Noticias en español
Differential-difference equations arising in number theory.
Abstract: In an attempt to find a more intuitive proof of the Prime Number Theorem, Lord Cherwell derived, through heuristic arguments, the equation: f'(x) = -(f(x) f(\sqrt{x})/(2x), where f(x) represents the “density of primes at x”. Through a simple change of variables, the differential equation can be rewritten as the following delay differential equation:h'(u) = -(ln 2)(h(u) + 1)h(u – 1) which marks the first appearance of this type of equation in number theory. In this talk, we present other families of differential equations, both with delay and advance, related to...
Read MoreDomain Branching in Micromagnetism.
Abstract: Nonconvex variational problems regularized by higher order terms have been used to describe many physical systems, including, for example, martensitic phase transformation, micromagnetics, and the Ginzburg–Landau model of nucleation. These problems exhibit microstructure formation, as the coefficient of the higher order term tends to zero. They can be naturally embedded in a whole family of problems of the form: minimize E(u)= S(u)+N(u) over an admissible class of functions u taking only two values, say -1 and 1, with a nonlocal interaction N favoring small-scale phase...
Read MoreRecent Progress on the Fractional Yamabe Problem.
Abstract: Let $(M^n, [\hat{g}])$ be the conformal infinity of an asymptotically hyperbolic Einstein (AHE) manifold $(X^{n+1},g^+).$ We will take the scattering operator associated to the AHE filling in as the fractional conformal Laplacian. Equipped with fractional conformal Laplacians defined via the AHE manifold, we can define a fractional Yamabe problem, looking for a conformal metric of $(M^n,[\hat{g}])$ which has constant fractional scalar curvature. We will present some new developments on the fractional Yamabe problem assuming an AHE filling in.
Read MoreAbstract: In this talk, we examine a concentration phenomenon for solutions to the constant Q-curvature equation, a critical fourth-order equation on a closed Riemannian manifold. The challenge of finding constant Q-curvature metrics is closely linked to the Yamabe problem and arises from the goal of identifying optimal metrics for a given compact, boundaryless manifold. In this work, we address the problem on a product Riemannian manifold. We will start by briefly introducing the concept of Q-curvature and then outline the main ideas for finding solutions that concentrate around specific...
Read MoreAsymptotic stability of kinks in the odd energy space.
Abstract: In this talk I will first present a 10 years old result about the asymptotic stability of the kink in the classical φ^4 model under the assumption of oddness of the initial perturbations. I will explain how the problem can be decomposed into radiation and internal modes and how the components can be controlled through virial estimates. This result depends on some numerical approximations and its proof can be viewed as computer assisted. Recently, we were able to generalize the asymptotic stability result to one dimensional scalar field models with one internal mode. I will show...
Read MoreBlow-up Analysis of Large Conformal Metrics With Prescribed Gaussian And Geodesic Curvatures
Abstract: In this talk, we consider a compact Riemannian surface (M,g) with nonempty boundary and negative Euler characteristic. Given two smooth non-constant functions f in M and h in the boundary of M with max f = max h = 0, under a suitable condition on the maximum points of f and h, we prove that for sufficiently small positive constants λ and μ, there exist at least two distinct conformal metrics g_{λ,μ}=e^{2u_{μ,λ}}g and g^{λ,μ}=e^{2u^{μ,λ}}g with prescribed sign-changing Gaussian and geodesic curvature equal to f+μ and h+λ, respectively. Additionally, we employ the method Borer et...
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