Current Year: 2017-18 Previous Years: 2016-17 2015-16 2014-15 2013-14 2012-13 |
Berkeley Topology SeminarThe Topology Seminar meets on Wednesdays from 4.10 to 5.00 in 3 Evans unless otherwise indicated.
Previous Semesters Fall 2012 *Special Room* August 22, 740 Evans Saul Schleimer (Warwick) The graph of handlebodies (Joint work with Joseph Maher.) We introduce the graph of handlebodies and prove that it is quasi-isometric to an electrification of the curve complex. We show that this graph is Gromov-hyperbolic and of infinite diameter. If time permits we will also discuss the construction of pseudo-Anosov maps acting on the graph with positive translation distance. *Special Seminar* August 23, 740 Evans, 2-3pm Michael Freedman (Microsoft Station Q) Closed 3-manifolds smoothly imbedded in R^{4}. Ian Agol and I have been working to sharpen up some tools for studing existance and uniqueness problems for smooth imbeddings of closed 3-manifolds in R^{4}. Potential target applications include great problems so it is worth getting the tools sharp. The mapping class group of a Heegaard surface for the 3-manifold appears to plays an important role. One result is that every imbedding of a closed 3-manifold in R^{4} is isotopic to one with a unique local minimum and a unique local maximum. August 29 Soren Galatius (Stanford) Homological stability for moduli spaces of high dimensional manifolds I will discuss recent joint work with Oscar Randal-Williams concerning the manifolds W_g^{2n} obtained as the connected sum of g copies of S^n x S^n. For n=1 this is a genus g surface, and there is a moduli space M_g parametrizing smooth surface bundles with genus g fibers. For higher n there is an analogous moduli space M_g^n parametrizing smooth fiber bundles with fibers W_g (although for n > 1 it is no longer finite dimensional). We prove that for n > 2 the cohomology groups H^k(M_g^n) are independent of g as long as g >> k, generalizing a result of John Harer and others for n=1. September 5 Jacob Lurie (Harvard) Ambidexterity Let M be a compact oriented manifold. Poincare duality establishes an isomorphism between the homology and cohomology of M. For compact orbifolds, there is a similar isomorphism if we take homology and cohomology with rational coefficients, but not if we use integer coefficients. In this lecture, I'll discuss some other examples of cohomology theories which satisfy Poincare duality for orbifolds, and related duality phenomena. September 12 Owen Gwilliam (Berkeley) Higher enveloping algebras of Lie algebras, topological field theories, and factorization algebras If we view an E_n algebra (an algebra over the little n-disks operad) as living on the n-disk, it is natural to ask for a local-to-global object on an n-manifold that is locally an E_n algebra. The notion of a factorization algebra makes this idea precise. The talk will revolve around a general construction taking as input a Lie algebra g and smooth manifold M, and it will motivate the definition of factorization algebra via this construction. Along the way, we will interpret this construction from various perspectives, such as deformation theory, quantization, and manifold calculus. At the end, we hope to sketch a relationship with vertex algebras and affine Kac-Moody algebras. September 19 Mohammed Abouzaid (Columbia/Simons Center) On embeddings and immersions of Lagrangians spheres A conjecture of Arnold would imply that every exact Lagrangian in a cotangent bundle is Hamiltonian isotopic to the zero section. We now know that every such Lagrangian is diffeomorphic to the zero section. I will explain how, combining the h-principle with the spectrum-valued invariants introduced by Kragh, one can hope to show that such Lagrangians are in fact isotopic to the zero section through Lagrangian immersions. I will discuss the case of spheres, where we can at least find obstructions to the classes of Lagrangian immersions which admit embeddings. This is joint work with T. Kragh. September 26 Dylan Thurston (MSRI/Berkeley/Indiana) 4-manifolds and the categorification of surfaces Smooth 4-manifolds are the last frontier of low-dimensional topology. For instance, four is the only dimension in which the smooth Poincaré conjecture is still open. Prominent in this problem is the search for non-trivial invariants of smooth 4-manifolds. There is essentially only one such invariant known, the Donaldson invariant and various other invariants conjectured to be equivalent to it. In order to get a handle on these invariants of 4-manifolds, it makes sense to get a handle on their dimensional reductions, which ought to assign some sort of vector space to 3-manifolds and some sort of category to surfaces. In this talk, we will see how this program is starting to shape up for Heegaard Floer