homogenous algebraic varieties''
 +
 +
Let Q be a homogenous integral polynomial of degree at least two. We consider certain results and questions concerning the distribution of the integral points on the level sets of Q.
 +
 +
===Alexei Poltoratski===
 +
 +
''Completeness of exponentials: Beurling-Malliavin and type problems''
 +
 +
This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin problem was solved in the early 1960s and I will present its classical solution along with modern generalizations and applications. I will then discuss history and recent progress in the type problem, which stood open for more than 70 years.
 +
 +
 +
===Shaoming Guo===
 +
 +
''Polynomial Roth theorems in Salem sets''
 +
 +
Let P(t) be a polynomial of one real variable. I will report a result on searching for patterns of the form (x, x+t, x+P(t)) within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik.
 +
 +
 +
 +
 +
===Dean Baskin===
 +
 +
''Radiation fields for wave equations''
 +
 +
Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.
 +
 +
===Lillian Pierce===
 +
 +
''Short character sums''
 +
 +
A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.
 +
 +
===Loredana Lanzani===
 +
 +
''On regularity and irregularity of the Cauchy-Szegő projection in several complex variables''
 +
 +
This talk is a survey of my latest, and now final, collaboration with Eli Stein.
 +
 +
It is known that for bounded domains $D$ in $\mathbb C^n$ that are of class $C^2$ and are strongly pseudo-convex, the Cauchy-Szegő projection is bounded in $L^p(\text{b}D, d\Sigma)$ for $1<p<\infty$. (Here $d\Sigma$ is induced Lebesgue measure.)  We show, using appropriate worm domains, that this fails for any $p\neq 2$, when we assume that the domain in question is only weakly pseudo-convex. Our starting point are the ideas of Kiselman-Barrett introduced more than 30 years ago in the analysis of the Bergman projection. However the study of the Cauchy-Szegő projection raises a number of new issues and obstacles that need to be overcome. We will also compare these results to the analogous problem for the Cauchy-Leray integral, where however the relevant counter-example is of much simpler nature.
 +
 +
===Trevor Leslie===
 +
 +
''Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time''
 +
 +
In this talk, we discuss the problem of energy equality for strong solutions of the Navier-Stokes Equations (NSE) at the first time where such solutions may lose regularity.  Our approach is motivated by a famous theorem of Caffarelli, Kohn, and Nirenberg, which states that the set of singular points associated to a suitable weak solution of the NSE has parabolic Hausdorff dimension of at most 1.  In particular, we furnish sufficient conditions for energy equality which depend on the dimension of the singularity set in addition to time and space integrability assumptions; in doing so we improve upon the classical results when attention is restricted to the first blowup time.  When our method is inconclusive, we are able to quantify the possible failure of energy equality in terms of the lower local dimension and the ''concentration dimension'' of a certain measure associated to the solution.  The work described is joint with Roman Shvydkoy (UIC).
 +
 +
===Stefan Steinerberger===
 +
 +
''Wasserstein Distance as a Tool in Analysis''
 +
 +
Wasserstein Distance is a way of measuring the distance between two probability distributions (minimizing it is a main problem in Optimal Transport). We will give a gentle Introduction into what it means and then use it to prove (1) a completely elementary but possibly new and quite curious inequality for real-valued functions and (2) a statement along the following lines: linear combinations of eigenfunctions of elliptic operators corresponding to high frequencies oscillate a lot and vanish on a large set of co-dimension 1 (this is already interesting for trigonometric polynomials on the 2-torus, sums of finitely many sines and cosines, whose sum has to vanish on long lines) and (3) some statements in Basic Analytic Number Theory that drop out for free as a byproduct.
 +
 +
===Franc Forstnerič===
 +
 +
''Minimal surfaces by way of complex analysis''
 +
 +
After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.
 +
 +
===Andrew Zimmer===
 +
 +
''The geometry of domains with negatively pinched Kaehler metrics''
 +
 +
Every bounded pseudoconvex domain in C^n has a natural complete metric: the Kaehler-Einstein metric constructed by Cheng-Yau. When the boundary of the domain is strongly pseudoconvex, Cheng-Yau showed that the holomorphic sectional curvature of this metric is asymptotically a negative constant. In this talk I will describe some partial converses to this result, including the following: if a smoothly bounded convex domain has a complete Kaehler metric with close to constant negative holomorphic sectional curvature near the boundary, then the domain is strongly pseudoconvex. This is joint work with F. Bracci and H. Gaussier.
 +
 +
 +
===Brian Street===
 +
 +
''Maximal Hypoellipticity''
 +
 +
In 1974, Folland and Stein introduced a generalization of ellipticity known as maximal hypoellipticity.  This talk will be an introduction to this concept and some of the ways it generalizes ellipticity.
 +
 +
 +
===Zhen Zeng===
 +
 +
''Decay property of multilinear oscillatory integrals''
 +
 +
In this talk, I will be talking about the conditions of the phase function $P$ and the linear mappings $\{\pi_i\}_{i=1}^n$ to ensure the asymptotic power decay properties of the following trilinear oscillatory integrals
 +
\[
 +
I_{\lambda}(f_1,f_2,f_3)=\int_{\mathbb{R}^m}e^{i\lambda P(x)}\prod_{j=1}^3 f_j(\pi_j(x))\eta(x)dx, 
 +
\]
 +
which falls into the broad goal in the previous work of Christ, Li, Tao and Thiele.
 +
 +
 +
 
