2013–2014 Abstracts
Fall 2013
6 September 2013
- Yuri Berest, Cornell University
- Dixmier Groups and the Dixmier Conjecture
- I will introduce a class of infinite-dimensional (ind-)algebraic groups closely related to the group of polynomial automorphisms of the affine plane. Originating in the theory of integrable systems these groups have an interesting structure and share many properties with finite-dimensional semisimple affine algebraic groups. In particular, we prove an infinite-dimensional analogue of the classical theorem of R. Steinberg giving an abstract characterization of Borel subgroups in affine algebraic groups, and we will use this last result to actually classify Borel subgroups up to conjugation. The proofs are not entirely geometric nor algebraic: the crucial ingredient is the Friedland-Milnor analytic classification of polynomial automorphisms in $\mathbf{C}^2$ (and its recent algebraic refinement due to S. Lamy). As a motivation I will explain a relation to the Jacobian Conjecture and its ‘quantum’ counterpart—the Dixmier Conjecture.
27 September 2013
- Geoffrey Scott, University of Michigan
- Torus actions on b-symplectic manifolds
- Delzant's theorem in symplectic geometry states that Delzant polytopes in $\mathbf{R}^n$ classify symplectic toric manifolds of dimension $2n$. In this talk, we generalize this result to b-symplectic manifolds, where the symplectic form is singular along a hypersurface. In particular, we show that b-symplectic toric manifolds are classified by Delzant polytopes in a certain enlargement of $\mathbf{R}^n$.
10 October 2013
- Toshiyuki Kobayashi, IPMU, University of Tokyo
- "Universal sounds" of anti-de Sitter manifolds
- In musical instruments, shorter strings produce a higher pitch than longer strings, and thinner strings produce a higher pitch than thicker strings. Similarly, in compact Riemann surfaces, any nonzero eigenvalue of the Laplacian varies as a function on Teichmüller space. In Riemannian geometry, the eigenvalues of the Laplacian of a compact manifold provide rich information about the original manifold, and some aspects of spectral geometry are well known from the article “Can one hear the shape of a drum?” (M. Kac, 1966).
- What properties can be found beyond Riemannian geometry? I plan to talk about a strange phenomenon in anti-de Sitter manifolds: there exist countably many stable eigenvalues of the Laplacian that do not vary under the deformation of geometric structure. The proof uses the geometry of discrete groups, partial differential equations, integral geometry (e.g., the idea of a CT scan), and various ideas coming from Lie groups.
- The talk will be aimed at a general mathematical audience, and a main part will be accessible to undergraduates with basic knowledge of advanced calculus, linear algebra, and (a little) topology. The non-expository part of the talk is based on joint work with F. Kassel.
11 October 2013
- Toshiyuki Kobayashi, IPMU, University of Tokyo
- Global Geometry and Analysis on Locally pseudo-Riemannian Symmetric Spaces
- The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure. Taking anti-de Sitter manifolds, which are locally modelled on AdS$^n$ as an example, I plan to explain two programs:
- 1. (global shape) Existence problem of compact locally homogeneous spaces, and defomation theory.
- 2. (spectral analysis) Construction of the spectrum of the Laplacian, and its stability under the deformation of the geometric structure.
18 and 25 October 2013
- Bruce Fontaine, Cornell University
- Cluster Algebras—Parts 1 and 2
- In 2001, while investigating the connection between total positivity and the canonical basis, Fomin and Zelevinsky developed the theory of cluster algebras. I will show how these two topics inspired the definitions of a cluster algebra and will show how one obtains a cluster structure on the coordinate ring of the base affine space, $\mathbf{C}[\mathbf{SL}_n/N]$.
1 and 8 November 2013
- Raúl Gómez, Cornell University
- Theta lifting of generalized Whittaker models associated with nilpotent orbits I and II
- Howe's theory of reductive dual pairs is one of the most useful tools for understanding representations of classical groups. In this series of talks we will explore the compatibility between Howe's Theta-lifting correspondence and some of the nilpotent invariants associated to a representation of a reductive group G. (Like wave front set, generalized Whittaker models, associated cycle...)
- In the first talk, I will review the classification of nilpotent orbits for a classical group and introduce the nilpotent invariants of interest. Then I will discuss a generalization of a result of Moeglin regarding the compatibility of Howe's Theta-lifting correspondence and the space of generalized Whittaker models.