homology (one of the family of invariants conjecturally-equivalent to the Donaldson invariants) and some glimmerings of how one might try to get another such invariant from categorifying surface cluster algebras. Portions of this talk are joint work with P. Ozsváth and R. Lipshitz. October 3 David Nadler (Berkeley) Lagrangian cobordisms The notion of semi-infinite homology of path spaces in symplectic manifolds is fundamental but mysterious. We will discuss a proposal (joint with Hiro Tanaka) to approach semi-infinite homology via Lagrangian cobordisms. Understanding the simplest examples leads to concrete questions about E_{∞}-ring spectra. *Special Time and Room* October 10, 939 Evans, 2-3pm Julie Bergner (UC Riverside) Comparing models for (∞, n)-categories With many definitions being given for (∞, n)-categories, one criterion to check is whether they can be thought of as categories enriched in (∞, n-1)-categories. In joint work with Charles Rezk, we are establishing a chain of Quillen equivalences from the model structure for Θ_{n}-spaces and the model structure for categories enriched in Θ_{n-1}-spaces. This comparison also gives insight into how to compare to other known models. October 17 Anatoly Preygel (Berkeley) Thom-Sebastiani and duality for matrix factorizations (via DAG) The Z/2-graded dg-category MF of matrix factorizations is a categorical singularity invariant appearing on the algebro-geometric side of mirror symmetry. Though it lives in commutative algebra, and the first definitions were very explicit, it turns out to admit a few descriptions very much in the spirit of homotopy theory. After recalling a bit about sheaf theories in derived algebraic geometry, we'll sketch at one of these definitions and show how it can be used to give conceptual ('homotopy theoretic') proofs of some fundamental properties of MF. Topology (mostly homotopy theory) will make regular side appearances throughout. *Special seminar joint with Representation Theory, Geometry and Combinatorics meets from 4-5.30 in 3 Evans* October 24 Kevin Costello (Northwestern) N=1 supersymmetric gauge theory and the Yangian I'll explain a dictionary relating a deformation of the N=1 supersymmetric gauge theory and the Yangian. This dictionary allows one to perform exact calculations in the deformed N=1 gauge theory using the representation theory of the Yangian. The talk will start with a brief discussion of what I mean by a quantum field theory. October 31 David Futer (Temple) The Jones polynomial and surfaces far from fibers This talk explores relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We also show that certain coefficients of the Jones and colored Jones polynomials measure how far this surface is from being a fiber in the knot complement. This is joint work with Effie Kalfagianni and Jessica Purcell. November 7 Ian Biringer (Boston College) Growth of Betti numbers and a probabilistic take on geometric convergence We will describe an asymptotic relationship between the volume and the Betti numbers of certain locally symmetric spaces. The proof uses a synthesis of Gromov-Hausdorff convergence of Riemannian manifolds and Benjamini-Schramm convergence from graph theory November 14 Marc Culler (UIC) Peripherally elliptic representations and character varieties I am interested in SL(2,C) representations of knot groups with the property that the peripheral elements are sent to elliptic elements. I will describe some computations, known results and questions about the structure of the set of these representations and connections with character varieties and A-polynomials. November 21 Jon Bloom (MIT) A bordered monopole Floer theory I will report on work-in-progress to develop a bordered monopole Floer theory. We associate an algebra to a surface, a module to 3-manifold with boundary, and a map of modules to a 4-manifold with corners (all in the A-infinity sense). These structures satisfy the natural gluing theorems inherent in a 4-dimensional TQFT with corners, and are closely related to Khovanov's invariant of tangles and Szabo's geometric spectral sequence. This is joint work with John Baldwin. November 28 Ko Honda (USC) A categorification of the quantum superalgebra U_q sl(1|1) In the framework of Reshetikhin-Turaev invariants, the quantum group U_q sl(1|1) of the super Lie algebra sl(1|1) gives rise to the Alexander polynomial of links just as the quantum group U_q sl(2) gives rise to the Jones and colored Jones polynomials. I will describe the work of my student Yin Tian which gives a categorification of U_q sl(1|1) which is motivated by contact geometry and in particular by the contact category. Spring 2012 January 18 Nathan Dunfield (UIUC) Integer homology 3-spheres with large injectivity radius Conjecturally, the amount of torsion in the first