Colloquia: Difference between revisions
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|Sara Maloni (UVA) | |[https://sites.google.com/view/sara-maloni Sara Maloni] (UVA) | ||
| | |Some new results in Higher Teichmüller Theory | ||
|Dymarz, Uyanik, GmMaW | |Dymarz, Uyanik, GmMaW | ||
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A celebrated theorem of Thurston tells us that among the many ways of filling in cusps of hyperbolic $3$--manfiolds, all but finitely many of them produce hyperbolic manifolds once again. This finiteness may be refined in a number of ways depending on the ``shape’’ of the cusp, and I’ll give a light and breezy discussion of joint work with K. Bromberg and Y. Minsky that allows shapes not covered by any of the previous theorems. This has applications such as answering questions asked in my 2010 job talk here at UW. | A celebrated theorem of Thurston tells us that among the many ways of filling in cusps of hyperbolic $3$--manfiolds, all but finitely many of them produce hyperbolic manifolds once again. This finiteness may be refined in a number of ways depending on the ``shape’’ of the cusp, and I’ll give a light and breezy discussion of joint work with K. Bromberg and Y. Minsky that allows shapes not covered by any of the previous theorems. This has applications such as answering questions asked in my 2010 job talk here at UW. | ||
''' Friday, October 20. Sara Maloni''' | |||
The Teichmüller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, the Teichmüller space can also be seen as a connected component of (conjugacy classes of) representations from the fundamental group of S into PSL(2,R), consisting entirely of discrete and faithful representations. Generalizing this point of view, Higher Teichmüller Theory studies connected components of (conjugacy classes of) representations from the fundamental group of S into more general semisimple Lie groups which consist entirely of discrete and faithful representations. | |||
We will give a survey of some aspects of Higher Teichmüller Theory and will make links with the recent powerful notion of Anosov representation. We will conclude by focusing on two separate questions: Do these representations correspond to deformation of geometric structures? | |||
Can we generalize the important notion of pleated surfaces to higher rank Lie groups like PSL(d, C)? | |||
The answer to question 1 is joint work with Alessandrini, Tholozan and Wienhard, while the answer to question 2 is joint work with Martone, Mazzoli and Zhang. | |||
'''Wednesday, October 25. Gigliola Staffilani''' | '''Wednesday, October 25. Gigliola Staffilani''' | ||
Revision as of 13:11, 13 October 2023
UW Madison mathematics Colloquium is on Fridays at 4:00 pm in Van Vleck B239 unless otherwise noted.
Fall 2023
| date | speaker | title | host(s) | |
|---|---|---|---|---|
| Sept 8 | Tushar Das (University of Wisconsin-La Crosse) | Playing games on fractals: Dynamical & Diophantine | Stovall | |
| Sept 15 | John Schotland (Yale) | Nonlocal PDEs and Quantum Optics | Li | |
| Sept 22 | David Dumas(University of Illinois Chicago) | Geometry of surface group homomorphisms | Zimmer | |
| Sept 29 | no colloquium (see Monday) | |||
| Monday Oct 2 at 4 pm | Edriss Titi (Texas A&M University) | Distinguished lectures: On the Solvability of the Navier-Stokes and Euler Equations, where do we stand? | Smith, Stechmann | |
| Oct 13 | Autumn Kent | The 0π Theorem | ||
| Oct 20 | Sara Maloni (UVA) | Some new results in Higher Teichmüller Theory | Dymarz, Uyanik, GmMaW | |
| Wednesday Oct 25 at 4 pm | Gigliola Staffilani (MIT) | The Schrödinger equations as inspiration of beautiful mathematics | Ifrim, Smith | |
| Oct 27 | Rodrigo Bañuelos (Purdue) | Probabilistic tools in discrete harmonic analysis | Stovall | |
| Tuesday Oct 31 at 4 pm | Irit Dinur (The Weizmann Institute of Science) | Distinguished lectures | Gurevich | |
| Wednesday Nov 1 at 4 pm | Irit Dinur (The Weizmann Institute of Science) | Distinguished lectures | Gurevich |
Abstracts
Friday, September 8. Tushar Das
Playing games on fractals: Dynamical & Diophantine 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.
Friday, September 15. John Schotland
Nonlocal PDEs and Quantum Optics Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics with applications to many body problems. The goal is to understand the transport of nonclassical states of light in random media. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.
Friday, September 22. David Dumas
The space of homomorphisms from the fundamental group of a compact surface to a Lie group is a remarkably rich and versatile object, playing a key role in mathematical developments spanning disciplines of algebra, analysis, geometry, and mathematical physics. In this talk I will discuss and weave together two threads of research within this larger story: 1) the study of manifolds that are obtained by taking quotients of symmetric spaces (the "inside view") and 2) those obtained as quotients of domains in flag varieties (the "boundary view"). This discussion will start with classical objects--hyperbolic structures on surfaces---and continue into topics of ongoing research.
Friday, October 13. Autumn Kent
A celebrated theorem of Thurston tells us that among the many ways of filling in cusps of hyperbolic $3$--manfiolds, all but finitely many of them produce hyperbolic manifolds once again. This finiteness may be refined in a number of ways depending on the ``shape’’ of the cusp, and I’ll give a light and breezy discussion of joint work with K. Bromberg and Y. Minsky that allows shapes not covered by any of the previous theorems. This has applications such as answering questions asked in my 2010 job talk here at UW.
Friday, October 20. Sara Maloni
The Teichmüller space of a surface S is the space of marked hyperbolic structure on S, up to equivalence. By considering the holonomy representation of such structures, the Teichmüller space can also be seen as a connected component of (conjugacy classes of) representations from the fundamental group of S into PSL(2,R), consisting entirely of discrete and faithful representations. Generalizing this point of view, Higher Teichmüller Theory studies connected components of (conjugacy classes of) representations from the fundamental group of S into more general semisimple Lie groups which consist entirely of discrete and faithful representations.
We will give a survey of some aspects of Higher Teichmüller Theory and will make links with the recent powerful notion of Anosov representation. We will conclude by focusing on two separate questions: Do these representations correspond to deformation of geometric structures? Can we generalize the important notion of pleated surfaces to higher rank Lie groups like PSL(d, C)? The answer to question 1 is joint work with Alessandrini, Tholozan and Wienhard, while the answer to question 2 is joint work with Martone, Mazzoli and Zhang.
Wednesday, October 25. Gigliola Staffilani
In the last two decades great progress has been made in the study of dispersive and wave equations. Over the years the toolbox used in order to attack highly nontrivial problems related to these equations has developed to include a collection of techniques: Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of results using as model problem mainly the periodic 2D cubic nonlinear Schrödinger equation. I will start by giving a physical derivation of the equation from a quantum many-particles system, I will introduce periodic Strichartz estimates along with some remarkable connections to analytic number theory, I will move on the concept of energy transfer and its connection to dynamical systems, and I will end with some results following from viewing the periodic nonlinear Schrödinger equation as an infinite dimensional Hamiltonian system.
Friday, October 27. Rodrigo Bañuelos
Probabilistic tools in discrete harmonic analysis
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.