Mon, Oct 16

Colloquium
Gurbir Dhillon, Yale
Sheaves on semi-infinite flag manifolds and Langlands duality Abstract : Let \(G\), \(G^L\) be Langlands-dual reductive groups. The geometric Satake equivalence is the wonderful fact that one can realize the monoidal category of \(G^L\)-representations as a category of constructible sheaves on a space built from \(G\), namely its affine Grassmannian. After recalling this story, we will present similar realizations of the representations of every parabolic and Levi subgroup of \(G^L\) as sheaves on the semi-infinite partial flag manifolds of \(G\). Time permitting, we will describe some natural extensions, including the realizations of the representations of various quantum groups. The contents of this talk build on work and conjectures of many people, notably Arkhipov, Bezrukavnikov, Braverman, Feigin, Finkelberg, Frenkel, Gaitsgory, Lusztig, Mirkovic, Vilonen, and Raskin, and is the subject of work in progress with Campbell, Chen, Lysenko, and Achar-Riche. 
4:00PM, Mathematics Building, room 250

Title: Sheaves on semi-infinite flag manifolds and Langlands duality

Abstract : Let \(G\), \(G^L\) be Langlands-dual reductive groups. The geometric Satake equivalence is the wonderful fact that one can realize the monoidal category of \(G^L\)-representations as a category of constructible sheaves on a space built from \(G\), namely its affine Grassmannian. After recalling this story, we will present similar realizations of the representations of every parabolic and Levi subgroup of \(G^L\) as sheaves on the semi-infinite partial flag manifolds of \(G\). Time permitting, we will describe some natural extensions, including the realizations of the representations of various quantum groups.

The contents of this talk build on work and conjectures of many people, notably Arkhipov, Bezrukavnikov, Braverman, Feigin, Finkelberg, Frenkel, Gaitsgory, Lusztig, Mirkovic, Vilonen, and Raskin, and is the subject of work in progress with Campbell, Chen, Lysenko, and Achar-Riche.

Thu, Oct 26

Colloquium
Lei Yang, Institute for Advanced Study
Effective versions of Ratner’s equidistribution theorem Speaker host: Hanfeng Li
4:00PM, Mathematics Building, room 250

Title: Effective versions of Ratner’s equidistribution theorem

Speaker host: Hanfeng Li

Fri, Nov 3

Applied Math Seminar
Yijun Sun, UB
TBA.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

TBA.

Fri, Nov 10

Applied Math Seminar
Dmitry Pelinovsky, Mcmaster University
Instability of peaked waves in hydrodynamical models.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

Stokes' wave is a traveling (periodic or solitary) wave with a stagnation point at its crest, where the surface of fluids has a peaked singularity. Stokes waves have been considered within Euler's equations and have been modeled by using reduced equations of motion such as the Camassa-Holm equation, the rotation-modified Ostrovsky equation, and other systems with wave breaking. I will overview the recent analysis of nonlinear, linear, and spectral instability of the traveling peaked waves in some reduced models such as the b-family of the Camassa-Holm equations. Well-posedness of the initial-value problem for this family holds in the energy space as long as the first spatial derivative (the wave slope) is bounded. We show that the travelling peaked waves are unstable due to wave breaking when the wave slope becomes unbounded in a finite time. This instability can also be studied by using the spectral stability analysis within the linearized equations of motion obtained consistently with the well-posedness results.

Fri, Nov 17

Applied Math Seminar
Giselle Sosa Jones, Oakland University
Discontinuous Galerkin discretizations of multiphase flow problems in porous media.

3:00PM, Math 122 and on Zoom - contact mbichuch@buffalo.edu for link

Modeling the flow of liquid, aqueous, and vapor phases through porous media is a complex and challenging task that requires solving nonlinear coupled partial differential equations. In this talk, we propose a second-order accurate and energy-stable time discretization method for the three-phase flow problem in porous media. We prove the convergence of the subiterations to resolve the nonlinearity, and show that the time-stepping method mimics the energy balance relation that the continuous problem satisfies. Our spatial discretization uses an interior penalty discontinuous Galerkin method, for which we establish the well-posedness of the discrete problem and provide error estimates under certain conditions on the data. We validate our method through numerical simulations, which show that our approach achieves the expected theoretical convergence rates. Furthermore, the numerical examples highlight the advantages of our time discretization over other time discretizations.

Wed, Apr 24

Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #1
4:00PM

Title: 2023-24 Myhill Lecture #1

Thu, Apr 25

Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #2
4:00PM

Title: 2023-24 Myhill Lecture #2

Fri, Apr 26

Colloquium
Tomasz Mroka, MIT
2023-24 Myhill Lecture #3
4:00PM

Title: 2023-24 Myhill Lecture #3