Current Year: 201516 Previous Years: 201415 201314 201213 
Berkeley Topology Seminar

Abstracts for Spring 2014  

Date  Time/Place  Details 
Jan 15, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Hongbin Sun (Princeton) Construction of almost totally geodesic surfaces in closed hyperbolic 3manifolds. (Part I: Introductory Talk) In this talk, we will review KahnMarkovic's and LiuMarkovic's work on constructing almost totally geodesic surfaces in closed hyperbolic 3manifolds. Kahn and Markovic's work gives immersed π_{1}injective closed almost totally geodesic surfaces in any closed hyperbolic 3manifolds, which is the first step of the proof of the virtual Haken and virtual fibered conjectures. Actually, Liu and Markovic's work is a 3dimensional version of KahnMarkovic's work on the good pants homology. In particular, for any union of closed geodesics L in a closed hyperbolic 3manifold M, they showed that any homology class in H_{2}(M, L) can be virtually realized by a connected immersed almost totally geodesic surface. 
4:00pm—5:00pm Room 3, Evans Hall 
Hongbin Sun (Princeton) Virtual Properties of closed hyperbolic 3manifolds (Part II: Main Talk) By using KahnMarkovic and LiuMarkovic's construction of almost totally geodesic quasiFuchsian surfaces (closed or bounded) , we can construct various immersed π_{1}injective 2complexes in any closed hyperbolic 3manifold M. By using Agol's result that the groups of hyperbolic 3manifolds are LERF, a "geometric neighborhood" of such an immersed π_{1}injective 2complex embeds into some finite cover of M, which gives some topological information of this finite cover (virtual property of M). By using the above idea, we prove the following two results:


Jan 22, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Theo JohnsonFreyd (Northwestern) Title: TBA (Part I: Introductory Talk) 
4:00pm—5:00pm Room 3, Evans Hall 
Theo JohnsonFreyd (Northwestern) ptohomotopy Frobenius structures on manifolds, and how they relate to perturbative QFT (Part II: Main Talk) The de Rham homology of an oriented manifold carries a wellknown gradedcommutative Frobenius algebra structure. Does this structure lift in a geometrically meaningful uptohomotopy way to de Rham chains? The answer depends on the meanings of "geometrically meaningful" and "uptohomotopy". I will describe two potential choices for the meanings of these words. Using the first choice, the answer to the question is always Yes. Using the second gives a more subtle situation, in which the answer is No in dimension 1, and related to the formality of the E_{n} operad in dimension n > 1. To explain this relationship (and my interest in the problem) requires a short sojourn in the world of perturbative topological quantum field theory. 

Jan 29, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Adam Levine (Princeton) Heegaard Floer homology and genus bounds (Part I: Introductory Talk) I will provide a brief overview of Heegaard Floer homology, with an emphasis on the ways in which it provides bounds on the genera of embedded surfaces in 3 and 4manifolds. 
4:00pm—5:00pm Room 3, Evans Hall 
Adam Levine (Princeton) Nonorientable surfaces in homology cobordisms (Part II: Main Talk) We study the minimal genus problem for embeddings of closed, nonorientable surfaces in a homology cobordism between rational homology spheres, using obstructions derived from Heegaard Floer homology and from the AtiyahSinger index theorem. For instance, we show that if a nonorientable surface embeds essentially in the product of a lens space with an interval, its genus and normal Euler number are the same as those of a stabilization of a nonorientable surface embedded in the lens space itself. This is joint work with Danny Ruberman and Saso Strle. 

Feb 12, 2014  4:00pm—5:00pm Room 3, Evans Hall 
Jenya Sapir (Stanford) Counting NonSimple Closed Geodesics on Surfaces We get coarse bounds on the number of (nonsimple) closed geodesics on a surface, given upper bounds on both length and selfintersection number. Recent work by Mirzakhani and by Rivin has given asymptotics for the growth of simple closed curves and curves with one selfintersection (respectively) with respect to length. No asymptotics for arbitrary selfintersection number are currently known, but we give coarse bounds for arbitrary selfintersection number and length. We show how to reduce this problem to counting curves on a pair of pants, and give explicit bounds with respect to both length and intersection number in that case. 
Feb 19, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Andre Henriques (Universiteit Utrecht) Factorization algebras, chiral CFTs, and topological states of matter We'll present a mathematical model for 2d topological states of matter with chiral edge modes (the fractional quantum Hall effect being the prototypical example). The formulation of our model combines factorization algebras and conformal nets. 
4:00pm—5:00pm Room 3, Evans Hall 
Ian Hambleton (McMaster) Recognizing products of surfaces and simply connected 4manifolds We give necessary and sufficient conditions for a closed smooth 6manifold N to be diffeomorphic to a product of a surface F and a simply connected 4manifold M in terms of basic invariants like the fundamental group and cohomological data. Any isometry of the intersection form of M is realized by a selfdiffeomorphism of M x F. 