= 2017-2018 Analysis Seminar Schedule =
 
= 2017-2018 Analysis Seminar Schedule =
 
{| cellpadding="8"
 
{| cellpadding="8"

Revision as of 15:40, 3 June 2019

Analysis Seminar

The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.

If you wish to invite a speaker please contact Brian at street(at)math

Previous Analysis seminars

Analysis Seminar Schedule

date speaker institution title host(s)
Sept 11 Simon Marshall UW Madison Integrals of eigenfunctions on hyperbolic manifolds
Wednesday, Sept 12 Gunther Uhlmann University of Washington Distinguished Lecture Series See colloquium website for location
Friday, Sept 14 Gunther Uhlmann University of Washington Distinguished Lecture Series See colloquium website for location
Sept 18 Grad Student Seminar
Sept 25 Grad Student Seminar
Oct 9 Hong Wang MIT About Falconer distance problem in the plane Ruixiang
Oct 16 Polona Durcik Caltech Singular Brascamp-Lieb inequalities and extended boxes in R^n Joris
Oct 23 Song-Ying Li UC Irvine Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold Xianghong
Oct 30 Grad student seminar
Nov 6 Hanlong Fang UW Madison A generalization of the theorem of Weil and Kodaira on prescribing residues Brian
Monday, Nov. 12, B139 Kyle Hambrook San Jose State University Fourier Decay and Fourier Restriction for Fractal Measures on Curves Andreas
Nov 13 Laurent Stolovitch Université de Nice - Sophia Antipolis Equivalence of Cauchy-Riemann manifolds and multisummability theory Xianghong
Nov 20 Grad Student Seminar
Nov 27 No Seminar
Dec 4 No Seminar
Jan 22 Brian Cook Kent Equidistribution results for integral points on affine homogenous algebraic varieties Street
Jan 29 No Seminar
Feb 5, B239 Alexei Poltoratski Texas A&M Completeness of exponentials: Beurling-Malliavin and type problems Denisov
Friday, Feb 8 Aaron Naber Northwestern University A structure theory for spaces with lower Ricci curvature bounds See colloquium website for location
Feb 12 Shaoming Guo UW Madison Polynomial Roth theorems in Salem sets
Wed, Feb 13, B239 Dean Baskin TAMU Radiation fields for wave equations Colloquium
Friday, Feb 15 Lillian Pierce Duke Short character sums Colloquium
Monday, Feb 18, 3:30 p.m, B239. Daniel Tataru UC Berkeley A Morawetz inequality for water waves PDE Seminar
Feb 19 Wenjia Jing Tsinghua University Periodic homogenization of Dirichlet problems in perforated domains: a unified proof PDE Seminar
Feb 26 No Seminar
Mar 5 Loredana Lanzani Syracuse University On regularity and irregularity of the Cauchy-Szegő projection in several complex variables Xianghong
Mar 12 Trevor Leslie UW Madison Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time
Mar 19 Spring Break!
Mar 26 No seminar
Apr 2 Stefan Steinerberger Yale Wasserstein Distance as a Tool in Analysis Shaoming, Andreas
Apr 9 Franc Forstnerič Unversity of Ljubljana Minimal surfaces by way of complex analysis Xianghong, Andreas
Apr 16 Andrew Zimmer Louisiana State University The geometry of domains with negatively pinched Kaehler metrics Xianghong
Apr 23 Brian Street University of Wisconsin-Madison Maximal Hypoellipticity Street
Apr 30 Zhen Zeng UPenn Decay property of multilinear oscillatory integrals Shaoming
*Madison Lectures in Fourier Analysis
Summer
Sept 10 Jose Madrid UCLA Andreas, David
Oct 15 Bassam Shayya American University of Beirut Andreas, Betsy

