Fall 2024 Analysis Seminar: Difference between revisions
No edit summary |
No edit summary |
||
| Line 347: | Line 347: | ||
| | | | ||
|} | |} | ||
=Abstracts= | |||
===[[Tushar Das]]=== | |||
Title: Playing games on fractals: Dynamical & Diophantine | |||
Abstract: We will present sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Wolfgang Schmidt and Leonhard Summerer) to Diophantine approximation for systems of m linear forms in n variables. Our variational principle (arXiv:1901.06602) provides a unified framework to compute Hausdorff and packing dimensions of a variety of sets of number-theoretic interest, as well as their dynamical counterparts via the Dani correspondence. Highlights include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty whose interests contain a convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry. | |||
===[[Rajula Srivastava]]=== | |||
Title: Counting Rational Points near Flat Hypersurfaces | |||
Abstract: How many rational points with denominator of a given size lie | |||
within a given distance from a compact hypersurface? In this talk, we | |||
shall describe how the geometry of the surface plays a key role in | |||
determining this count, and present a heuristic for the same. In a | |||
recent breakthrough, J.J. Huang proved that this guess is indeed true | |||
for hypersurfaces with non-vanishing Gaussian curvature. What about | |||
hypersurfaces with curvature only vanishing up to a finite order, at a | |||
single point? We shall offer a new heuristic in this regime which also | |||
incorporates the contribution arising from "local flatness". Further, we | |||
will describe how ideas from Harmonic Analysis can be used to establish | |||
the indicated estimates for hypersurfaces of this type immersed by | |||
homogeneous functions. In particular, we shall use a powerful | |||
bootstrapping argument relying on Poisson summation, duality between | |||
flat and "rough" hypersurfaces, and the method of stationary phase. A | |||
crucial role is played by a dyadic scaling argument exploiting the | |||
homogeneous nature of the hypersurface. Based on joint work with N. Technau. | |||
===[[Niclas Technau]]=== | |||
Title: Oscillatory Integrals Count | |||
Abstract: This talk is about phrasing (number theoretic) counting problems in terms oscillatory integrals. | |||
We shall provide a simple introduction to the topic, mention open questions, and | |||
report on joint work with Sam Chow, as well as on joint work with Chris Lutsko. | |||
===[[Terry Harris]]=== | |||
Title: Horizontal Besicovitch sets of measure zero and some related problems | |||
Abstract: It is shown that horizontal Besicovitch sets of measure zero exist in R^3. The proof is constructive and uses point-line duality analogously to Kahane’s construction of measure zero Besicovitch sets in the plane. Some consequences and related examples are shown for the SL_2 Kakeya maximal function. | |||
===[[Tristan Leger]]=== | |||
Title: L^p bounds for spectral projectors on hyperbolic surfaces | |||
Abstract: In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces. I will focus on the case where the spectral window has small width. Indeed the negative curvature assumption leads to improvements over the universal bounds of C.Sogge, thus illustrating how these objects are sensitive to the global geometry of the underlying manifold. I will also explain how this relates to the general theme of delocalization of eigenfunctions of the Laplacian on hyperbolic surfaces. | |||
This is based on joint work with Jean-Philippe Anker and Pierre Germain. | |||
===[[Bingyang Hu]]=== | |||
Title: On the curved Trilinear Hilbert transform | |||
Abstract: The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator | |||
$$ | |||
H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R | |||
$$ | |||
is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1<p_1, p_3<\infty$, $1<p_2 \le \infty$ and $1 \le r <\infty$. | |||
The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements: | |||
1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space; | |||
2). a structural analysis of suitable maximal "joint Fourier coefficients"; | |||
3). a level set analysis with respect to the time-frequency correlation set. | |||
This is a joint work with my postdoc advisor Victor Lie from Purdue. | |||
===[[Rodrigo Bañuelos]]=== | |||
Title: Probabilistic tools in discrete harmonic analysis | |||
Abstract: The discrete Hilbert transform was introduced by David Hilbert at the beginning of the 20th century as an example of a singular quadratic form. Its boundedness on the space of square summable sequences appeared in H. Weyl’s doctoral dissertation (under Hilbert) in 1908. In 1925, M. Riesz proved that the continuous version of this operator is bounded on L^p(R), 1 < p < \infty, and that the same holds for the discrete version on the integers. Shortly thereafter (1926), E. C. Titchmarsh gave a different proof and from it concluded that the operators have the same p-norm. Unfortunately, Titchmarsh’s argument for equality was incorrect. The question of equality of the norms had been a “simple tantalizing" problem ever since. | |||