Feb 26, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Stefan Schwede (Bonn) Introduction to global equivariant homotopy theory (Part I: Introductory Talk) Global homotopy theory studies equivariant phenomena that exist for all compact Lie groups in a uniform way. In this talk I present a rigorous formalism for this and discuss example of global homotopy types. 
4:00pm—5:00pm Room 3, Evans Hall 
Stefan Schwede (Bonn) Equivariant properties of symmetric products (Part II: Main Talk) The filtration on the infinite symmetric product of spheres by number of factors provides a sequence of spectra between the sphere spectrum and the integral EilenbergMac Lane spectrum. This filtration has received a lot of attention and the subquotients are interesting stable homotopy types. In this talk I will discuss the equivariant stable homotopy types, for finite groups, obtained from this filtration for the infinite symmetric product of representation spheres. The filtration is more complicated than in the nonequivariant case, and already on the zeroth homotopy groups an interesting filtration of the augmentation ideal of the Burnside rings arises. Our method is by `global' homotopy theory, i.e., we study the simultaneous behaviour for all finite groups at once. In this context, the equivariant subquotients are no longer rationally trivial, nor even concentrated in dimension 0. 

Mar 5, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Matt Hedden (Michigan State) Some background on contact geometry, classification of surface diffeomorphisms, and Heegaard Floer homology (Part I: Introductory Talk) I'll offer the contents of the title in an informal way, with the discussion being led by the interest of the students and what they would like to hear about. The aim will be to supply the background necessary for better appreciation of the afternoon talk, but perhaps will still be of interest to someone planning on skipping the afternoon talk. 
4:00pm—5:00pm Room 3, Evans Hall 
Matt Hedden (Michigan State) Floer homology and Fractional Dehn twists (Part II: Main Talk) There is a rational valued invariant of an automorphism of a surface with a single boundary component called the fractional Dehn twist coefficient. Roughly, it measures the twisting of the automorphism around the boundary. The fractional Dehn twist coefficient is related to the theory of taut foliations, essential laminations, and contact structures on 3manifolds obtained by performing Dehn surgery on fibered knots. These connections arise by associating to a fibered knot in a 3manifold the fractional Dehn twist coefficient of the monodromy of the fibration on its complement. I'll describe how the Heegaard Floer homology of a 3manifold bounds the fractional Dehn twist coefficient of any of its fibered knots, and some consequences this has for contact structures on 3manifolds. This is joint work with Tom Mark. 

Mar 12, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Clark Barwick (MIT) Algebraic Ktheory as categorified stable homotopy theory (Part I: Introductory Talk) This is a nontechnical introduction to the new idea of algebraic Ktheory as a homology theory for (higher) categories that plays the role of stable homotopy theory in this context. 
4:00pm—5:00pm Room 3, Evans Hall 
Clark Barwick (MIT) (Equivariant) algebraic Ktheory of rings and ring spectra (Part II: Main Talk) This is a survey of some relatively concrete advances in algebraic Ktheory resulting from the new perspective afforded by higher categorical technology. 

Mar 19, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Vigleik Angeltveit (ANU) What is topological Hochschild homology and how can we compute it? (Part I: Introductory Talk) I will give a "naive" definition of topological Hochschild homology (THH) and compute THH(A) for some rings A. Time permitting I will also define topological cyclic homology (TC) and indicate how one goes about computing TC(A). 
4:00pm—5:00pm Room 3, Evans Hall 
Vigleik Angeltveit (ANU) Topological Hochschild homology and topological cyclic homology via the HillHopkinsRavenel norm (Part II: Main Talk) Bökstedt defined topological Hochschild homology (THH) before the advent of the modern categories of spectra in use today, and he had to invent some rather complicated coherence machinery to mimic the algebraic definition of Hochschild homology. It was thought that a "naive" definition of THH using a modern category of spectra could never give the correct equivariant homotopy type. We compare the Bökstedt smash product to the norm construction from the HillHopkinsRavenel proof of the Kervaire Invariant One problem, and use that to show that using the category of orthogonal spectra we do get a sensible definition of THH. This simplifies the foundations and makes it possible to define things like Adams operations on THH(A) and TC(A). This is joint work with Blumberg, Gerhardt, Hill, Lawson and Mandell. 

