Abstract: To a probability-measure-preserving action of a countable group, the transformation-groupoid is associated. This talk is focused on the Baumslag-Solitar group. After reviewing invariants and structure of the associated groupoids, I will report that some of them admit a quotient with a treeing, which is an analogue of a free-generating-set of a group, and its application.
Abstract: In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to $C^*$-algebra $K$-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of $p$-adic groups.
This talk will present an extension of the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces which relies on the beautiful geometry of nonpositively curved spaces. The utility of this construction is demonstrated through a new proof of $K$-amenability for groups that act properly on bounded-geometry CAT(0)-cubical spaces. This talk is based on joint work with Nigel Higson and Erik Guentner.
Abstract: During this seminar we shall present the most recent findings about Levinson's theorem. This theorem, first obtained by Levinson in 1949, turns out to be an index theorem in scattering theory. Recently, its topological version has been extended for systems having either a finite number of complex eigenvalues or an infinite number of bound states. Part of these developments are based on a recent work about Schroedinger operators with inverse square potential on the half-line.
Abstract: An orientifold is a proper Lie groupoid $(X_1, X_0, \epsilon)$ with a groupoid homomorphism $\epsilon$ from the space $X_1$ of arrows to $Z_2 = {\pm 1}$. It is a generalization of the notion of orbifold and manifold with involution. An arrow is called even or odd if its value under $\epsilon$ is $+1$ or $-1$. Orientifold vector bundle is a complex vector bundle over $X_0$ on which the even arrows in $X_1$ act complex linearly and the odd arrows act complex anti-linearly. In this talk, I will focus on one particular example (an action orientifold) coming from an action of a Lie group $G$ on a manifold $X$ with a group homomorphism $\epsilon: G \to Z_2$. I will discuss a notion of Dirac operator on a compact Spin^c orientifold whose index provides an element in the unitary-antiunitary representation ring of $(G, \epsilon)$.
Abstract: These lectures are a part of a course on noncommutative geometry and tempered representation theory, but they will be independent of the preceding parts of the course.
The Dirac operator has been studied in representation theory for a long time, beginning with work of Bott on compact groups (if not before) and then continuing with Parthasarathy's construction of the discrete series as spaces of harmonic spinors, then the closely related work of Atiyah and Schmid, and finally (so far) the Connes-Kasparov isomorphism. I shall survey this work. But in addition I will sketch a quite different role for the Dirac operator in the study of discrete series representations.
Abstract: (1) Commutative case. We define twisted Donaldson invariants using the Dirac operators twisted by flat connections when the fundamental group of a four manifold is free abelian. (2) Non commutative case. We generalize the above construction to the non commutative case in the spirit of non commutative geometry. It involves a homomorphism between group cohomology of the fundamental group.
Abstract: I will review a version of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion Spinc structure, and a gluing formula for certain 4-dimensional manifolds splitting along an embedded 3-manifold. As an application, I will discuss a result joint with Vicente Munoz about a product structure on the Seiberg-Witten-Floer homology of a surface times a circle. It was conjectured that this product structure is isomorphic to a product coming from the quantum cohomology of the symmetric product of the surface.
Abstract: In his seminal work on cocycle superrigidity theorem, Popa introduced the class of finite type Polish groups. A Polish group is said to be of finite type if it is embeddable into the unitary group of a II_1 factor. A finite type Polish group must be unitarily representable (i.e. it is embeddable into the unitary group of a Hilbert space with the strong operator topology) and SIN (i.e., it admits a two-sided invariant metric compatible with the topology). He asked a question whether the converse holds, i.e., if a unitarily representable SIN Polish group is necessarily of finite type.
In a joint work with Y. Matsuzawa, A. Thom and A. Tornquist, we show that this is not always the case. A counterexample is constructed from a uniformly bounded non-unitarizable representation (i.e., the one which is not similar to a unitary representation) of a countable discrete group on a Hilbert space. As a byproduct our construction gives a new characterization of unitarizable groups.
Abstract: In this talk we describe a framework of local theory of symplectic geometry and holomorphic maps on the space of oriented immersed loops in a three manifold. These structures are interpreted as transverse objects on the base three manifolds. This is a joint work with Mathai Varghese.
Last updated on May 1, 2017.