In this general colloquium talk the speaker will discuss a probabilistic construction, based on Doob’s “h-Brownian motion," that leads to sharp inequalities for a collection of discrete operators on the d-dimensional lattice Z^d, d ≥ 1. The case d = 1 verifies equality of the norms for the discrete and continuous Hilbert transforms. The case d > 1 leads to similar questions and conjectures for other Calderón-Zygmund singular integrals in higher dimensions. | |||
===[[Shaoming Guo]]=== | |||
Title: Oscillatory integrals on Riemannian manifolds, and related Kakeya and Nikodym problems | |||
Abstract: The talk is about oscillatory integral operators on manifolds. Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures. | |||
===[[Lechao Xiao]]=== | |||
Title: Some connections between Harmonic Analysis and Theory of Deep Learning | |||
Abstract: The past decade has witnessed a remarkable surge in breakthroughs in artificial intelligence (AI), with the potential to profoundly impact various aspects of our lives. However, the fundamental mathematical principles underlying the success of deep learning, the core technology behind these breakthroughs, is still far from well-understood. In this presentation, I will share some interesting connections between the theory of deep learning and harmonic analysis. | |||
The first half provides a gentle introduction to machine learning and deep learning. The second half focuses on two technical topics: | |||
An uncertainty principle between space and frequency and its significance in overcoming the curse of dimensionality. | |||
The multi-scale Marchenko-Pastur law and its interplay with the multiple-descent learning curve phenomenon. | |||
===[[Neeraja Kulkarni]]=== | |||
Title: An improved Minkowski dimension estimate for Kakeya sets in higher dimensions using planebrushes. | |||
Abstract: A Kakeya set is defined as a compact subset of $\mathbb{R}^n$ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Kakeya set has Minkowski and Hausdorff dimensions equal to $n$. In this talk, I will discuss an improved Minkowski dimension estimate for Kakeya sets in dimensions $n\geq 5$. The improved estimate comes from using a geometric argument called a ``$k$-planebrush'', which is a higher dimensional analogue of Wolff's ``hairbrush'' argument from 1995. The $k$-planebrush argument is used in conjunction with a previously known "k-linear" result on Kakeya sets proved by Hickman-Rogers-Zhang (and concurrently by Zahl) in 2019 along with an x-ray transform estimate which is a corollary of Hickman-Rogers-Zhang (and Zahl). The x-ray transform estimate is used to deduce that the Kakeya set has a structural property called ``stickiness,'' which was first introduced in a paper by Katz-Laba-Tao in 1999. Sticky Kakeya sets exhibit a self-similar structure which is exploited by the $k$-planebrush argument. | |||
===[[Changkeun]]=== | |||
Title: Discrete restriction estimates for manifolds avoiding a line. | |||
Abstract: We identify a new way to divide the d-neighborhood of surfaces in R^3. We decompose the d-neighborhood of surfaces into a finitely-overlapping collection of rectangular boxes S. We obtain an (l^2,L^p) decoupling estimate using this decomposition, for the sharp range of exponents. The decoupling theorem we prove is new for the hyperbolic paraboloid, and recovers the Tomas-Stein restriction inequality. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line. In this talk, I'll focus on explaning backgrounds and theorems rather than giving proofs. This is joint work with Larry Guth and Dominique Maldague. | |||
===[[Ryan Bushlin]]=== | |||
Friday, December 8, 1:20-2:10 in B107 | |||
'''Some variational problems characterizing families of convex domains''' | |||
We prove a singular integral identity for the surface measure of (n - 1)-rectifiable sets in Euclidean n-space satisfying the orientation cancellation condition. In particular, sets of finite perimeter enjoy this property, and from this observation follows a geometric inequality in which equality is attained precisely by the convex sets. More generally, the integral identity has anisotropic analogues whose corresponding inequalities characterize some geometrically simple subfamilies of the family of convex sets. | |||
===[[Jing-Jing Huang]]=== | |||
'''Diophantine approximation on affine subspaces''' | |||
We establish a clear-cut criterion for an affine subspace of R^n to be extremal (i.e. the Dirichlet exponent 1/n is best possible almost everywhere), confirming a conjecture of Kleinbock. We also extend the classical theorem of Khintchine on metric diophantine approximation to affine subspaces. Moreover, an exact formula for the Hausdorff dimension of the set of very well approximable vectors on the affine subspace is obtained. The above results are proved as consequences of our novel estimates for the number of rational points lying close to the affine subspace. The proof of this counting estimate is Fourier analytic in nature and in particular utilizes the large sieve inequality. | |||
[[Previous Analysis seminars|Links to previous seminars]] | |||
[[Spring 2024 Analysis Seminar|Link to the analysis seminar in 2024 Spring]] | |||
Revision as of 08:59, 9 August 2024
Organizers: Shengwen Gan, Terry Harris and Andreas Seeger
Emails:
- Shengwen Gan: sgan7 at math dot wisc dot edu
- Terry Harris: tlharris4 at math dot wisc dot edu
- Andreas Seeger: seeger at math dot wisc dot edu
Time: Wed 3:30--4:30
Room: B223
We also have room B211 reserved at 4:25-5:25 for discussions after talks.
