Geometry and Topology Seminar 2010-11

Fall 2010

This semester, the seminar usually meets on Mondays, from 4:30 to 5:30 in KAP 245.

Monday 8/30, 3:30-4:30	Joint with Algebra Seminar Ben Williams (USC): The Cohomology of Varieties of Chain Complexes Let S denote the polynomial ring in n variables over a field. There are similar conjectures due to Buchsbaum-Eisenbud & Horrocks, and Carlsson, which assert that if certain differential-graded objects (chain complexes, resolutions, DGMs) have small (artinian) homology, then they themselves must be large. We will explain the various conjectures, as well as an application to topology of Carlsson's conjecture. We will then reinterpret the conjectures as problems in the topology of varieties, and from there, use cohomological methods to deduce constraints on some of the differential graded objects that can arise.
Monday 9/6	Labor Day
Monday 9/13	Joint with Algebra Seminar Ben Antieau (UCLA): The etale index of division algebras After introducing the period-index problem for division algebras and the notion of twisted algebraic K-theory, we introduce the etale index of a division algebra. If D is a central division algebra over the field k, then the etale index is defined as the (positive) generator of the image of the rank map on the etale K_0-theory of D. When k is of finite etale cohomological dimension d, an upper bound is given on the etale index of D which depends on d and on the period of D (the order of the class of D in the Brauer group of k). This bound is expressed with the exponents of the stable homotopy groups of spheres and classifying spaces of finite abelian groups. These appear due to a twisted form of the unit morphism from the sphere spectrum to algebraic K-theory. The upper bound shows that the etale index differs from the index in general.
Friday 9/17, 2:00-2:50 (Intro) 3:30-4:30 (Formal)	Tsuyoshi Kato (Kyoto University): Growth of Casson handles and complexity of smooth structures on 4-manifolds
Monday 9/20	Francis Bonahon (USC): Kauffman brackets, character varieties, and triangulations of surfaces A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. I will show how an irreducible representation of the skein algebra usually specifies a point of the character variety of homeomorphisms from the fundamental group of the surface to PSL_2(C), as well as  certain weights associated to the punctures of the surface. I will then sketch a proof of a partial converse, which states that each point of the character variety and each choice of appropriate puncture weights uniquely determine a Kauffman bracket. This is joint work with Helen Wong.
Monday 9/27, 3:30-5:30 KAP 414	The Joint Los Angeles Topology Seminar Joint with Algebra Seminar Anthony Licata (Stanford): Heisenberg Algebras, Hilbert Schemes, and Categorification Representations of Heisenberg algebras appear naturally in many mathematical contexts.  In geometry, one notable appearance is in the work of Nakajima and Grojnowski relating Heisenberg algebras and Hilbert schemes of points on a complex surface.  We will describe a categorification of the Nakajima-Grojnowski construction.  This is joint work with Sabin Cautis. Joan Licata (Stanford): Legendrian contact homology for Seifert fibered spaces In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form.  A Legendrian knot in such a manifold is described by a specially labeled Lagrangian diagram, and from this data one can compute both the ``classical" invariants for Legendrian knots in rational homology three-spheres and also a new invariant which takes the form of a differential graded algebra. This work is joint with J. Sabloff. Special Geomtry/Topology Seminar
Friday 10/1, 3:30-4:30, KAP 414	Dave Auckly (Mathematical Sciences Research Institute): Two-fold branched covers It is possible to give a census of 2-fold branched covering behavior for surgeries on knots of 10 or fewer crossings. There are integral homology spheres that are not 2-fold branched covers of any 3-manifold, there are examples that are 2-fold branched covers of $S^3$ but no other manifold, examples that cover some manifold but not $S^3$, and examples that cover several different manifolds. This talk will present these examples. Thurston asked if any every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle. This is the virtual fibering conjecture. The presentation will also include a virtual virtual fibering result showing that the answer is yes if cover is replaced by branched cover.
Monday 10/4	Pan Peng (University of Arizona, Tucson): Integrality structure in Chern-Simons/Topological string duality Duality is one of the core ideas in string theory. In this talk, we will focus on the duality between Chern-Simons gauge theory and topological string theory. Several geometric and topological problems will be discussed from the duality point of view.
Thursday 10/7, 3:30-4:30, KAP414	Special Geomtry/Topology Seminar Tamas Kalman (Tokyo Institute of Technology): Heegaard Floer homology and the Homfly polynomial I will report on joint work in progress with Andras Juhasz and Jake Rasmussen. If the oriented link K has a special alternating diagram, we take the complement of the associated Seifert surface and its sutured Floer homology, or rather only its Euler characteristic which can be viewed as a set of points in an integer lattice. We show that the set is convex by providing an alternative description as the "hypertree polytope" of a certain hypergraph. We also establish a way of reading a polynomial I off of the hypertree polytope. (This is a partial generalization of the Tutte polynomial. Much of the talk will concentrate on this discrete mathematical construction.) Then, our main conjecture is that I appears inside the Homfly polynomial of the original link K, namely it is the part that corresponds to the leading coefficient of the Alexander polynomial.
