MenuEvents for 04/11/2025 from all calendars
Nonlinear Waves and Microlocal Analysis
Time: 1:50PM - 2:50PM
Location: BLOC 302
Speaker: Jonas Luhrmann
Title: Asymptotic stability of kinks outside symmetry
Abstract: We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the \phi^4 model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes. We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving kinks and other Klein-Gordon solitons. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known \phi^4 model. This is joint work with Gong Chen (GeorgiaTech).
Mathematical Physics and Harmonic Analysis Seminar
Time: 1:50PM - 2:50PM
Location: BLOC 302
Speaker: Jonas Luhrmann
Title: Asymptotic stability of kinks outside symmetry
Abstract: We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the \\phi^4 model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes. We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving kinks and other Klein-Gordon solitons. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known \\phi^4 model. This is joint work with Gong Chen (GeorgiaTech).
Departmental Colloquia
Time: 3:00PM - 4:00PM
Location: BLOC 302
Speaker: Victor Reiner, University of Minnesota
Title: Ehrhart theory and a q-analogue
Abstract: Classical Ehrhart theory begins with this fact: for a convex polytope P whose vertices lie in the integer lattice Z^n, the number of lattice points in the integer dilates mP grow as a polynomial function of m. We will review some highlights of the classical theory, and explain a new "q-analogue": it replaces the number of lattice points in mP by a polynomial in q that specializes to the lattice point count at q=1. There are q-analogues for several classical Ehrhart theory results, some proven, others conjectural. In particular, a certain new commutative algebra, and the theory of Macaulay's inverse systems, play a prominent role. (Based on arXiv:2407.06511, with Brendon Rhoades)
Algebra and Combinatorics Seminar
Time: 3:00PM - 4:00PM
Title: Colloquium by Victor Reiner
Geometry Seminar
Time: 4:00PM - 5:00PM
Location: BLOC 302
Speaker: K. Ganapathy, UCSD
Title: Non-noetherianity of GL-varieties
Abstract: GL-varieties are infinite-dimensional varieties equipped with an action of the infinite general linear group GL, satisfying certain mildness conditions. Draisma proved that the topology of GL-varieties exhibits a striking noetherian property. However, whether a scheme-theoretic analogue of this property holds remains a longstanding open question. In this talk, I will present the first counterexample to this problem over fields of characteristic two. This counterexample is closely connected to recent work aimed at salvaging Weyl's theorem on polarization in positive characteristics, which I will also discuss.