All talks will be in-person unless otherwise specified.
In some cases the seminar may be scheduled at different time to accommodate speakers.
If you would like to subscribe to the Analysis seminar list, send a blank email to analysis+subscribe (at) g-groups (dot) wisc (dot) edu
| Date | Speaker | Institution | Title | Host(s) | Notes (e.g. unusual room/day/time) | |
|---|---|---|---|---|---|---|
| 9-11 | ||||||
| 9-25 | ||||||
| 10-2 | ||||||
| 10-9 | ||||||
| 10-16 | ||||||
| 10-23 | ||||||
| 10-30 | Burak Hatinoglu | Michigan State University | ||||
| 11-6 | ||||||
| 11-13 | ||||||
| 11-20 | ||||||
| 11-27 | No seminar | |||||
| 12-4 | ||||||
Abstracts
Tushar Das
Title: Playing games on fractals: Dynamical & Diophantine
Abstract: We will present sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Wolfgang Schmidt and Leonhard Summerer) to Diophantine approximation for systems of m linear forms in n variables. Our variational principle (arXiv:1901.06602) provides a unified framework to compute Hausdorff and packing dimensions of a variety of sets of number-theoretic interest, as well as their dynamical counterparts via the Dani correspondence. Highlights include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty whose interests contain a convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry.
Rajula Srivastava
Title: Counting Rational Points near Flat Hypersurfaces
Abstract: How many rational points with denominator of a given size lie within a given distance from a compact hypersurface? In this talk, we shall describe how the geometry of the surface plays a key role in determining this count, and present a heuristic for the same. In a recent breakthrough, J.J. Huang proved that this guess is indeed true for hypersurfaces with non-vanishing Gaussian curvature. What about hypersurfaces with curvature only vanishing up to a finite order, at a single point? We shall offer a new heuristic in this regime which also incorporates the contribution arising from "local flatness". Further, we will describe how ideas from Harmonic Analysis can be used to establish the indicated estimates for hypersurfaces of this type immersed by homogeneous functions. In particular, we shall use a powerful bootstrapping argument relying on Poisson summation, duality between flat and "rough" hypersurfaces, and the method of stationary phase. A crucial role is played by a dyadic scaling argument exploiting the homogeneous nature of the hypersurface. Based on joint work with N. Technau.
Niclas Technau
Title: Oscillatory Integrals Count
Abstract: This talk is about phrasing (number theoretic) counting problems in terms oscillatory integrals. We shall provide a simple introduction to the topic, mention open questions, and report on joint work with Sam Chow, as well as on joint work with Chris Lutsko.
Terry Harris
Title: Horizontal Besicovitch sets of measure zero and some related problems
Abstract: It is shown that horizontal Besicovitch sets of measure zero exist in R^3. The proof is constructive and uses point-line duality analogously to Kahane’s construction of measure zero Besicovitch sets in the plane. Some consequences and related examples are shown for the SL_2 Kakeya maximal function.
Tristan Leger
Title: L^p bounds for spectral projectors on hyperbolic surfaces
Abstract: In this talk I will present L^p boundedness results for spectral projectors on hyperbolic surfaces. I will focus on the case where the spectral window has small width. Indeed the negative curvature assumption leads to improvements over the universal bounds of C.Sogge, thus illustrating how these objects are sensitive to the global geometry of the underlying manifold. I will also explain how this relates to the general theme of delocalization of eigenfunctions of the Laplacian on hyperbolic surfaces. This is based on joint work with Jean-Philippe Anker and Pierre Germain.
Bingyang Hu
Title: On the curved Trilinear Hilbert transform
Abstract: The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator $$ H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R $$ is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1<p_1, p_3<\infty$, $1<p_2 \le \infty$ and $1 \le r <\infty$.
The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:
1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;
2). a structural analysis of suitable maximal "joint Fourier coefficients";
3). a level set analysis with respect to the time-frequency correlation set.
This is a joint work with my postdoc advisor Victor Lie from Purdue.
Rodrigo Bañuelos
Title: Probabilistic tools in discrete harmonic analysis
Abstract: The discrete Hilbert transform was introduced by David Hilbert at the beginning of the 20th century as an example of a singular quadratic form. Its boundedness on the space of square summable sequences appeared in H. Weyl’s doctoral dissertation (under Hilbert) in 1908. In 1925, M. Riesz proved that the continuous version of this operator is bounded on L^p(R), 1 < p < \infty, and that the same holds for the discrete version on the integers. Shortly thereafter (1926), E. C. Titchmarsh gave a different proof and from it concluded that the operators have the same p-norm. Unfortunately, Titchmarsh’s argument for equality was incorrect. The question of equality of the norms had been a “simple tantalizing" problem ever since.