homology group of a hyperbolic 3-manifold must grow rapidly in any exhaustive tower of covers (see Bergeron-Venkatesh and F. Calegari-Venkatesh). In contrast, the first betti number can stay constant (and zero) in such covers. Here "exhaustive" means that the injectivity radius of the covers goes to infinity. In this talk, I will explain how to construct hyperbolic 3-manifolds with trivial first homology where the injectivity radius is big almost everywhere by using ideas from Kleinian groups. I will then relate this to the recent work of Abert, Bergeron, Biringer, et. al. In particular, these examples show a differing approximation behavior for L^2 torsion as compared to L^2 betti numbers. This is joint work with Jeff Brock. January 25 Tom Church (Stanford) Invariant invariants of surface bundles The Morita-Mumford-Miller classes are the quintessential characteristic classes of surface bundles. These characteristic classes depend on a space E together with a fibering S_g -> E -> B as a surface bundle. Or do they? Well, the Hirzebruch signature formula implies that for a fibered 4-manifold, the first MMM class is just the signature -- so it doesn't depend on the fibering, only on the 4-manifold E. It turns out that this is not a coincidence. Say that a characteristic class is "geometric" if it only depends on the total space of the surface bundle, and not on the fibering. I will explain the surprising result that all the odd MMM classes are geometric, joint with Benson Farb and Matthew Thibault. Time permitting, I will describe a complete classification of which characteristic classes are geometric w.r.t. cobordism, joint with Martin Crossley and Jeffrey Giansiracusa. February 1 Dan Margalit (Georgia Tech) Hyperelliptic curves, braid groups, and congruence subgroups The hyperelliptic Torelli group is the subgroup of the mapping class group of a surface consisting of elements that act trivially on the homology of the surface and also commute with some fixed hyperelliptic involution. This group can also be characterized as the fundamental group of the branch locus of the period mapping, as well as the kernel of the (specialized) Burau representation of the pure braid group. Hain has conjectured that the hyperelliptic Torelli group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. His conjecture gives a meaningful description of the topology of the branch locus of the period mapping. We present some evidence for the conjecture, as well as progress towards its resolution. Some of the results rely on some new, intricate relations in the pure braid group. This is joint work with Tara Brendle. February 8 Robert Lipshitz (Columbia) A spectrification of a categorification of the Jones polynomial In the early 1980's, Jones introduced a new knot invariant, now called the Jones polynomial. Roughly ten years ago, Khovanov gave a refinement -- or categorification -- of the Jones polynomial; this refinement is now called Khovanov homology. In this talk we will sketch definitions of the Jones polynomial and Khovanov homology, and mention some of their most spectacular applications. We will then discuss a recent space-level refinement of Khovanov homology; this refinement is joint work with S. Sarkar. February 15 Gabriel C. Drummond-Cole (Northwestern) Formal formality of low dimensional Calabi-Yau manifolds Barannikov and Konsevich defined a "hypercommutative" algebra structure on the cohomology of a Calabi-Yau variety. It is natural to consider this hypercommutative algebra as the first piece of a homotopy hypercommutative algebra structure. This homotopy structure can be truncated to Barannikov-Kontsevich's. I'll talk a bit about how to see that this truncation loses no information for some cases, including all Calabi-Yau 3-folds. February 22 John Pardon (Stanford) Totally disconnected groups (not) acting on three-manifolds Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert-Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery-Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three. The proof uses tools from low-dimensional topology, for example incompressible surfaces, minimal surfaces, and a property of the mapping class group. February 29 Nathaniel Rounds (Indiana) Compactifying string topology String topology studies the algebraic topology of the free loop space of a manifold. In this talk, we describe a compact space of graphs and show how this space gives algebraic operations on the singular chains of the free loop space. In particular, our chain level operations induce Cohen and Godin's "positive boundary TQFT" on the homology the free loop space. This project is joint work with Kate Poirier. March 7 Curt McMullen (Harvard) Cascades in the dynamics of foliations on surfaces We will discuss harmonic foliations on Riemann