Group Actions, Geometry and Dynamics [3] | Exponential mixing via additive combinatorics |
4:00pm -
LOM 206
|
The Bowen-Ruelle conjecture predicts that geodesic flows on negatively curved manifolds are exponentially mixing with respect to all their equilibrium states. In a breakthrough in ‘98, Dolgopyat pioneered a method rooted in the thermodynamic formalism that settled the conjecture for flows satisfying certain strong regularity hypotheses. Soon after, Liverani introduced a more intrinsic refinement of Dolgopyat’s method which overcame these regularity limitations while simultaneously producing more precise rates of mixing, albeit at the price of being limited to smooth invariant measures. Despite these important developments, the conjecture remains open in general even for the measure of maximal entropy. In this talk, I will describe a new approach leveraging inverse theorems in additive combinatorics to overcome the limitations in Liverani’s approach in a concrete algebraic setting, namely the setting of geometrically finite quotients of rank one symmetric spaces. |
Geometry, Symmetry and Physics [4] | Kazhdan-Laumon categories, semi-infinite flags, and the algebra of braids and ties |
4:30pm -
LOM 214
|
We study D. Kazhdan and G. Laumon's 1988 gluing construction for perverse sheaves on the basic affine space G/U and explore unexpected connections to other interesting objects in representation theory. We first define an analogue of Category O in the context of Kazhdan-Laumon categories and explicitly classify its simple objects, and then use this combinatorial data to discuss its connections to Braverman-Kazhdan's Schwartz space on G/U and perverse sheaves on the semi-infinite flag variety. Finally, we study the action of the braid group appearing in the definition of Kazhdan-Laumon categories and give a categorification of the "algebra of braids and ties" occuring in the context of knot theory. |
Geometry & Topology [5] | Circle homeomorphisms with square summable diamond shears |
4:15pm -
LOM 206
|
The shear coordinate is a countable coordinate system to describe increasing self-maps of the unit circle, which is furthermore invariant under modular transformations. Characterizations of circle homeomorphism, quasisymmetric homeomorphisms were obtained by D.Šarić. We are interested in characterizing Weil-Petersson circle homeomorphisms using shears. This class of homeomorphisms arises from the study of Kähler geometry on the universal Teichmüller space and connects various distant fields that will be mentioned briefly. For this, we introduce diamond shear which is the minimal combination of shears producing WP homeomorphisms. Diamond shears are closely related to log-Lambda length introduced by R. Penner. We obtain sharp results comparing the class of circle homeomorphisms with square summable diamond shears with the Weil-Petersson class and Hölder classes. We also express the Weil-Petersson metric tensor and symplectic form in terms of infinitesimal shears and diamond shears. This talk is based on joint work with Dragomir Šarić and Catherine Wolfram. See https://arxiv.org/abs/2211.11497 [6]. |
Applied Mathematics [7] | Is polynomial interpolation in the monomial basis unstable? |
1:00pm -
AKW 200
|
In this talk, we will show that the monomial basis is generally as good as a well-conditioned polynomial basis for interpolation, provided that the condition number of the Vandermonde matrix is smaller than the reciprocal of machine epsilon. We will also show that the monomial basis is more advantageous than other polynomial bases in a number of applications. |
Colloquium [8] | Moduli spaces in tropical geometry |
4:15pm -
LOM 214
|
I will give a hopefully accessible introduction to some work on |
Analysis [9] | The desingularization of small moving corners for the Muskat equation | 4:00pm - |
The Muskat equation models the interaction of two incompressible fluids with equal viscosity propagating in porous medium, governed by Darcy’s law. In this talk, we investigate the small data critical regularity theory for this equation, and in particular, the desingularization of interfaces with small moving corners. This is a joint work with Eduardo Garcia-Juarez (Universidad de Sevilla), Javier Gomez-Serrano (Brown University) and Benoit Pausader (Brown University) |
Friday Morning Seminar [10] | Friday Morning Seminar |
9:30am -
LOM 215
|
A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more! |
Geometric Analysis and Application [11] | TBA |
2:00pm -
LOM 215
|
TBA |
Links
[1] https://math.yale.edu/list/calendar/grid/week/abstract/2023-W13
[2] https://math.yale.edu/list/calendar/grid/week/abstract/2023-W15
[3] https://math.yale.edu/seminars/group-actions-geometry-and-dynamics
[4] https://math.yale.edu/seminars/geometry-symmetry-and-physics
[5] https://math.yale.edu/seminars/geometry-topology
[6] https://arxiv.org/abs/2211.11497
[7] https://math.yale.edu/seminars/applied-mathematics
[8] https://math.yale.edu/seminars/colloquium
[9] https://math.yale.edu/seminars/analysis
[10] https://math.yale.edu/seminars/friday-morning-seminar
[11] https://math.yale.edu/seminars/geometric-analysis-and-application