In this general colloquium talk the speaker will discuss a probabilistic construction, based on Doob’s “h-Brownian motion," that leads to sharp inequalities for a collection of discrete operators on the d-dimensional lattice Z^d, d ≥ 1. The case d = 1 verifies equality of the norms for the discrete and continuous Hilbert transforms. The case d > 1 leads to similar questions and conjectures for other Calderón-Zygmund singular integrals in higher dimensions.
Shaoming Guo
Title: Oscillatory integrals on Riemannian manifolds, and related Kakeya and Nikodym problems
Abstract: The talk is about oscillatory integral operators on manifolds. Manifolds of constant sectional curvatures are particularly interesting, and we will see that very good estimates on these manifolds can be expected. We will also discuss Kakeya and Nikodym problems on general manifolds, in particular, manifolds satisfying Sogge’s chaotic curvatures.
Lechao Xiao
Title: Some connections between Harmonic Analysis and Theory of Deep Learning
Abstract: The past decade has witnessed a remarkable surge in breakthroughs in artificial intelligence (AI), with the potential to profoundly impact various aspects of our lives. However, the fundamental mathematical principles underlying the success of deep learning, the core technology behind these breakthroughs, is still far from well-understood. In this presentation, I will share some interesting connections between the theory of deep learning and harmonic analysis.
The first half provides a gentle introduction to machine learning and deep learning. The second half focuses on two technical topics:
An uncertainty principle between space and frequency and its significance in overcoming the curse of dimensionality.
The multi-scale Marchenko-Pastur law and its interplay with the multiple-descent learning curve phenomenon.
Neeraja Kulkarni
Title: An improved Minkowski dimension estimate for Kakeya sets in higher dimensions using planebrushes.
Abstract: A Kakeya set is defined as a compact subset of $\mathbb{R}^n$ which contains a line segment of length 1 in every direction. The Kakeya conjecture says that every Kakeya set has Minkowski and Hausdorff dimensions equal to $n$. In this talk, I will discuss an improved Minkowski dimension estimate for Kakeya sets in dimensions $n\geq 5$. The improved estimate comes from using a geometric argument called a ``$k$-planebrush, which is a higher dimensional analogue of Wolff's ``hairbrush argument from 1995. The $k$-planebrush argument is used in conjunction with a previously known "k-linear" result on Kakeya sets proved by Hickman-Rogers-Zhang (and concurrently by Zahl) in 2019 along with an x-ray transform estimate which is a corollary of Hickman-Rogers-Zhang (and Zahl). The x-ray transform estimate is used to deduce that the Kakeya set has a structural property called ``stickiness, which was first introduced in a paper by Katz-Laba-Tao in 1999. Sticky Kakeya sets exhibit a self-similar structure which is exploited by the $k$-planebrush argument.
Changkeun
Title: Discrete restriction estimates for manifolds avoiding a line.
Abstract: We identify a new way to divide the d-neighborhood of surfaces in R^3. We decompose the d-neighborhood of surfaces into a finitely-overlapping collection of rectangular boxes S. We obtain an (l^2,L^p) decoupling estimate using this decomposition, for the sharp range of exponents. The decoupling theorem we prove is new for the hyperbolic paraboloid, and recovers the Tomas-Stein restriction inequality. Our decoupling inequality leads to new exponential sum estimates where the frequencies lie on surfaces which do not contain a line. In this talk, I'll focus on explaning backgrounds and theorems rather than giving proofs. This is joint work with Larry Guth and Dominique Maldague.
Ryan Bushlin
Friday, December 8, 1:20-2:10 in B107
Some variational problems characterizing families of convex domains
We prove a singular integral identity for the surface measure of (n - 1)-rectifiable sets in Euclidean n-space satisfying the orientation cancellation condition. In particular, sets of finite perimeter enjoy this property, and from this observation follows a geometric inequality in which equality is attained precisely by the convex sets. More generally, the integral identity has anisotropic analogues whose corresponding inequalities characterize some geometrically simple subfamilies of the family of convex sets.
Jing-Jing Huang
Diophantine approximation on affine subspaces
We establish a clear-cut criterion for an affine subspace of R^n to be extremal (i.e. the Dirichlet exponent 1/n is best possible almost everywhere), confirming a conjecture of Kleinbock. We also extend the classical theorem of Khintchine on metric diophantine approximation to affine subspaces. Moreover, an exact formula for the Hausdorff dimension of the set of very well approximable vectors on the affine subspace is obtained. The above results are proved as consequences of our novel estimates for the number of rational points lying close to the affine subspace. The proof of this counting estimate is Fourier analytic in nature and in particular utilizes the large sieve inequality.