Abstracts

Simon Marshall

Integrals of eigenfunctions on hyperbolic manifolds

Let X be a compact hyperbolic manifold, and let Y be a totally geodesic closed submanifold in X. I will discuss the problem of bounding the integral of a Laplace eigenfunction on X over Y, as the eigenvalue tends to infinity. I will present an upper bound for these integrals that is sharp on average, and briefly describe ongoing work with Farrell Brumley in which we attempt to produce eigenfunctions with very large periods.


Hong Wang

About Falconer distance problem in the plane

If E is a compact set of Hausdorff dimension greater than 5/4 on the plane, we prove that there is a point x\in E such that the set of distances between x and E has positive Lebesgue measure. Our result improves upon Wolff's theorem for dim E> 4/3. This is joint work with Larry Guth, Alex Iosevich and Yumeng Ou.

Polona Durcik

Singular Brascamp-Lieb inequalities and extended boxes in R^n

Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms in R^n with a certain cubical structure and discuss their connection to some patterns in positive density subsets in R^n. Based on joint works with V. Kovac and C. Thiele.


Song-Ying Li

Estimates for the first positive eigenvalue of Kohn Laplacian on a pseudo-Hermitian manifold

In this talk, I will present my recent works with my collaborators on the lower bound and upper bounds estimates for the first positive eigenvalues of Kohn Laplacian and sub-Laplacian on a strictly pseudoconvex pseudo-Hermitian CR manifold, which include CR Lichnerowicz-Obata theorem for the lower and upper bounds for the first positive eigenvalue for the Kohn Laplacian on strictly pseudoconvex hypersurfaces.


Hanlong Fan

A generalization of the theorem of Weil and Kodaira on prescribing residues

An old theorem of Weil and Kodaira says that: For a K\"ahler manifold X, there exists a closed meromorphic one-form with residue divisor D if and only if D is homologous to zero. In this talk, I will generalize Weil and Kodaira's criterion to non-K\"ahler manifolds.

Kyle Hambrook

Fourier Decay and Fourier Restriction for Fractal Measures on Curves

I will discuss my recent work on some problems concerning Fourier decay and Fourier restriction for fractal measures on curves.

Laurent Stolovitch

Equivalence of Cauchy-Riemann manifolds and multisummability theory

We apply the multisummability theory from Dynamical Systems to CR-geometry. As the main result, we show that two real-analytic hypersurfaces in $\mathbb C^2$ are formally equivalent, if and only if they are $C^\infty$ CR-equivalent at the respective point. As a corollary, we prove that all formal equivalences between real-algebraic Levi-nonflat hypersurfaces in $\mathbb C^2$ are algebraic (and in particular convergent). This is a joint work with I. Kossovskiy and B. Lamel.


Brian Cook

Equidistribution results for integral points on affine homogenous algebraic varieties

Let Q be a homogenous integral polynomial of degree at least two. We consider certain results and questions concerning the distribution of the integral points on the level sets of Q.

Alexei Poltoratski

Completeness of exponentials: Beurling-Malliavin and type problems

This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin problem was solved in the early 1960s and I will present its classical solution along with modern generalizations and applications. I will then discuss history and recent progress in the type problem, which stood open for more than 70 years.


Shaoming Guo

Polynomial Roth theorems in Salem sets

Let P(t) be a polynomial of one real variable. I will report a result on searching for patterns of the form (x, x+t, x+P(t)) within Salem sets, whose Hausdorff dimension is sufficiently close to one. Joint work with Fraser and Pramanik.



Dean Baskin

Radiation fields for wave equations

Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.

Lillian Pierce

Short character sums

A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.

Loredana Lanzani

On regularity and irregularity of the Cauchy-Szegő projection in several complex variables

This talk is a survey of my latest, and now final, collaboration with Eli Stein.