Apr 2, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Justin Noel (Regensburg) Prerequisites on ring spectra and nilpotence (Part I: Introductory Talk) If you know the fundamentals about ring spectra and the chromatic picture, then you should stay away from this talk. If you do not know about these things, then this is the talk for you. 
4:00pm—5:00pm Room 3, Evans Hall 
Justin Noel (Regensburg) On a nilpotence conjecture of J.P. May (Part II: Main Talk) In 1986 Peter May made the following conjecture: Suppose that R is a ring spectrum with power operations (e.g., an E_{∞} ring spectrum/ commutative Salgebra). Then the torsion elements in the kernel of the integral Hurewicz homomorphism π_{∗} R → H_{∗}(R;𝕫) are nilpotent. If R is the sphere spectrum, this is Nishida's nilpotence theorem. If we strengthen the condition on the integral homology to a condition about the complex bordism of R, then this is a special case of the nilpotence theorem of Devinatz, Hopkins, and Smith. The proof is short and simple, using only results that have been around since the late 90's. As a corollary we obtain results on the nonexistence of commutative Salgebra structures on various quotients of MU. For example MU / (p^{i}) or ku / (p^{i} v) for i > 0. We also obtain new results about the behavior of the Adams spectral sequence for Thom and THH spectra. This project is joint with Akhil Mathew and Niko Naumann. I will fill any remaining time with some fun results about ring spectra with power operations. 

Apr 9, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Greg Kuperberg (UC Davis) Computational complexity for geometric topologists (Part I: Introductory Talk) This talk will be an introduction to questions in the complexity theory of geometric topology problems. I will give a brief review of some complexity classes (P, NP, and all that), and turn to known and conjectured complexity results for distinguishing manifolds and computing topological invariants. 
4:00pm—5:00pm Room 3, Evans Hall 
Greg Kuperberg (UC Davis) Knottedness is in NP, modulo GRH (Part II: Main Talk) In this seminar I will discuss the details of the result that knottedness is in NP assuming the generalized Riemann hypothesis. The main part of the work is to properly understand Koiran's construction that solvability of a system of algebraic equations is in AM. It also uses on major result in geometric topology due to Kronheimer and Mrowka which can be accepted without serious study, although it is worth appreciating where it came from. 

Apr 16, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Wolfgang Lück (Bonn) Introduction to the middle Ktheory of group rings and their relevance in topology (Part I: Introductory Talk) We give a basic introduction to the projective class group of a group ring and the Whitehead group of a group and discuss applications to topology such as Wall's finiteness obstruction, the sCobordism Theorem and topological rigidity. 
4:00pm—5:00pm Room 3, Evans Hall 
Wolfgang Lück (Bonn) Algebraic K and Ltheory of groups rings and their applications to topology and geometry (Part II: Main Talk) We give an introduction to the K and Ltheoretic FarrellJones Conjecture and discuss its status. e.g, recently it has been proved for all lattices in almost connected Lie groups. We give a panorama of its large variety of applications, for instance to the Novikov Conjecvture about the homotopy invariance of higher signatures, the Borel Conjecture about the topological rigidity of aspherical manifolds and to hyperbolic groups with spheres as boundary. Finally we dicsuss some connections to equivariant homotopy and homology for proper actions of infinite groups. 

