GRASP Lecture Series: Geometry, Representations, and some Physics

The Hodge theory arising in homological mirror symmetry. Pure nc Hodge structures and their Betti and de Rham aspects. Categorical nc geometry, variations of nc Hodge structures, nc Deligne cohomology, Griffiths groups, and normal functions. Examples from singularity theory, symplectic topology, and complex geometry. Fukaya and matrix factorization categories. Chern-Simons functionals. Motivic local systems and nc Hodge structures: the Fano case. Mirror maps for del Pezzo surfaces.

The Hodge theory arising in homological mirror symmetry. Pure nc Hodge structures and their Betti and de Rham aspects. Categorical nc geometry, variations of nc Hodge structures, nc Deligne cohomology, Griffiths groups, and normal functions. Examples from singularity theory, symplectic topology, and complex geometry. Fukaya and matrix factorization categories. Chern-Simons functionals. Motivic local systems and nc Hodge structures: the Fano case. Mirror maps for del Pezzo surfaces.

The Hodge theory arising in homological mirror symmetry. Pure nc Hodge structures and their Betti and de Rham aspects. Categorical nc geometry, variations of nc Hodge structures, nc Deligne cohomology, Griffiths groups, and normal functions. Examples from singularity theory, symplectic topology, and complex geometry. Fukaya and matrix factorization categories. Chern-Simons functionals. Motivic local systems and nc Hodge structures: the Fano case. Mirror maps for del Pezzo surfaces.

The Hodge theory arising in homological mirror symmetry. Pure nc Hodge structures and their Betti and de Rham aspects. Categorical nc geometry, variations of nc Hodge structures, nc Deligne cohomology, Griffiths groups, and normal functions. Examples from singularity theory, symplectic topology, and complex geometry. Fukaya and matrix factorization categories. Chern-Simons functionals. Motivic local systems and nc Hodge structures: the Fano case. Mirror maps for del Pezzo surfaces.

The Hodge theory arising in homological mirror symmetry. Pure nc Hodge structures and their Betti and de Rham aspects. Categorical nc geometry, variations of nc Hodge structures, nc Deligne cohomology, Griffiths groups, and normal functions. Examples from singularity theory, symplectic topology, and complex geometry. Fukaya and matrix factorization categories. Chern-Simons functionals. Motivic local systems and nc Hodge structures: the Fano case. Mirror maps for del Pezzo surfaces.

The Hodge theory arising in homological mirror symmetry. Pure nc Hodge structures and their Betti and de Rham aspects. Categorical nc geometry, variations of nc Hodge structures, nc Deligne cohomology, Griffiths groups, and normal functions. Examples from singularity theory, symplectic topology, and complex geometry. Fukaya and matrix factorization categories. Chern-Simons functionals. Motivic local systems and nc Hodge structures: the Fano case. Mirror maps for del Pezzo surfaces.

I'll discuss some conjectures (joint with T. Braden, A. Licata and N. Proudfoot) relating S-duality in 3-dimensional field theory to the geometry of certain symplectic varieties and the representation theory of "universal enveloping algebras" attached to these varieties. We will start with a very down to earth description of the abelian case (which can be stripped down to some combinatorics of hyperplane arrangements), and proceed toward a broader and more geometric view. These conjectures also relate to my other talk on the categorification of quantum groups, and they hope to relate the two of the great universes of geometric representation theory: quiver varieties and the affine Grassmannian.

The Fundamental Lemma, abstract: I will give a gentle overview of the ideas surrounding the Fundamental Lemma and its solution by Ngo Bao-Chau (recently ranked number 7 in Time Magazine's Top 10 Scientific Discoveries of 2009). The Fundamental Lemma is a key ingredient in the Arthur-Selberg Trace Formula and the entire Langlands program, with deep implications for number theory. Its conjectural status (to quote Langlands) "rendered progress almost impossible for nearly twenty years". Ngo's solution is a stunning application of the analogy between Riemann surfaces and number fields: it revolves around Hitchin's integrable system, a construction in the geometry of bundles on surfaces motivated by physics.

These talks will be about the C*-algebra approach to index theory and K-theory that was proposed by Atiyah and worked out in detail by Kasparov. In the last lecture I'll discuss the most famous application of Kasparov's work - to the Novikov higher signature conjecture and the Baum-Connes conjecture. I'll sketch the proof of both conjectures for Gromov's a-T-menable groups (these are groups that act properly on an infinite-dimensional Euclidean space, and include amenable groups, free groups, Coxeter groups and others). The argument uses an interesting noncommutative C*-algebra that serves as a proxy for the commutative algebra of continuous functions on a Euclidean space (which isn't itself very useful when the space is infinite-dimensional). This algebra, together with a closely related Bott-Dirac operator on the Euclidean space, may have other applications.

These talks will be about the C*-algebra approach to index theory and K-theory that was proposed by Atiyah and worked out in detail by Kasparov. What is K-homology good for? I'll try to answer with examples in this and the previous talks. There is an obvious connection to the Atiyah-Singer index theorem, and roughly speaking K-homology provides a context in which to consider the index theorem's many elaborations. A general theme is that while two operators may look rather different, for example when studied using a symbol calculus, functional analysis and the framework of K-homology can sometimes give a means to identify their Fredholm index theories.

