I will offer some new perspectives on the coarse geometric approach to index theory on noncompact manifolds. In real-world applications, coarse indices capture the rigidity of certain macroscopic observables of quantum systems. For example, the quantum Hall effect will be examined from the large-scale spectral geometry and gauge theory perspective, and its role in the 2019 redefinition of the kilogram will be discussed.
For a proper action of a Lie group on a manifold, I will explain how to define the index of an orbital elliptic operator as an element of K-theory of the appropriate crossed product C*-algebra. The orbital index theorem is similar to the index theorem for usual elliptic operators but involves the orbital Dirac operator instead of the Dolbeault element.
The knot signature and more generally the Levine–Tristram signature function provide important classical invariants of knots. The Levine–Tristram signature of a knot can be computed by a signed count of the irreducible SU(2) representations, mod conjugacy, of the knot complement with prescribed meridional traces. This point of view on the Levine–Tristram signature together with the theory of singular instantons can be used to obtain categorifications of Levine–Tristram signature. In fact, such categorifications are one of the various homology theories that is obtained from a more universal object, which is called the singular instanton S-complex of a knot. In this series of talks, I'll discuss the theory of singular instanton S-complexes and some of the additional structures that they admit (including an unoriented Skein exact triangle and Alexander differential operators associated to Seifert surfaces). I will also discuss some topological applications of singular instanton S-complexes.
In this lecture series, I will describe a simplified version of the pseudo-differential calculus which is sufficient in many cases to deal with the index theory problems. This simplified version is compatible with the Hörmander’s PDO calculus.
After that, I will give a brief introduction to K-theory and K-homology for C*-algebras and a sketch of proof of the Atiyah-Singer index theorem. Depending on the time remaining, I plan also to discuss the index theory for hypoelliptic operators on contact manifolds.
Requirements : Some knowledge of the theory of C*-algebras.
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The subject of ''Quantum chaos'' studies the spectrum of Schrödinger operators, when the
underlying classical hamiltonian system is chaotic. How are chaotic properties in classical mechanics mirrorred in quantum mechanics ?
The question is generally treated in the ''semiclassical approximation'', that is, in the regime of small wavelengths, where it is known that quantum mechanics converges to classical mechanics. A typical question is the study of the spectrum of the laplacian on negatively curved manifolds, in relation to the ergodic properties of the geodesic flow. This series of lectures will survey the main conjectures in the domain (most of them still open), and advances of the last 20 years regarding the eigenfunctions of the laplacian on negatively curved manifolds.
Lecture I : Introduction to quantum chaos. Pseudodifferential operators and their use to study the semiclassical approximation.
Lecture II : The Shnirelman theorem
Lectures III, IV : Entropy and support of semiclassical measures (after Anantharaman-Nonnenmacher, Dyatlov-Jin).
Lecture V : Quantum ergodicity on large graphs.
The goal of this lecture series is to give an introduction to the program for understanding mirror symmetry introduced by myself and Siebert. In particular, we have recently given general constructions for mirrors of log Calabi-Yau manifolds with maximally degenerate boundary and Calabi-Yau manifolds equipped with a maximally unipotent degeneration. The key ingredient of these constructions is logarithmic Gromov-Witten theory, constructed by myself and Siebert as well as Abramovich and Chen.
The lectures will begin with a general introduction to logarithmic algebraic geometry and log and punctured Gromov-Witten theory. I will then explain the mirror constructions in terms of canonical wall structures as well as the more direct construction of "Intrinsic Mirror Symmetry".
Finally, I will discuss ongoing work with Keel, Hacking and Siebert, applying these constructions to understand compactifications of the moduli space of K3 surfaces .
A (convex) polytope is the convex hull of finitely many points. Given that the main prerequisite to define and understand a polytope is linear algebra, this topic is suitable for first-year graduate students. This course will give an introduction to the theory of polytopes and their applications to algebraic combinatorics as well as algebraic geometry. During the course we will discuss an important class of polytopes called generalized permutohedra, which has been receiving a lot of attention recently. We will also cover Newton–Okounkov bodies, which allow us to study the geometry of varieties using polytopes.