Difference between revisions of "Analysis Seminar"
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The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated. | The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated. | ||
− | If you wish to invite a speaker please contact | + | If you wish to invite a speaker please contact Brian at street(at)math |
===[[Previous Analysis seminars]]=== | ===[[Previous Analysis seminars]]=== | ||
− | = | + | = Analysis Seminar Schedule = |
{| cellpadding="8" | {| cellpadding="8" | ||
!align="left" | date | !align="left" | date | ||
Line 16: | Line 16: | ||
!align="left" | host(s) | !align="left" | host(s) | ||
|- | |- | ||
− | | | + | |Sept 11 |
− | | | + | | Simon Marshall |
| UW Madison | | 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 | | UW Madison | ||
− | |[[# | + | |[[#Hanlong Fang | A generalization of the theorem of Weil and Kodaira on prescribing residues ]] |
− | + | | Brian | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | | | ||
|- | |- | ||
− | |Monday, | + | ||'''Monday, Nov. 12, B139''' |
− | | | + | | Kyle Hambrook |
− | | | + | | San Jose State University |
− | |[[# | + | |[[#Kyle Hambrook | Fourier Decay and Fourier Restriction for Fractal Measures on Curves ]] |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
| Andreas | | 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 | ||
+ | |[[#lDaniel 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 |
− | | [[# | + | |[[#linktoabstract | Title ]] |
| | | | ||
|- | |- | ||
− | | | + | |Mar 19 |
+ | |Spring Break!!! | ||
| | | | ||
− | | | + | | |
− | |||
| | | | ||
|- | |- | ||
− | | | + | |Mar 26 |
− | | | + | | Person |
− | | | + | | Institution |
− | | [[# | + | |[[#linktoabstract | Title ]] |
− | | | + | | Sponsor |
+ | |- | ||
+ | |Apr 2 | ||
+ | | Stefan Steinerberger | ||
+ | | Yale | ||
+ | |[[#linktoabstract | Title ]] | ||
+ | | Shaoming, Andreas | ||
|- | |- | ||
− | | | + | |
− | | | + | |Apr 9 |
− | | | + | | Franc Forstnerič |
− | | [[# | + | | Unversity of Ljubljana |
− | | | + | |[[#linktoabstract | Title ]] |
+ | | Xianghong, Andreas | ||
|- | |- | ||
− | | | + | |Apr 16 |
− | | | + | | Andrew Zimmer |
− | | | + | | Louisiana State University |
− | | [[# | + | |[[#linktoabstract | Title ]] |
− | | | + | | Xianghong |
|- | |- | ||
− | | | + | |Apr 23 |
− | | | + | | Person |
− | | | + | | Institution |
− | | [[# | + | |[[#linktoabstract | Title ]] |
− | | | + | | Sponsor |
|- | |- | ||
− | | | + | |Apr 30 |
− | | | + | | Person |
− | | | + | | Institution |
− | | [[# | + | |[[#linktoabstract | Title ]] |
− | | | + | | Sponsor |
|- | |- | ||
|} | |} | ||
=Abstracts= | =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. | |
+ | |||
=Extras= | =Extras= | ||
[[Blank Analysis Seminar Template]] | [[Blank Analysis Seminar Template]] |
Latest revision as of 18:41, 21 February 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 | Title | |
Mar 19 | Spring Break!!! | |||
Mar 26 | Person | Institution | Title | Sponsor |
Apr 2 | Stefan Steinerberger | Yale | Title | Shaoming, Andreas |
Apr 9 | Franc Forstnerič | Unversity of Ljubljana | Title | Xianghong, Andreas |
Apr 16 | Andrew Zimmer | Louisiana State University | Title | Xianghong |
Apr 23 | Person | Institution | Title | Sponsor |
Apr 30 | Person | Institution | Title | Sponsor |
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.