- In the second talk I will describe further improvements to this result involving invariant linear functionals and look at some applications.
15 November 2013
- Yoshiki Oshima, University of Tokyo and IAS
- Discrete branching laws for some unitary representations
- The Zuckerman functor provides a certain important class of unitary representations of real reductive Lie groups. We consider the restriction of these representations to reductive subgroups. T. Kobayashi introduced a nice framework for branching problems, the notion of "discretely decomposable restrictions". In this framework we obtain explicit branching formulas by using D-modules on the flag variety.
22 November and 6 December 2013
- Yuri Berest, Cornell University
- Representation homology and strong Macdonald conjectures I and II
- In the early 1980s, I. Macdonald discovered a number of highly non-trivial combinatorial identities related to a semisimple complex Lie algebra $\mathfrak{g}$. These identities were under intensive study for a decade until they were proved by I.Cherednik using his theory of double affine Hecke algebras. One of the key identities in Macdonald's list—the so-called constant term identity—has a natural homological interpretation: it formally follows from the fact that the Lie algebra cohomology of a truncated current Lie algebra over $\mathfrak{g}$ is a free exterior algebra with generators of prescribed degree (depending on $\mathfrak{g}$). This last fact (called the strong Macdonal conjecture) was proposed by P. Hanlon and B. Feigin in the 80s and proved only recently by S. Fishel, I. Grojnowski and C. Teleman.
- I will present analogues (in fact, generalizations) of strong Macdonald conjectures arising from homology of derived representation schemes. In the first talk, I will explain what classical and derived representation schemes are and how they can be constructed. In the second talk, I will explain the relation to Lie algebra cohomology and focus on conjectures and examples.
Spring 2014
14 and 28 February 2014
- Reyer Sjamaar, Cornell University
- Equivariant cohomology
- Equivariant cohomology is a tool for studying group actions on topological spaces invented by A. Borel. I will give an introduction to the subject accessible to those with a basic knowledge of fibre bundles, cohomology theory and Lie groups.
21 February 2014
- Alexander Caviedes Castro, University of Toronto
- Upper bounds for the Gromov width of coadjoint orbits
- I will show how to find an upper bound for the Gromov width of coadjoint orbits with respect to the Kirillov-Kostant-Souriau symplectic form by computing certain Gromov-Witten invariants. The approach presented here is closely related to the one used by Gromov in his celebrated non-squeezing theorem.
7 and 21 March 2014
- Bruce Fontaine, Cornell University
- An introduction to perverse sheaves
- These are two expository lectures on perverse sheaves and their uses. I plan to discuss the definition and properties of perverse sheaves along with the most important result concerning them: the decomposition theorem of Beilinson, Bernstein and Deligne. The goal of the second lecture will be to understand some of the uses of this result in the context of Lie groups, such as Lusztig's construction of the canonical basis and the geometric Satake correspondence.
18 and 25 April 2014
- Marcelo Aguiar, Cornell University
- A peek into Lie theory relative to a hyperplane arrangement
- A result due to Joyal, Klyachko, and Stanley relates free Lie algebras to partition lattices. We will discuss the precise relationship and interpret the result in terms of the braid hyperplane arrangement. We will then extend the result to arbitrary (finite, real, and central) hyperplane arrangements. Important elements are work of Björner and Wachs on the topology of geometric lattices and a generalization of the classical Dynkin idempotent that we introduce.
- This is part of joint work with Swapneel Mahajan in which we extend several aspects of the classical theory of the free Lie algebra to the setting of hyperplane arrangements and develop a corresponding Hopf-Lie dictionary.
2 May 2014
- Mahdi Asgari, Cornell University
- The Langlands $L$-group
- In his 1967 letter to Andre Weil, Robert Langlands outlined, among other things, what came to be known as the Langlands $L$-group. It is a semi-direct product of a Galois piece and a complex reductive group and it turns out to play a crucial role in the Langlands conjectures, both locally and globally and may other places. I will attempt to introduce this construct and illustrate its importance in the representation theory of p-adic groups (of characteristic zero—for this talk), and perhaps also some global aspects of its role if there is time. This will be a survey talk.