These talks will be about the C*-algebra approach to index theory and K-theory that was proposed by Atiyah and worked out in detail by Kasparov. What is K-homology good for? I'll try to answer with examples in this and the following talks. There is an obvious connection to the Atiyah-Singer index theorem, and roughly speaking K-homology provides a context in which to consider the index theorem's many elaborations. A general theme is that while two operators may look rather different, for example when studied using a symbol calculus, functional analysis and the framework of K-homology can sometimes give a means to identify their Fredholm index theories.

These talks will be about the C*-algebra approach to index theory and K-theory that was proposed by Atiyah and worked out in detail by Kasparov. Atiyah pointed out that an elliptic operator on a manifold can be viewed as cycles for the homology theory that is dual to K-theory. This led him to suggest a functional-analytic definition for K-homology, and Kasparov later worked out the complete theory. In the first talk I shall tell more of this story.

The moduli space of complex curves of genus g has been intensively studied since Riemann. On the other hand, moduli spaces of complex surfaces are only poorly understood at present. I will survey what is known and discuss some examples. In particular I will describe the natural compactification analogous to the Deligne--Mumford compactification of the moduli space of curves.

Let A be an abelian category which is linear over a finite field IF_q. I will review the construction of the Hall algebra H(A) of A. In examples coming from quivers, H(A) is typically a quantum group, where the role of the deformation parameter is played by the cardinality q of the field. I will also explain how one may set q=1 and obtain the enveloping algebras of semisimple Lie algebras.

The classical Schur-Weyl duality relates modules for the general linear Lie algebra with modules over the symmetric group. I first explain a higher level version of this where cyclotomic versions of degenerate Hecke algebras occur. Afterwords I will indicate how this picture can be categorified. The combinatorics of crystal graphs plays an important role here. Finally I want to illustrate in two examples how this setup can be used to derive equivalences of categories where the aforementioned Hecke algebras play the key role.

This talk will try to explain the (surprisingly strong) connections between the topology of the moduli spaces of surfaces and the homological algebra of non-commutative Frobenius algebras.

This will be a general talk about derived categories of coherent sheaves. I'll describe some of the basic examples of derived equivalences: Fourier-Mukai transforms, the McKay correspondence and threefold flops, and indicate some possible directions for further research.

This lecture is based on joint work in progress with H.Derksen and J.Weyman. We obtain a far-reaching generalization of classical Bernstein-Gelfand-Ponomarev reflection functors playing a fundamental role in the theory of quiver representations. These functors are defined only at a source or a sink of the quiver in question. We introduce a class of quivers with relations of a special kind given by non-commutative analogs of Jacobian ideals in the path algebra. We then define the mutations at arbitrary vertices for these quivers and their representations. If the vertex in question is a source or a sink, our mutations specialize to reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras. We will keep the exposition elementary, with all necessary background explained from scratch.

The Physics of Knot Homologies presented by Sergei Gukov (University of California Santa Barbara) December 13th, 2006

Abstract: I will give a symplectic geometer's answer to the question "What can you do with a representation of a group?" This construction comes up repeatedly in geometric representation theory,and I will describe a situation in which it can be used to produce bases for representations of infinite dimensional Lie algebras.

In recent years, a surprising number of significant insights and results in noncommutative algebra have been obtained by using the global techniques of projective algebraic geometry. This talk will survey some of these results. Thus we will be interested in using geometric techniques to study graded noncommutative rings.

A fundamental combinatorial question concerning Lie algebras is to calculate weight and tensor product multiplicities for their representations. The modern approach to this question is to construct special bases which are adapted to these calculations and then describe their combinatorics. There have been a number of constructions of such bases in the past 15 years, crystal bases by Kashiwara, canonical bases by Lusztig, the MV basis by Mirkovic-Vilonen, the basis given by components of quiver varieties by Nakajima, etc. Some of these constructions are more representation theoretic, while others more geometric, however all are quite non-trivial. We will survey some of these constructions and discuss the resulting combinatorics.

Abstract: We'll explore the intimate relationship between the representation theory of Lie algebras and the geometry of flag varieties. In particular we'll consider the Borel-Weil theorem, identifying irreducible finite dimensional representations with line bundles, and the Beilinson-Bernstein theorem, identifying arbitrary representations with "sheaves with flat connection" (D-modules).

Abstract: We will begin with a review of the classical theorem of Bezout, which computes the number of intersection points of two algebraic curves in the projective plane, provided that they meet transversely. In the case of nontransverse intersections, one can make a similar assertion provided that one counts the intersection points with the correct multiplicities. The search for the correct intersection multiplicities will lead us into the world of "nonabelian homological algebra", a theory which is a mixture of classical algebra and homotopy theory.

Abstract: A quiver is simply a directed graph. By a representation of a quiver we mean a collection of vector spaces indexed by the nodes of the graph, together with linear maps corresponding to the arrows of the graph. We will introduce the notions of simple and indecomposable representations of a quiver (which may be considered part of the Abelian category structure of quiver representations). Gabriel asked which quivers have "finite representation type", that is, which quivers have finitely many indecomposables, and discovered a beautiful connection to Lie theory: The quivers with this property are precisely those whose underlying undirected graph is the Dynkin diagram of a Lie algebra. Moreover the indecomposable objects themselves are indexed by the roots of the associated Lie algebra. We will give an account of this theorem.