Fall 2023 Analysis Seminar: Difference between revisions
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===[[Tushar Das]]=== | ===[[Tushar Das]]=== | ||
Title: | Title: Playing games on fractals: Dynamical & Diophantine | ||
Abstract: | 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. | ||
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]]=== | ===[[Rajula Srivastava]]=== | ||
Revision as of 15:28, 5 September 2023
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 | analysis group | ||
| 4 | We, Sept. 27 | |||||
| 5 | We, Oct. 4 | Tristan Leger | Princeton | Simon | ||
| 6 | We, Oct. 11 | Bingyang Hu | Auburn | Brian | ||
| 7 | We, Oct. 18 | |||||
| 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.