Fall 2023 Analysis Seminar

Organizer: Shaoming Guo

Email: shaomingguo (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+join (at) g-groups (dot) wisc (dot) edu

Week	Date	Speaker	Institution	Title	Host(s)	Notes (e.g. unusual room/day/time)
1	We, Sep. 6
	Fr, Sept. 8	Tushar Das	UW La Crosse	Playing games on fractals: Dynamical and Diophantine	Betsy	Colloquium, 4-5pm in B239
2	Tu, Sept. 12	Rajula Srivastava	Hausdorff Center of Mathematics, Bonn	Counting Rational Points near Flat Hypersurfaces	Andreas	Tuesday 4:00 pm in VV B135
	We, Sept. 13	Niclas Technau	University of Graz	Oscillatory Integrals Count	Andreas
3	We, Sept. 20	Terry Harris	UW Madison	Horizontal Besicovitch sets of measure zero and some related problems	analysis group
4	We, Sept. 27	Danold Stull	U Chicago		Jake
5	We, Oct. 4	Tristan Leger	Princeton		Simon
6	We, Oct. 11	Bingyang Hu	Auburn		Brian
7	We, Oct. 18	Ashley Zhang	Vanderbilt		Alexei
8	We, Oct. 25	Gigliola Staffilani	MIT		Mihaela and Leslie	Special Colloquium 4-5pm in B239
	Fr, Oct 27	Rodrigo Bañuelos	Purdue		Betsy	Colloquium, 4-5pm in B239
9	We, Nov. 1	Tent scheduled distinguished lecture				Distinguished lecture 4-5pm in B239
10	We, Nov. 8	Lechao Xiao	Google deepmind		Shaoming
11	We, Nov. 15	Neeraja Kulkarni	Caltech		Jacob
12	We, Nov. 22	No talk	No talk		No talk	Thanksgiving week
13	We, Nov. 29	Changkeun Oh	MIT		Shaoming
14	We, Dec. 6
15	We, Dec. 13
1	We, Jan. 24, 2024
2	We, Jan. 31
3	We, Feb. 7
4	We, Feb. 14
5	We, Feb. 21
6	We, Feb. 28	Alex Rutar	University of St. Andrews		Andreas
7	We, Mar. 6	Song-Ying Li	UC-Irvine		Xianghong
8	We, Mar. 13
9	We, Mar. 20
	Fr, Mar. 22	Jack Lutz	Iowa State University		Shaoming	department colloquium, 4-5pm
10	We, Mar. 27	Spring recess	spring recess		spring recess
11	We, Apr. 3
12	We, Apr. 10
13	We, Apr. 17
14	We, Apr. 24
15	We, May 1

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.

Links to previous seminars