It is known that for bounded domains $D$ in $\mathbb C^n$ that are of class $C^2$ and are strongly pseudo-convex, the Cauchy-Szegő projection is bounded in $L^p(\text{b}D, d\Sigma)$ for $1<p<\infty$. (Here $d\Sigma$ is induced Lebesgue measure.) We show, using appropriate worm domains, that this fails for any $p\neq 2$, when we assume that the domain in question is only weakly pseudo-convex. Our starting point are the ideas of Kiselman-Barrett introduced more than 30 years ago in the analysis of the Bergman projection. However the study of the Cauchy-Szegő projection raises a number of new issues and obstacles that need to be overcome. We will also compare these results to the analogous problem for the Cauchy-Leray integral, where however the relevant counter-example is of much simpler nature.

Trevor Leslie

Energy Equality for the Navier-Stokes Equations at the First Possible Blowup Time

In this talk, we discuss the problem of energy equality for strong solutions of the Navier-Stokes Equations (NSE) at the first time where such solutions may lose regularity. Our approach is motivated by a famous theorem of Caffarelli, Kohn, and Nirenberg, which states that the set of singular points associated to a suitable weak solution of the NSE has parabolic Hausdorff dimension of at most 1. In particular, we furnish sufficient conditions for energy equality which depend on the dimension of the singularity set in addition to time and space integrability assumptions; in doing so we improve upon the classical results when attention is restricted to the first blowup time. When our method is inconclusive, we are able to quantify the possible failure of energy equality in terms of the lower local dimension and the concentration dimension of a certain measure associated to the solution. The work described is joint with Roman Shvydkoy (UIC).

Stefan Steinerberger

Wasserstein Distance as a Tool in Analysis

Wasserstein Distance is a way of measuring the distance between two probability distributions (minimizing it is a main problem in Optimal Transport). We will give a gentle Introduction into what it means and then use it to prove (1) a completely elementary but possibly new and quite curious inequality for real-valued functions and (2) a statement along the following lines: linear combinations of eigenfunctions of elliptic operators corresponding to high frequencies oscillate a lot and vanish on a large set of co-dimension 1 (this is already interesting for trigonometric polynomials on the 2-torus, sums of finitely many sines and cosines, whose sum has to vanish on long lines) and (3) some statements in Basic Analytic Number Theory that drop out for free as a byproduct.

Franc Forstnerič

Minimal surfaces by way of complex analysis

After a brief historical introduction, I will present some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. The emphasis will be on results pertaining to the global theory of minimal surfaces including Runge and Mergelyan approximation, the conformal Calabi-Yau problem, properly immersed and embedded minimal surfaces, and a new result on the Gauss map of minimal surfaces.

Andrew Zimmer

The geometry of domains with negatively pinched Kaehler metrics

Every bounded pseudoconvex domain in C^n has a natural complete metric: the Kaehler-Einstein metric constructed by Cheng-Yau. When the boundary of the domain is strongly pseudoconvex, Cheng-Yau showed that the holomorphic sectional curvature of this metric is asymptotically a negative constant. In this talk I will describe some partial converses to this result, including the following: if a smoothly bounded convex domain has a complete Kaehler metric with close to constant negative holomorphic sectional curvature near the boundary, then the domain is strongly pseudoconvex. This is joint work with F. Bracci and H. Gaussier.


Brian Street

Maximal Hypoellipticity

In 1974, Folland and Stein introduced a generalization of ellipticity known as maximal hypoellipticity. This talk will be an introduction to this concept and some of the ways it generalizes ellipticity.


Zhen Zeng

Decay property of multilinear oscillatory integrals

In this talk, I will be talking about the conditions of the phase function $P$ and the linear mappings $\{\pi_i\}_{i=1}^n$ to ensure the asymptotic power decay properties of the following trilinear oscillatory integrals \[ I_{\lambda}(f_1,f_2,f_3)=\int_{\mathbb{R}^m}e^{i\lambda P(x)}\prod_{j=1}^3 f_j(\pi_j(x))\eta(x)dx, \] which falls into the broad goal in the previous work of Christ, Li, Tao and Thiele.