Apr 23, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Mike Hill (UVA) Equivariant Homotopy and Mackey Functors (Part I: Introductory Talk) I'll describe the computational tools needed to understand the proof of the results in the main talk. 
4:00pm—5:00pm Room 3, Evans Hall 
Mike Hill (UVA) A variant of Rohlin's Theorem: on eta cubed (Part II: Main Talk) Rohlin's theorem on the signature of Spin 4manifolds can be restated in terms of the connection between real and complex Ktheory given by homotopy fixed points. This comes from a bordism result about Real manifolds versus unoriented manifolds, which in turn, comes from a C_{2}equivariant story. I'll describe a surprising analogue of this for larger cyclic 2 groups, showing that the element eta cubed is never detected! In particular, for any bordism theory orienting these generalizations of Real manifolds, the three torus is always a boundary. 

Apr 30, 2014  2:00pm—3:00pm Room 740, Evans Hall 
Nathalie Wahl (University of Copenhagen) Introduction to homological stability (Part I: Introductory Talk) I'll give an introduction to homological stability for families of groups, and in particular go through Quillen's extremely useful argument to prove stability theorems. 
4:00pm—5:00pm Room 3, Evans Hall 
Nathalie Wahl (University of Copenhagen) Homological stability for families of groups (Part II: Main Talk) Many families of groups, such as symmetric groups, general linear groups, mapping class groups, satisfy that their homology stabilises with the rank or genus. I'll describe a general framework that encompasses all these examples, and show how it can be used to prove stability for the automorphism groups of right angled Artin groups. 

May 7, 2014  2:00pm—3:00pm Room 939, Evans Hall 
Jesper Grodal (University of Copenhagen) Fixed points, homotopy fixed points, and all that... (Part I: Introductory Talk) Many deep results in homotopy theory build around comparing fixed points and homotopy fixed points. I'll try to give a quick "best of" tour of some highlights from the last thirty+ years: The Segal conjecture, the Sullivan conjecture, etc, theorems so fundamental that they retain the name "conjecture" as a honorary title, long after their proof... NOTE: different room 
4:00pm—5:00pm Room 9, Evans Hall 
Jesper Grodal (University of Copenhagen) Uncompleting the Segal conjecture, and homotopical representation theory (Part II: Main Talk) The Segal conjecture identifies maps from the classifying space of a finite group to the plus construction on the classifying space of the infinite symmetric group with the completed Burnside ring. I'll give an uncompleted version of this result where we instead consider maps to the plus construction on the classifying space of the finite symmetric groups. I'll also explore further results in this direction, that may trace an outline of a homotopical representation theory. NOTE: different room 

May 16, 2014  2:00pm—3:00pm Room 736, Evans Hall 
POSTPONED: MOVED TO FRIDAY (NOTE DIFFERENT ROOM) Josh Greene (Boston College) Branched double covers in knot theory (Part I: Introductory Talk) This basic talk will focus on the utility of the branched double cover construction for approaching questions in knot theory. Such questions include: when does a knot have unknotting number one, and when does a knot bound a properly embedded disk in the fourball? 
4:00pm—5:00pm Room 736 Evans Hall 
POSTPONED: MOVED TO FRIDAY (NOTE DIFFERENT ROOM) Josh Greene (Boston College) Conway mutation and alternating links (Part II: Main Talk) I will prove that a pair of reduced, alternating link diagrams D and D' represent links with homeomorphic branched double covers iff D and D' are Conway mutants. The proof relies on combinatorial arguments and a black box from gauge theory (Donaldson's diagonalization theorem). I will also draw mutant knots simultaneously and ambidextrously. 
Abstracts for Fall 2013  

Date  Time/Place  Details 
Sep 4, 2013  4:00pm—5:00pm Room 3, Evans Hall 
Alexander Coward (Berkeley) Crossing changes and circular Heegaard splittings Twenty years ago Scharlemann and Thompson used deep results from sutured manifold theory to prove that a genus reducing crossing change on a knot maybe be realized as untwisting a Hopf band plumbed onto a minimal genus Seifert surface. This gives a hint that understanding genus reducing crossing changes is closely related to understanding how a compact surface in S^{3} changes when it is twisted. In this talk we use modern technology from the theory of Heegaard splittings to show that understanding when two surfaces are related by a single twist implies the existence of an algorithm to determine when two (hyperbolic or fibered) knots of different genus are related by a single crossing change. 
Sep 11, 2013  2:00pm—3:00pm Room 740, Evans Hall 
Ulrich Pennig (Muenster) Ktheory, BU and C*algbras (Part I: Introductory Talk) This is a quick refresher of the generalized cohomology known as topological Ktheory, constructed from complex vector bundles. We will review its relation to the classifying space BU of the unitary group and the relation to C∗algebras. The talk will be accessible to graduate students and others with no prior exposure to the subject, and also serves to provide background for the exotic version GL_{1}(BU) that I will discuss in the topology seminar later in the day. 
4:00pm—5:00pm Room 3, Evans Hall 
Ulrich Pennig (Muenster) Unit spectra of Ktheory via strongly selfabsorbing C*algebras (Part II: Main Talk) I will speak about an operator algebraic model for the first space of the unit spectrum of complex topological Ktheory, i.e. BGL_{1}(KU), and related infinite loop spaces via bundles of stabilized strongly selfabsorbing C*algebras. The proof that the classifying space of these bundles has the right homotopy type is based on the Imonoid model for GL_{1}(KU) developed by Sagave and Schlichtkrull. I will try to keep the material selfcontained, so no prior knowledge of C*algebras is required to follow the talk. The results are joint work with Marius Dadarlat from Purdue. 

