MIT PDE/Analysis Seminar

Building 2, 2-105

Special seminar with two speakers, starting at 3 PM.Speakers: Francois Pagano (University of Geneva) and Luiz Hartmann (Federal University of São Carlos, Brazil)________________________________Speaker: Francois Pagano (University of Geneva)Title: Localization and Eigenfunctions to Second-Order Elliptic PDEsAbstract: In the 70’s, Anderson studied the motion of electrons in materials. If the atomic structure is periodic, electrons can travel freely: the material conducts electricity. On the other hand, if the material has impurities or if the atomic structure is more random, electrons can get trapped: the material is now an insulator. Anderson received the Nobel Prize in Physics for this discovery in ’77. Understanding this question mathematically amounts to understanding the nature of the spectrum for a periodic or random Schrödinger operator. In this talk, we will first illustrate, using results from Kuchment (’12) and Bourgain-Kenig (’05), how this problem is related to the following (deterministic) question going back to Landis (late 60’s): given A elliptic, C^1 (or smoother) and V bounded, how rapidly can a non-trivial solution to −div(A∇u) + V u = 0 decay to zero at infinity?We will discuss the construction of an operator on the cylinder T^2 × R with an eigenfunction div(A∇u) = −μu, which has double exponential decay at both ±∞, where A is uniformly elliptic and uniformly C^1 smooth in the cylinder. Joint work with S. Krymskii and A. Logunov._____________________________Speaker: Luiz Hartmann (Federal University of São Carlos, Brazil)Title: An Atiyah-Bott formula for the Lefschetz number of a singular flotationAbstract: Let N be a closed Riemannian manifold, and T a smooth nowhere-vanishing Killing vector field. The closure of the orbits of T forms a torus denoted as G, which acts on N . In this talk, I will explore the Lefschetz number of a smooth map f : N → N that preserves T and aim to derive a formula inspired by Atiyah-Bott’s style. To achieve this, we will examine the de Rham complex of smooth forms on N in the kernel of the interior multiplication by T while being invariant under T ’s action. The cohomology of this complex is finite-dimensional, ensuring the welldefined nature of the Lefschetz number. It is worth noting that these cohomology groups are related to, but not identical to, those of the basic cohomology. In conclusion, I will provide an overview of the proof strategy for the Atiyah-Bott formula. This is a joint work in progress with Gerardo Mendoza (Tempe University).