Monday 10/11	An Huang (UC-Berkeley): An Invitation to Nahm's Conjecture A conjecture due to Werner Nahm relates the modularity of a class of q-hypergeometric series to torsion elements in the Bloch group of certain number fields. The main idea arises from studying certain integrable quantum field theories in two dimensions. I will introduce some of these background, and report on a recent result with Chul-hee Lee concerning rank 2 case of this conjecture.
Monday 10/18	Hisham Sati (University of Maryland): M-theory, higher structures, and generalized cohomology Anomalies and partition functions have played a major role in uncovering geometric and topological structures behind physical theories, such as quantum field theory. We will consider the case of string theory and M-theory and show how their anomalies and partition functions point to generalized cohomology theories such as elliptic cohomology. Viewed as obstructions, anomalies also give rise to structures such as String structures and even lead to generalizations. We will also show that in cases of interest such structures, as well as the corresponding generalized cohomology theories, are in fact twisted.
Monday 10/25	No Seminar
Monday 11/1, 4:00-6:00, UCLA MS6229	The Joint Los Angeles Topology Seminar Allison Gilmore (Columbia): An algebraic proof of invariance for knot Floer homology We investigate the algebraic structure of knot Floer homology in the context of categorification. Ozsvath and Szabo gave the first completely algebraic description of knot Floer homology via a cube of resolutions construction. Starting with a braid diagram for a knot, one singularizes or smooths each crossing, then associates an algebra to each resulting singular braid. These can be arranged into a chain complex that computes knot Floer homology. Using this construction, we give a fully algebraic proof of invariance for knot Floer homology that avoids any mention of holomorphic disks or grid diagrams. We close with an alternative description of knot Floer homology in terms of Soergel bimodules that suggests a close relationship with HOMFLY-PT homology. John Baldwin (Princeton): A combinatorial spanning tree model for knot Floer homology I'll describe an ongoing project with Adam Levine to iterate Manolescu's exact triangle in knot Floer homology. This iteration results in a spectral sequence which converges to (a stabilized version of) knot Floer homology. With coefficients in a Novikov ring, the E_2 term of this spectral sequence is roughly generated by Kauffman states of the knot or, equivalently, by spanning trees of its black graph. One can describe the d_2 differential combinatorially and can prove that this spectral sequence collapses at E_3. Therefore (E_2,d_2) provides a combinatorial chain complex for delta-graded knot Floer homology.
Monday 11/8, KAP414	Roland van der Veen (KDV Institute, University of Amsterdam): Hyperbolic polyhedra and the Jones polynomial For knots the hyperbolic geometry of the complement is known to be related to its Jones polynomial in various ways. We propose to study this relationship more closely by extending the Jones polynomial to graphs. For a planar graph we will show how its Jones polynomial then gives rise to the hyperbolic volume of the polyhedron whose 1-skeleton is the graph. Joint with Francois Gueritaud and Francois Costantino
Monday 11/15	No Seminar
Monday 11/22	Aaron Lauda (Columbia University): A diagrammatic categorification of quantum groups The Jones polynomial can be understood in terms of the representation theory of the quantum group associated to sl2.  This description facilitated a vast generalization of the Jones polynomial to other quantum link and tangle invariants called Reshetikhin-Turaev invariants. These invariants, which arise from representations of quantum groups associated to simple Lie algebras, subsequently led to the definition of quantum 3-manifold invariants.  In this talk we categorify quantum groups using a simple diagrammatic calculus that requires no previous knowledge of quantum groups.  These diagrammatically categorified quantum groups not only lead to a representation theoretic explanation of Khovanov homology but also inspired Webster's recent work categorifying all Reshetikhin-Turaev invariants of tangles.
Monday 11/29	Yi-Jen Lee (Purdue University): Seiberg-Witten-Floer homology of certain sutured manifolds I will discuss the Seiberg-Witten-Floer homology of some sutured manifolds not covered by the Kronheimer-Mrowka definition of sutured monopole Floer homology.
Friday 12/3, 4:00-6:00, Caltech Sloan257	The Joint Los Angeles Topology Seminar Joshua Greene (Columbia): Mutation and alternating links I will discuss the proof and consequences of the following result. Suppose that D_1 and D_2 are reduced alternating diagrams for a pair of links whose branched double-covers have isomorphic Heegaard Floer homology groups. Then the spaces are diffeomorphic, and moreover D_1 and D_2 are related by a sequence of Conway mutations. Andrew Putman (Rice): Equivariant homological stability for congruence subgroups An important structural feature of the kth homology group of SL_n(Z) is that it is independent of n once n is sufficiently large.  This property is called "homological stability" for SL_n(Z).  Congruence subgroups of SL_n(Z) do not satisfy homological stability; however, I will discuss a theorem that says that they do satisfy a certain equivariant version of homological stability.