Sep 18, 2013  4:00pm—5:00pm Room 3, Evans Hall 
David Carchedi (Max Planck) A differential graded approach to derived manifolds Given two smooth maps of manifolds f : M → L and g : N → L, if they are not transverse, the fibered product M x_{L} N may not exist, or may not have the expected dimension. In the world of derived manifolds, such a fibered product always exists as a smooth object, regardless of transversality. In fact, every derived manifold is locally of this form. In this talk, we briefly explain what derived manifolds ought to be, why one should care about them, and how one can describe them. We end by explaining a bit of our joint work with Dmitry Roytenberg in which we make rigorous some ideas of Kontsevich to give a model for derived manifolds as certain differential graded manifolds. 
Sep 25, 2013  4:00pm—5:00pm Room 3, Evans Hall 
Tian Yang (Stanford) Hyperbolic cone metrics on 3manifolds with boundary In this joint work with Feng Luo, we prove that a hyperbolic cone metric on an ideally triangulated compact 3manifold with boundary consisting of surfaces of negative Euler characteristic is determined by its combinatorial curvature. The proof uses a convex extension of the Legendre transformation of the volume function. Depending on the time, several related results on maximum volumed semiangle structures will also be mentioned. 
Oct 2, 2013  2:00pm—3:00pm Room 740, Evans Hall 
John Pardon (Stanford) Title: TBA (Part I: Introductory Talk) 
4:00pm—5:00pm Room 3, Evans Hall 
John Pardon (Stanford) Virtual fundamental cycles on moduli spaces of Jholomorphic curves (Part II: Main Talk) Invariants of symplectic manifolds based on "counting" Jholomorphic curves are much more difficult to define when the relevant moduli spaces of such curves are not cut out transversally (it requires a theory of "virtual fundamental cycles"). I will discuss work on a certain algebraic "VFC package" which is applicable in many such cases. Theories of "virtual fundamental cycles" have two parts: (1) One needs to construct (and properly organize together!) some extra local charts for the moduli space (usually "finitedimensional reductions"), and (2) One needs to define a virtual fundamental cycle from this data. Our approach to (1) is closely related to recent work of McDuffWehrheim on "Kuranishi atlases". Our approach to (2) is algebraic, in contrast to the geometric "perturbation" methods used previously. I will give an overview of the construction of this algebraic "VFC package" and discuss its advantages and disadvantages. 

Oct 16, 2013  2:00pm—3:00pm Room 740, Evans Hall 
Kenji Kozai (Berkeley) Deforming geometric structures and halfpipe geometry (Part I: Introductory Talk) We will give an introduction to geometric structures on manifolds using the formalism of (X,G)structures. We will also discuss how to deform geometric structures and give a simple example of a geometric transition from hyperbolic to spherical geometry. This will be followed by an overview of Danciger's halfpipe geometry in dimension three, a transitional geometry between hyperbolic and anti de Sitter geometry. 
4:00pm—5:00pm Room 3, Evans Hall 
Kenji Kozai (Berkeley) Regenerating hyperbolic structures from Sol (Part II: Main Talk) The invariant measured foliations of a pseudoAnosov map of a hyperbolic surface S define a (singular) Euclidean metric on S. Moreover, the corresponding pseudoAnosov flow gives the mapping torus a Sol structure. We will give sufficient conditions on the pseudoAnosov map for finding nearby, nearly collapsed hyperbolic structures that limit (up to rescaling) to the Sol structure, generalizing results of Hodgson and HeusenerPortiSuarez for punctured torus bundles. The proof uses the transition geometry of halfpipe geometry, which was introduced by Danciger. 

