A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side
A Local Relative Trace Formula for the Ginzburg-Rallis Model: The Geometric Side

Author: Chen Wan

Publisher: American Mathematical Soc.

Published: 2019-12-02

Total Pages: 102

ISBN-13: 1470436868

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Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.

Geometric Optics for Surface Waves in Nonlinear Elasticity
Geometric Optics for Surface Waves in Nonlinear Elasticity

Author: Jean-François Coulombel

Publisher: American Mathematical Soc.

Published: 2020-04-03

Total Pages: 164

ISBN-13: 1470440377

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This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. The authors consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which is referred to as “the amplitude equation”, is an integrodifferential equation of nonlocal Burgers type. The authors begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions uε to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength ε, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to uε on a time interval independent of ε. This paper focuses mainly on the case of Rayleigh waves that are pulses, which have profiles with continuous Fourier spectrum, but the authors' method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.

Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces

Author: Luigi Ambrosio

Publisher: American Mathematical Soc.

Published: 2020-02-13

Total Pages: 134

ISBN-13: 1470439131

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The aim of this paper is to provide new characterizations of the curvature dimension condition in the context of metric measure spaces (X,d,m). On the geometric side, the authors' new approach takes into account suitable weighted action functionals which provide the natural modulus of K-convexity when one investigates the convexity properties of N-dimensional entropies. On the side of diffusion semigroups and evolution variational inequalities, the authors' new approach uses the nonlinear diffusion semigroup induced by the N-dimensional entropy, in place of the heat flow. Under suitable assumptions (most notably the quadraticity of Cheeger's energy relative to the metric measure structure) both approaches are shown to be equivalent to the strong CD∗(K,N) condition of Bacher-Sturm.

Global Smooth Solutions for the Inviscid SQG Equation
Global Smooth Solutions for the Inviscid SQG Equation

Author: Angel Castro

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 89

ISBN-13: 1470442140

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In this paper, the authors show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.

Degree Theory of Immersed Hypersurfaces
Degree Theory of Immersed Hypersurfaces

Author: Harold Rosenberg

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 62

ISBN-13: 1470441853

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The authors develop a degree theory for compact immersed hypersurfaces of prescribed $K$-curvature immersed in a compact, orientable Riemannian manifold, where $K$ is any elliptic curvature function.

Rigid Character Groups, Lubin-Tate Theory, and (φ,Γ)-Modules
Rigid Character Groups, Lubin-Tate Theory, and (φ,Γ)-Modules

Author: Laurent Berger

Publisher: American Mathematical Soc.

Published: 2020-04-03

Total Pages: 92

ISBN-13: 1470440733

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The construction of the p-adic local Langlands correspondence for GL2(Qp) uses in an essential way Fontaine's theory of cyclotomic (φ,Γ)-modules. Here cyclotomic means that Γ=Gal(Qp(μp∞)/Qp) is the Galois group of the cyclotomic extension of Qp. In order to generalize the p-adic local Langlands correspondence to GL2(L), where L is a finite extension of Qp, it seems necessary to have at our disposal a theory of Lubin-Tate (φ,Γ)-modules. Such a generalization has been carried out, to some extent, by working over the p-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of this article is to carry out a Lubin-Tate generalization of the theory of cyclotomic (φ,Γ)-modules in a different fashion. Instead of the p-adic open unit disk, the authors work over a character variety that parameterizes the locally L-analytic characters on oL. They study (φ,Γ)-modules in this setting and relate some of them to what was known previously.

Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on R
Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on R

Author: Peter Poláčik

Publisher: American Mathematical Soc.

Published: 2020-05-13

Total Pages: 100

ISBN-13: 1470441128

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The author considers semilinear parabolic equations of the form ut=uxx+f(u),x∈R,t>0, where f a C1 function. Assuming that 0 and γ>0 are constant steady states, the author investigates the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x,0) are near γ for x≈−∞ and near 0 for x≈∞. If the steady states 0 and γ are both stable, the main theorem shows that at large times, the graph of u(⋅,t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). The author proves this result without requiring monotonicity of u(⋅,0) or the nondegeneracy of zeros of f. The case when one or both of the steady states 0, γ is unstable is considered as well. As a corollary to the author's theorems, he shows that all front-like solutions are quasiconvergent: their ω-limit sets with respect to the locally uniform convergence consist of steady states. In the author's proofs he employs phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories {(u(x,t),ux(x,t)):x∈R}, t>0, of the solutions in question.

Affine Flag Varieties and Quantum Symmetric Pairs
Affine Flag Varieties and Quantum Symmetric Pairs

Author: Zhaobing Fan

Publisher: American Mathematical Soc.

Published: 2020-09-28

Total Pages: 123

ISBN-13: 1470441756

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The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$.