2017-2018 Analysis Seminar Schedule

date speaker institution title host(s)
September 8 in B239 (Colloquium) Tess Anderson UW Madison A Spherical Maximal Function along the Primes Tonghai
September 19 Brian Street UW Madison Convenient Coordinates Betsy
September 26 Hiroyoshi Mitake Hiroshima University Derivation of multi-layered interface system and its application Hung
October 3 Joris Roos UW Madison A polynomial Roth theorem on the real line Betsy
October 10 Michael Greenblatt UI Chicago Maximal averages and Radon transforms for two-dimensional hypersurfaces Andreas
October 17 David Beltran Basque Center of Applied Mathematics Fefferman-Stein inequalities Andreas
Wednesday, October 18, 4:00 p.m. in B131 Jonathan Hickman University of Chicago Factorising X^n Andreas
October 24 Xiaochun Li UIUC Recent progress on the pointwise convergence problems of Schroedinger equations Betsy
Thursday, October 26, 4:30 p.m. in B139 Fedor Nazarov Kent State University The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp Sergey, Andreas
Friday, October 27, 4:00 p.m. in B239 Stefanie Petermichl University of Toulouse Higher order Journé commutators Betsy, Andreas
Wednesday, November 1, 4:00 p.m. in B239 (Colloquium) Shaoming Guo Indiana University Parsell-Vinogradov systems in higher dimensions Andreas
November 14 Naser Talebizadeh Sardari UW Madison Quadratic forms and the semiclassical eigenfunction hypothesis Betsy
November 28 Xianghong Chen UW Milwaukee Some transfer operators on the circle with trigonometric weights Betsy
Monday, December 4, 4:00, B139 Bartosz Langowski and Tomasz Szarek Institute of Mathematics, Polish Academy of Sciences Discrete Harmonic Analysis in the Non-Commutative Setting Betsy
Wednesday, December 13, 4:00, B239 (Colloquium) Bobby Wilson MIT Projections in Banach Spaces and Harmonic Analysis Andreas
Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar) Andreas Seeger UW Singular integrals and a problem on mixing flows
February 6 Dong Dong UIUC Hibert transforms in a 3 by 3 matrix and applications in number theory Betsy
February 13 Sergey Denisov UW Madison Spectral Szegő theorem on the real line
February 20 Ruixiang Zhang IAS (Princeton) The (Euclidean) Fractal Uncertainty Principle Betsy, Jordan, Andreas
February 27 Detlef Müller University of Kiel On Fourier restriction for a non-quadratic hyperbolic surface Betsy, Andreas
Wednesday, March 7, 4:00 p.m. Winfried Sickel Friedrich-Schiller-Universität Jena On the regularity of compositions of functions Andreas
March 20 Betsy Stovall UW Two endpoint bounds via inverse problems
April 10 Martina Neuman UC Berkeley Gowers-Host-Kra norms and Gowers structure on Euclidean spaces Betsy
Friday, April 13, 4:00 p.m. (Colloquium, 911 VV) Jill Pipher Brown Mathematical ideas in cryptography WIMAW
April 17 Title
April 24 Lenka Slavíková University of Missouri [math]L^2 \times L^2 \to L^1[/math] boundedness criteria Betsy, Andreas
May 1 at 3:30pm Xianghong Gong UW Smooth equivalence of deformations of domains in complex euclidean spaces
May 2 in B239 at 4pm Keith Rush senior data scientist with the Milwaukee Brewers Guerilla warfare: ruling the data jungle
May 7 in B223 Ebru Toprak UIUC Dispersive estimates for massive Dirac equations Betsy
May 15 Gennady Uraltsev Cornell TBA Andreas, Betsy
May 16-18, Workshop in Fourier Analysis Betsy, Andreas

Abstracts

Brian Street

Title: Convenient Coordinates

Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".

Hiroyoshi Mitake

Title: Derivation of multi-layered interface system and its application

Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.

Joris Roos

Title: A polynomial Roth theorem on the real line

Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.

Michael Greenblatt

Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces

Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.

David Beltran

Title: Fefferman Stein Inequalities

Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.

Jonathan Hickman

Title: Factorising X^n.

Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.

Xiaochun Li

Title: Recent progress on the pointwise convergence problems of Schrodinger equations

Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.

Fedor Nazarov

Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden conjecture is sharp.

Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for the norm of the Hilbert transform on the line as an operator from $L^1(w)$ to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work with Andrei Lerner and Sheldy Ombrosi.

Stefanie Petermichl

Title: Higher order Journé commutators

Abstract: We consider questions that stem from operator theory via Hankel and Toeplitz forms and target (weak) factorisation of Hardy spaces. In more basic terms, let us consider a function on the unit circle in its Fourier representation. Let P_+ denote the projection onto non-negative and P_- onto negative frequencies. Let b denote multiplication by the symbol function b. It is a classical theorem by Nehari that the composed operator P_+ b P_- is bounded on L^2 if and only if b is in an appropriate space of functions of bounded mean oscillation. The necessity makes use of a classical factorisation theorem of complex function theory on the disk. This type of question can be reformulated in terms of commutators [b,H]=bH-Hb with the Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such as in the real variable setting, in the multi-parameter setting or other, these classifications can be very difficult.