Oct 23, 2013  2:00pm—3:00pm Room 740, Evans Hall 
Christine Lescop (Institut Fourier) An introduction to invariants of links and 3–manifolds obtained by counting graph configurations (Part I: Introductory Talk) We will explain how to count graph configurations in 3manifolds in order to obtain invariants of knots, links and 3manifolds, following Gauss (1833), and, more recently, Witten, BarNatan, Kontsevich and others. We will warm up with several equivalent definitions of the simplest of these invariants that is the Gauss linking number of twocomponent links, and pursue with a definition of the CassonWalker invariant of rational homology spheres as an algebraic count of configurations of the θgraph. 
4:00pm—5:00pm Room 3, Evans Hall 
Christine Lescop (Institut Fourier) On a cube of the equivariant linking pairing and a universal equivariant finite type knot invariant (Part II: Main Talk) We will describe an invariant of knots in rational homology 3spheres and some of its properties. Our invariant is an equivariant algebraic intersection of three representatives of the knot Blanchfield pairing in an equivariant configuration space of pairs of points of the knot exterior. We will also outline generalizations of this "cubic" topological construction that produce a "universal equivariant finite type knot invariant". Our invariant is conjecturally equivalent to the Kricker lift of the Kontsevich integral (generalized by Le, Murakami and Ohtsuki) and is indeed equivalent to this lift for knots with trivial Alexander polynomial. 

Oct 30, 2013  2:00pm—3:00pm Room 740, Evans Hall 
Steven Kerckhoff (Stanford) Hyperbolic and AdS Geometry in Dimension 3 (Part I: Introductory Talk) Basic properties and operations in hyperbolic and antideSitter (AdS) geometry will be described. The geometry and classification of "quasiFuchsian" structures in the two geometries, due to AhlforsBers in the hyperbolic case and Mess in the AdS case, will be discussed. There is a strong analogy but no obvious geometric connection. 
4:00pm—5:00pm Room 3, Evans Hall 
Steven Kerckhoff (Stanford) Hyperbolic and AdS Geometry in Dimension 3 (Part II: Main Talk) There's a suggestive analogy between the classical theories of hyperbolic and of AdS quasiFuchsian manifolds in dimension 3. Theorems about their convex hulls by Series and Bonahon in the hyperbolic and by BonsanteSchlenker in AdS settings provide further analogies. I'll discuss joint work with Jeff Danciger that provides a direct geometric context linking the two theories. 

Nov 13, 2013  2:00pm—3:00pm Room 740, Evans Hall 
Pere MenalFerrer (Georgia Tech) Reidemesiter Torsion for Hyperbolic 3Manifolds (Part I: Introductory Talk) In this talk, I will review the definition of Reidemeister torsion and some of its classical applications and results, such as the classification of lens spaces. I will also try to describe some of its applications to 3manifolds. 
4:00pm—5:00pm Room 3, Evans Hall 
Pere MenalFerrer (Georgia Tech) Reidemesiter Torsion for Hyperbolic 3Manifolds (Part II: Main Talk) The aim of this talk is to introduce a certain class of invariants {T_{n}(M)} attached to a finitevolume hyperbolic manifold M. Roughly speaking, T_{n}(M) is the Reidemeister torsion of M obtained from its holonomy representation and the ndimensional fundamental representation of SL(n, C). I will show that the sequence { logT_{n}(M) / n^{2} } converges to Vol(M)/4π; this is an extension of a result by W. Müller which deals with closed manifolds. Finally, I will discuss how the sequence { T_{n} (M) } determines and is determined by the complex length spectrum of M. Joint work with Joan Porti. 

