Spring 2024

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Colloquium**Tomasz Mrowka, MIT**

2023-24 Myhill Lecture #3

4:00PM

Algebra Seminar**Vasily Krylov, MIT**

Around the Hikita-Nakajima conjecture

4:00PM, Zoom (please email achirvas@buffalo.edu

Colloquium**Dr Willy Hereman, Colorado School of Mines**

** **Symbolic** **computation of solitary wavesolutions and solitons through homogenization of degree

4:00PM, Mathematics Building room 250

A simplified version of Hirota's method for thecomputation of solitary waves and solitons of nonlinear PDEs will be presented.The approach requires a change of dependent variable so that the transformedPDE is homogenous of degree in the new variable.

The resulting homogenous PDE does not have tobe quadratic and the method still applies if its bilinear form is not known.Solitons are then computed using a perturbation scheme involving linear andnonlinear operators. For soliton equations the scheme terminates after a finitenumber of steps. To illustrate the approach, solitons are computed for a classof fifth-order KdV equations due to Lax, Sawada-Kotera, and Kaup-Kupershmidt.

Homogenization of degree also allows one tofind solitary wave solutions of nonlinear PDEs that are not completelyintegrable. Examples include the Fisher and FitzHugh-Nagumo equations, and acombined KdV-Burgers equation. When applied to a wave equation with a cubicsource term, the method leads to a `bi-soliton' solution which describes thecoalescence of two wavefronts.

The method is largely algorithmic andimplemented in Mathematica. A demonstration of the software packagePDESolitonsSolutions will be given.

Applied Math Seminar**Willy Hereman, Colorado School of Mines**

Symbolic computation of conservation laws of nonlinear partial differential equations.

3:00PM, Math 122

Algebra Seminar**Ivan Loseu, Yale**

Harish-Chandra centers for affine Kac-Moody algebras in positive characteristic Abstract : This talk is based on a joint work in progress with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90s describes the center of the universal enveloping algebra of an(untwisted) affine Kac-Moody Lie algebra at the so-called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite-dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic \(p\) at an arbitrary non-critical level. Namely, we prove that the algebra of loop-group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite-dimensional affine space that is ``\(p\) times smaller than the Feigin-Frenkel center''. In my talk I will introduce all necessary notions, state the result, explain motivations and examples.

4:00PM, Zoom (please email achirvas@buffalo.edu)

Title: Harish-Chandra centers for affine Kac-Moody algebras in positive characteristic

Abstract : This talk is based on a joint work in progress with Gurbir Dhillon. A remarkable theorem of Feigin and E. Frenkel from the early 90s describes the center of the universal enveloping algebra of an(untwisted) affine Kac-Moody Lie algebra at the so-called critical level proving a conjecture of Drinfeld: the center in question is the algebra of polynomial functions on an infinite-dimensional affine space known as the space of opers. In our work we study a part of the center in positive characteristic \(p\) at an arbitrary non-critical level. Namely, we prove that the algebra of loop-group invariants in the completed universal enveloping algebra is still the algebra of polynomials on an infinite-dimensional affine space that is ``\(p\) times smaller than the Feigin-Frenkel center''. In my talk I will introduce all necessary notions, state the result, explain motivations and examples.

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