surfaces and the notion of "zero flux". In genus two we will see such foliations go through `period-doubling' in self-similar cascades that can be explained through homological invariants. March 14 Peter Teichner (Berkeley/Max Planck-Bonn) Whitney towers for classical links We'll give a survey of recent joint work with Jim Conant and Rob Schneiderman on a quantitative way of measuring the failure of the Whitney trick in dimension four. March 12 Motohico Mulase (UC Davis) The mirror dual of the Catalan numbers and the Eynard-Orantin topological recursion What is the mirror dual to the Catalan numbers? By answering this question, I will explain the mathematical framework of the Eynard-Orantin topological recursion theory. This theory is a proposal for the universal B-model that is mirror to "integrable" A-model (or counting) problems. In this talk I will be focused on presenting the simplest mathematical example of the theory for which we can actually calculate everything. This example illustrates the general theory behind the scene. April 4 Ina Petkova (Columbia) Bordered Heegaard Floer homology - recent developments and applications Bordered Heegaard Floer homology is a Floer theory over Z/2 for manifolds with boundary. After gluing, it recovers Heegaard Floer homology for closed manifolds. I will briefly describe the bordered Floer package and some applications to knot theory. Then I will discuss its decategorification, and some progress towards extending the theory over the integers. April 11 Shelly Harvey (Rice) Filtering smooth concordance classes of topologically slice knots The n-solvable filtration of the smooth knot concordance group, suggested by work of Cochran-Orr-Teichner, is slightly flawed in the sense that any topologically slice knot lies in every term of the filtration. To repair this we investigate a new filtration, {B_n}, that is simultaneously a refinement of the n-solvable filtration and a generalization of notions of positivity studied by Gompf and Cochran. As is the case for the n-solvable filtration, each B_n/B_{n+1} has infinite rank. Our primary interest is in the induced filtration, {T_n}, on the subgroup, T, of knots that are topologically slice. We prove that T/T_0 is large, detected by gauge-theoretic invariants and the tau, s, and epsilon-invariants; while the non-triviality of T_0/T_1 can be detected by certain Heegaard Floer d-invariants. All of these concordance obstructions vanish for knots in T_1. Nonetheless, going beyond this, our main result is that T_1/T_2 has positive rank. Moreover under a "weak homotopy-ribbon" condition, we show that each T_n/T_{n+1} has positive rank. April 18 Ralph Cohen (Stanford) Topological Field Theories, Koszul Duality, and String Topology In this talk I will describe the role of generalized Fukaya categories in the Costello-Lurie classification of two dimensional topological field theories. Then I will switch gears and describe a variety of properties of Koszul dual differential graded algebras. The archetypical examples are the algebra of cochains, C^*(X), and the Pontrjagin algebra C_*(\Omega X), when X is simply connected. I will emphasize the effect of Koszul duality when one of the algebras satisfies Poincare duality up to homotopy. Finally I will connect these topics by discussing the effect of Koszul duality in the classification of topological field theories, while continuing to emphasize the example of the string topology of a manifold. April 25 Douglas LaFountain (Aarhus/Berkeley) Decorated Teichmuller theory and compactifications of moduli space For a genus g surface with s>0 punctures and 2g+s>2, decorated Teichmuller space (DTeich) is a trivial R_+^s-bundle over the usual Teichmuller space, where the fiber corresponds to families of horocycles peripheral to each puncture. As proved by R. Penner, DTeich admits a mapping class group-invariant cell decomposition, which then descends to a cell decomposition of Riemann's moduli space. In this talk we introduce a new cellular bordification of DTeich which is also MCG-invariant, namely the space of filtered screens. After an appropriate quotient, we obtain a cell decomposition for a new compactification of moduli space, which is shown to be homotopy equivalent to the Deligne-Mumford compactification. This work is joint with R. Penner. May 2 Shea Vela-Vick (Columbia) The equivalence of transverse link invariants in knot Floer homology The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces. Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined. The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams. The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo. We show that these two previously defined invariant agree. Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition. |
Last modified 29 July 2014.