Nov 20, 2013  2:00pm—3:00pm Room 740, Evans Hall 
Krzysztof Putyra (Columbia) A geometric chain complex for odd Khovanov homology (Part I: Introductory Talk) The (even) Khovanov homology is a link invariant that categorifies the Jones polynomial, i.e. the polynomial is the graded Euler characteristic of the homology. The odd Khovanov homology is a distinct theory, which also categorifies the Jones polynomial and both theories agree over Z/2, but they are different otherwise. In 2004 D.BarNatan constructed an invariant complex with objects given by Kauffman states. This complex extends naturally to tangles and it leads to the universal (even) sl_{2} link theory. One can recover the Khovanov's homology by applying the Khovanov's TQFT functor to it, but the odd theory does not fit in this framework. In my talk I will generalize this construction to include the odd theory. This requires a 2categorical structure on cobordisms given by framed Morse functions, called chronologies. Then I will analyze possible targets for TQFT 2functors producing link invariants, obtaining an odd version of dotted cobordisms. There is also a functor producing link homology over a ring Z [X, Y, Z^{ ±1}] / (X^{2} = Y^{2} = 1),which covers both theories (the odd one for X = Y = Z = 1 and the even for X = Z = 1, but Y = 1). 
4:00pm—5:00pm Room 3, Evans Hall 
Krzysztof Putyra (Columbia) Towards the odd Khovanov homology for tangles (Part II: Main Talk) In 2004 M.Khovanov extended the sl_{2} link homology to tangles, using a 2functor from the TemperleyLieb 2category (it has points on a horizontal line as objects, crossingless string diagrams in a plain as morphisms and cobordisms as 2morphisms) to the 2category of rings and bimodules. In my talk I will discuss a generalization of this 2functor, which is expected to produce the odd Khovanov homology, using the geometric construction of the odd theory based on chronological cobordisms (a chronology is a framed Morse function). However, there are a few substantial differences making the generalization nontrivial: the odd versions of arc rings are no longer associative and neither are they actions on bimodules, but both relations hold up to signs, which can be controlled by certain gradings by groupoids and sets on which they act. 

Dec 4, 2013  2:00pm—3:00pm Room 740, Evans Hall 
Jeffrey Danciger (UT Austin) Lorentzian geometry in dimension 2+1 and the group SO(2,1) (Part I: Introductory Talk) We discuss some basic tools for studying constant curvature Lorentzian structures on threemanifolds. These tools exploit a certain lowdimensional coincidence that allows questions about Lorentzian threemanifolds to be translated into questions about hyperbolic surfaces. We will focus on the problem of determining when a group action by Lorentzian isometries is properly discontinuous. 
4:00pm—5:00pm Room 3, Evans Hall 
Jeffrey Danciger (UT Austin) Moduli spaces of constant curvature spacetimes (Part II: Main Talk) A Margulis spacetime is the quotient of threedimensional space by a free group of affine transformations acting properly discontinuously. Each of these manifolds is equipped with a flat Lorentzian metric compatible with the affine structure. I will survey some recent results, joint with Francois Gueritaud and Fanny Kassel, about the geometry, topology, and deformation theory of these flat spacetimes. In particular, we give a parameterization of the moduli space in the same spirit as Penner's cell decomposition of the decorated Teichmuller space of a punctured surface. I will also discuss connections with the negative curvature (AdS geometry) setting. 

Dec 11, 2013  2:00pm—3:00pm Room 740, Evans Hall 
John Francis (Northwestern) Poincaré/Koszul duality (Part I: Introductory Talk) A factorization homology theory—after Lurie, Beilinson & Drinfeld—gives an invariant of an nmanifold M given a choice of coefficient system, which is an ndisk algebra. These factorization homology theories simultaneously generalize singular/generalized homology theories, Hochschild homology, and the observables in topological quantum field theories. The first talk will introduce factorization homology and its basic features, such the EilenbergSteenrod axioms for factorization homology, and essential calculations, where the coefficient system is an nfold loop space, a commutative algebra, or an ndisk enveloping algebra of a Lie algebras. 
4:00pm—5:00pm Room 3, Evans Hall 
John Francis (Northwestern) Poincaré/Koszul duality (Part II: Main Talk) The second talk, after reintroducing factorization homology, will answer the question of what form Poincaré duality should take for factorization homology theories. will focus on a theorem which simultaneously generalizes Poincaré duality for usual homology theories and Koszul duality for ndisk algebras. This duality has some consequences – when the manifold is the circle, in particular, new results are obtained about Hochschild homology of associative algebras. This work is joint with David Ayala. 
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Last modified 29 July 2014.