Such lines were begun by Coifman, Rochberg, Weiss (real variables) and by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of spaces of bounded mean oscillation via L^p boundedness of commutators. We present here an endpoint to this theory, bringing all such characterisation results under one roof.

The tools used go deep into modern advances in dyadic harmonic analysis, while preserving the Ansatz from classical operator theory.

Shaoming Guo

Title: Parsell-Vinogradov systems in higher dimensions

Abstract: I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions. Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed. Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.

Naser Talebizadeh Sardari

Title: Quadratic forms and the semiclassical eigenfunction hypothesis

Abstract: Let [math]Q(X)[/math] be any integral primitive positive definite quadratic form in [math]k[/math] variables, where [math]k\geq4[/math], and discriminant [math]D[/math]. For any integer [math]n[/math], we give an upper bound on the number of integral solutions of [math]Q(X)=n[/math] in terms of [math]n[/math], [math]k[/math], and [math]D[/math]. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus [math]\mathbb{T}^d[/math] for [math]d\geq 5[/math]. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.

Xianghong Chen

Title: Some transfer operators on the circle with trigonometric weights

Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer.

Bobby Wilson

Title: Projections in Banach Spaces and Harmonic Analysis

Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.

Andreas Seeger

Title: Singular integrals and a problem on mixing flows

Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.

Dong Dong

Title: Hibert transforms in a 3 by 3 matrix and applications in number theory

Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.

Sergey Denisov

Title: Spectral Szegő theorem on the real line

Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.

Ruixiang Zhang

Title: The (Euclidean) Fractal Uncertainty Principle

Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).

Detlef Müller

Title: On Fourier restriction for a non-quadratic hyperbolic surface

Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.

Winfried Sickel

Title: On the regularity of compositions of functions

Abstract: Let [math]E[/math] denote a Banach space of locally integrable functions on [math]\mathbb{R}[/math]. To each continuous function [math]f:\mathbb{R} \to \mathbb{R}[/math] we associate the composition operator [math]T_f(g):= f\circ g[/math], [math]g\in E[/math]. The properties of [math]T_f[/math] strongly depend on the chosen function space [math]E[/math]. In my talk I will concentrate on Sobolev spaces [math]W^m_p[/math] and Slobodeckij spaces [math]W^s_p[/math]. The main aim will consist in giving a survey on necessary and sufficient conditions on [math]f[/math] such that the composition operator maps such a space [math]E[/math] into itself.

Martina Neuman

Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces

Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.

Jill Pipher

Title: Mathematical ideas in cryptography

Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research, including homomorphic encryption.

Lenka Slavíková

Title: [math]L^2 \times L^2 \to L^1[/math] boundedness criteria

Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function [math]m[/math] is bounded from [math]L^2[/math] to itself if and only if [math]m[/math] belongs to the space [math]L^\infty[/math]. In this talk we will investigate the [math]L^2 \times L^2 \to L^1[/math] boundedness of bilinear multiplier operators which is as central in the bilinear theory as the [math]L^2[/math] boundedness is in the linear multiplier theory. We will present a sharp [math]L^2 \times L^2 \to L^1[/math] boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the [math]L^q[/math] integrability of this function; precisely we will show that boundedness holds if and only if [math]q\lt 4[/math]. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.

Xianghong Gong

Title: Smooth equivalence of deformations of domains in complex euclidean spaces

Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.

Keith Rush

Title: Guerilla warfare: ruling the data jungle

Abstract: Einstein said ‘As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.’ In this epistemological chaos, the world turns to those experienced with mathematical truth to apply their reasoning powers in the uncertain domain of existence. This talk will describe the fact and fiction of this business reality, the pitfalls (intellectual, moral, and social) and the opportunities. I will discuss the state of business analytics today, at least in sports, the relationship of a pure mathematician to it, and what it is like to help lead the charge as applied mathematics eats the world.

Ebru Toprak

Title: Dispersive estimates for massive Dirac equations

Abstract: In this talk, I will cover some existing L^1 \rightarrow L^\infty dispersive estimates for the linear Schr\"odinger equation with potential and present a related study on the two and three dimensional massive Dirac equation. In two dimension, we show that the t^{-1} decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t^{-1}(\log t)^{-2} is attained for large t, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate t^{-1/2} plus a term that decays at the rate t^{-3/2}. This is a joint work with M.Burak Erdo\u{g}an and William Green.