Fri, Apr 12
Applied Math Seminar
Alexandr Chernyavskiy, UB
Dark-bright soliton perturbation theory for the Manakov system.
3:00PM, Math 122
A direct perturbation method for studying dynamics of
dark-bright solitons of the Manakov system in the presence of
perturbations is presented. We combine multiscale expansion method,
perturbed conservation laws, and a boundary layer approach, which breaks
the problem into an inner region, where the bulk of the soliton resides,
and an outer region, which evolves independently of the soliton. We show
that a shelf develops around the dark soliton component, with speed of the
shelf proportional to the background intensity. Conservation laws of
the Manakov system are used to determine the properties of the shelf and
perturbed solutions. Our analytical predictions are corroborated by
numerical simulations..
Fri, Apr 12
Geometry and Topology Seminar
Indira Chatterji (University of Côte d'Azur / Fields Institute)
Property T versus aTmenability
Abstract : A group has property T if any action on a Hilbert space has a fixed point, and a group is called aTmenable if it admits a proper action on a Hilbert space by affine isometries. I will review what classical groups have property T, or aTmenability, or neither, how those notions appeared and what they are good for.
4:00PM, 122 Mathematics Building
Title: Property T versus aTmenability
Abstract : A group has property T if any action on a Hilbert space has a fixed point, and a group is called aTmenable if it admits a proper action on a Hilbert space by affine isometries. I will review what classical groups have property T, or aTmenability, or neither, how those notions appeared and what they are good for.
Mon, Apr 15
Algebra Seminar
Dani Szpruch, Open University of Israel
An analog of the Hasse-Davenport product relation for \(\epsilon\)-factors and an application
3:00PM, 250 Mathematics Building
The classical Hasse-Davenport product relation is an identity involving products of Gauss sums defined over a finite field. In this talk we shall introduce some generalizations of this classical result for Tate \(\epsilon\)-factors and closely related arithmetic factors defined over a p-adic field. We will then show that these generalizations are equivalent to a certain representation-theoretic identity involving an analog of Shahidi local coefficients for covering groups.
Wed, Apr 17
Analysis Seminar
Andy Zucker, University of Waterloo
Ultracoproducts and weak containment for flows of topological groups
4:00PM, 250 Math Building
We develop the theory of ultracoproducts and weak containment for flows of arbitrary topological groups. This provides a nice complement to corresponding theories for p.m.p. actions and unitary representations of locally compact groups. We isolate a new class of topological groups, which we call Fubini groups, for which iterated ultracopowers of certain G-flows behave nicely. Among the Fubini groups are the class of locally Roelcke precompact groups, for which the theory is especially rich. For these groups, we can define for certain families of G-flows a suitable compact space of weak types. When G is locally compact, all G-flows belong to one such family, yielding a single compact space describing all weak types of G-flows.
Fri, Apr 19
Applied Math Seminar
Anna Vainchtein, University of Pittsburgh
Supersonic fronts and pulses in a lattice with hardening-softening interactions.
3:00PM, Math 122
This talk is based on recent joint work with Lev Truskinovsky (ESPCI ParisTech). We consider a version of the classical Hamiltonian Fermi-Pasta-Ulam problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink-antikink bundle. We investigate stability of the obtained solutions via Floquet analysis and direct numerical simulations. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem..
Tue, Apr 23
Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #1
4:00PM
Title: 2023-24 Myhill Lecture #1
Wed, Apr 24
Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #2
4:00PM
Title: 2023-24 Myhill Lecture #2
Thu, Apr 25
Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #3
4:00PM
Title: 2023-24 Myhill Lecture #3
Thu, May 2
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.
Fri, May 3
Applied Math Seminar
Willy Hereman, Colorado School of Mines
Symbolic computation of conservation laws of nonlinear partial differential equations.
3:00PM, Math 122
A method will be presented for the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs) involving multiple space variables and time.
Using the scaling symmetries of the PDE, the conserved densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative is used to compute the undetermined coefficients. The homotopy operator is used to invert the divergence operator, leading to the analytic expression of the flux vector.
The method is algorithmic and has been implemented in Mathematica. The software is being used to compute conservation laws of nonlinear PDEs occurring in the applied sciences and engineering.
The software package will be demonstrated for PDEs that model shallow water waves, ion-acoustic waves in plasmas, sound waves in nonlinear media, and transonic gas flow. Equations featured in this talk include the Korteweg-de Vries and Zakharov-Kuznetsov equations..