Current Year:
2017-18

Previous Years:
2016-17
2015-16
2014-15
2013-14
2012-13

Berkeley Topology Seminar
Fall 2013 and Spring 2014


Organizers (for Fall 2013—Spring 2014)
Kenji Kozai
Rumen Zarev
Usual time and place
Introductory talk accessible to graduate students: Wednesday, 2pm to 3pm, Evans Hall, Room 740
Specialized research talk: Wednesday, 4pm to 5pm, Evans Hall, Room 3
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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 3-manifolds.
(Part I: Introductory Talk)

In this talk, we will review Kahn-Markovic's and Liu-Markovic's work on constructing almost totally geodesic surfaces in closed hyperbolic 3-manifolds. Kahn and Markovic's work gives immersed π1-injective closed almost totally geodesic surfaces in any closed hyperbolic 3-manifolds, which is the first step of the proof of the virtual Haken and virtual fibered conjectures. Actually, Liu and Markovic's work is a 3-dimensional version of Kahn-Markovic's work on the good pants homology. In particular, for any union of closed geodesics L in a closed hyperbolic 3-manifold M, they showed that any homology class in H2(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 3-manifolds
(Part II: Main Talk)

By using Kahn-Markovic and Liu-Markovic's construction of almost totally geodesic quasi-Fuchsian surfaces (closed or bounded) , we can construct various immersed π1-injective 2-complexes in any closed hyperbolic 3-manifold M. By using Agol's result that the groups of hyperbolic 3-manifolds are LERF, a "geometric neighborhood" of such an immersed π1-injective 2-complex 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:
  1. For any closed hyperbolic 3-manifold M, and any finite abelian group A, there is a finite cover M' of M, such that A is a direct summand of Tor(H1(M, Z)).
  2. For any closed hyperbolic 3-manifold M, and any closed oriented 3-manifold N, there is a finite cover M'' of M, such that there exists a non-zero degree map from M'' to N.
Jan 22, 2014 2:00pm—3:00pm
Room 740,
Evans Hall
Theo Johnson-Freyd (Northwestern)
Title: TBA
(Part I: Introductory Talk)
4:00pm—5:00pm
Room 3,
Evans Hall
Theo Johnson-Freyd (Northwestern)
p-to-homotopy 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 well-known graded-commutative Frobenius algebra structure. Does this structure lift in a geometrically meaningful up-to-homotopy way to de Rham chains? The answer depends on the meanings of "geometrically meaningful" and "up-to-homotopy". 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 En 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 4-manifolds.
4:00pm—5:00pm
Room 3,
Evans Hall
Adam Levine (Princeton)
Non-orientable surfaces in homology cobordisms
(Part II: Main Talk)

We study the minimal genus problem for embeddings of closed, non-orientable surfaces in a homology cobordism between rational homology spheres, using obstructions derived from Heegaard Floer homology and from the Atiyah-Singer index theorem. For instance, we show that if a non-orientable 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 non-orientable 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 Non-Simple Closed Geodesics on Surfaces

We get coarse bounds on the number of (non-simple) closed geodesics on a surface, given upper bounds on both length and self-intersection number. Recent work by Mirzakhani and by Rivin has given asymptotics for the growth of simple closed curves and curves with one self-intersection (respectively) with respect to length. No asymptotics for arbitrary self-intersection number are currently known, but we give coarse bounds for arbitrary self-intersection 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 4-manifolds

We give necessary and sufficient conditions for a closed smooth 6-manifold N to be diffeomorphic to a product of a surface F and a simply connected 4-manifold 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 self-diffeomorphism 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 Eilenberg-Mac 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 non-equivariant 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 3-manifolds obtained by performing Dehn surgery on fibered knots. These connections arise by associating to a fibered knot in a 3-manifold the fractional Dehn twist coefficient of the monodromy of the fibration on its complement. I'll describe how the Heegaard Floer homology of a 3-manifold bounds the fractional Dehn twist coefficient of any of its fibered knots, and some consequences this has for contact structures on 3-manifolds. This is joint work with Tom Mark.
Mar 12, 2014 2:00pm—3:00pm
Room 740,
Evans Hall
Clark Barwick (MIT)
Algebraic K-theory as categorified stable homotopy theory
(Part I: Introductory Talk)

This is a non-technical introduction to the new idea of algebraic K-theory 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 K-theory of rings and ring spectra
(Part II: Main Talk)

This is a survey of some relatively concrete advances in algebraic K-theory 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 Hill-Hopkins-Ravenel norm
(Part II: Main Talk)

B?kstedt 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 B?kstedt smash product to the norm construction from the Hill-Hopkins-Ravenel 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 S-algebra). 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 non-existence of commutative S-algebra structures on various quotients of MU. For example MU / (pi) or ku / (pi 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 Luck (Bonn)
Introduction to the middle K-theory 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 s-Cobordism Theorem and topological rigidity.
4:00pm—5:00pm
Room 3,
Evans Hall
Wolfgang Luck (Bonn)
Algebraic K- and L-theory of groups rings and their applications to topology and geometry
(Part II: Main Talk)

We give an introduction to the K- and L-theoretic Farrell-Jones 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 4-manifolds can be restated in terms of the connection between real and complex K-theory given by homotopy fixed points. This comes from a bordism result about Real manifolds versus unoriented manifolds, which in turn, comes from a C2-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 four-ball?
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 S3 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)
K-theory, BU and C*-algbras
(Part I: Introductory Talk)

This is a quick refresher of the generalized cohomology known as topological K-theory, 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 GL1(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 K-theory via strongly self-absorbing 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 K-theory, i.e. BGL1(KU), and related infinite loop spaces via bundles of stabilized strongly self-absorbing C*-algebras. The proof that the classifying space of these bundles has the right homotopy type is based on the I-monoid model for GL1(KU) developed by Sagave and Schlichtkrull. I will try to keep the material self-contained, 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 xL 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 3-manifolds with boundary

In this joint work with Feng Luo, we prove that a hyperbolic cone metric on an ideally triangulated compact 3-manifold 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 semi-angle 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 J-holomorphic curves
(Part II: Main Talk)

Invariants of symplectic manifolds based on "counting" J-holomorphic 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 "finite-dimensional 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 McDuff--Wehrheim 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 half-pipe 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 half-pipe 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 pseudo-Anosov map of a hyperbolic surface S define a (singular) Euclidean metric on S. Moreover, the corresponding pseudo-Anosov flow gives the mapping torus a Sol structure. We will give sufficient conditions on the pseudo-Anosov map for finding nearby, nearly collapsed hyperbolic structures that limit (up to rescaling) to the Sol structure, generalizing results of Hodgson and Heusener-Porti-Suarez for punctured torus bundles. The proof uses the transition geometry of half-pipe 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 3-manifolds in order to obtain invariants of knots, links and 3-manifolds, following Gauss (1833), and, more recently, Witten, Bar-Natan, Kontsevich and others. We will warm up with several equivalent definitions of the simplest of these invariants that is the Gauss linking number of two-component links, and pursue with a definition of the Casson-Walker invariant of rational homology spheres as an algebraic count of configurations of the θ-graph.
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 3-spheres 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 anti-deSitter (AdS) geometry will be described. The geometry and classification of "quasi-Fuchsian" structures in the two geometries, due to Ahlfors-Bers 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 quasi-Fuchsian manifolds in dimension 3. Theorems about their convex hulls by Series and Bonahon in the hyperbolic and by Bonsante-Schlenker 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 Menal-Ferrer (Georgia Tech)
Reidemesiter Torsion for Hyperbolic 3-Manifolds
(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 3-manifolds.
4:00pm—5:00pm
Room 3,
Evans Hall
Pere Menal-Ferrer (Georgia Tech)
Reidemesiter Torsion for Hyperbolic 3-Manifolds
(Part II: Main Talk)

The aim of this talk is to introduce a certain class of invariants {Tn(M)} attached to a finite-volume hyperbolic manifold M. Roughly speaking, Tn(M) is the Reidemeister torsion of M obtained from its holonomy representation and the n-dimensional fundamental representation of SL(n, C). I will show that the sequence { log|Tn(M)| / n2 } converges to -Vol(M)/4π; this is an extension of a result by W. Muller which deals with closed manifolds. Finally, I will discuss how the sequence { Tn (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.Bar-Natan constructed an invariant complex with objects given by Kauffman states. This complex extends naturally to tangles and it leads to the universal (even) sl2 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 2-categorical structure on cobordisms given by framed Morse functions, called chronologies. Then I will analyze possible targets for TQFT 2-functors 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] / (X2 = Y2 = 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 sl2 link homology to tangles, using a 2-functor from the Temperley-Lieb 2-category (it has points on a horizontal line as objects, crossingless string diagrams in a plain as morphisms and cobordisms as 2-morphisms) to the 2-category of rings and bimodules. In my talk I will discuss a generalization of this 2-functor, 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 non-trivial: 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 three-manifolds. These tools exploit a certain low-dimensional coincidence that allows questions about Lorentzian three-manifolds 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 three-dimensional 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)
Poincare/Koszul duality
(Part I: Introductory Talk)

A factorization homology theory after Lurie, Beilinson & Drinfeld gives an invariant of an n-manifold M given a choice of coefficient system, which is an n-disk 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 Eilenberg-Steenrod axioms for factorization homology, and essential calculations, where the coefficient system is an n-fold loop space, a commutative algebra, or an n-disk enveloping algebra of a Lie algebras.
4:00pm—5:00pm
Room 3,
Evans Hall
John Francis (Northwestern)
Poincare/Koszul duality
(Part II: Main Talk)

The second talk, after reintroducing factorization homology, will answer the question of what form Poincare duality should take for factorization homology theories. will focus on a theorem which simultaneously generalizes Poincare duality for usual homology theories and Koszul duality for n-disk 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.