Topology Seminar

Seminars During Academic Year 2005-06

Fall 2005

Wednesday August 3, 2005 in Rawles Hall 104

Eli Grigsby, Torsions and Knot Floer Homology

The knot Floer homology groups of a knot K in a closed, connected, oriented, 3-manifold Y were developed several years ago by Ozsvath and Szabo and independently by Rasmussen. They proved that if K is a knot in S^3, the Euler characteristic of the knot Floer homology group gives the Alexander polynomial of K. This relationship is most easily understood via the relationship of both invariants to the torsion of the infinite cyclic cover of the knot complement. We will warm up by explaining this relationship, and go on to prove a generalization of this result which holds for a nullhomologous knot in a rational homology sphere. Along the way, we will focus on the example of the preimage of a knot in its double branched cover.

Thursday August 4, 2005 in Rawles Hall 104

Eli Grigsby, Torsions and Knot Floer Homology (continued)

The knot Floer homology groups of a knot K in a closed, connected, oriented, 3-manifold Y were developed several years ago by Ozsvath and Szabo and independently by Rasmussen. They proved that if K is a knot in S^3, the Euler characteristic of the knot Floer homology group gives the Alexander polynomial of K. This relationship is most easily understood via the relationship of both invariants to the torsion of the infinite cyclic cover of the knot complement. We will warm up by explaining this relationship, and go on to prove a generalization of this result which holds for a nullhomologous knot in a rational homology sphere. Along the way, we will focus on the example of the preimage of a knot in its double branched cover.

Monday September 5, 2005 in Rawles Hall 104

Oleg Viro, Khovanov Homology

Tuesday September 13, 2005 in RH316

Chuck Livingston, Mapping Class Group Seminar - Organizational Meeting

Thursday September 15, 2005 in Rawles Hall 104

Mike Mandell, Ring spectra and module spectra

Wednesday September 21, 2005 in Rawles Hall 104

Mike Mandell, Ring spectra and module spectra (continued)

Wednesday September 28, 2005 in Rawles Hall 104

Mike Mandell, Ring spectra and module spectra (continued)

Wednesday October 5, 2005 in Rawles Hall 104

Eric Wilson, Exotic Lens Spaces

Wednesday October 12, 2005 in Rawles Hall 104

Eric Wilson, Exotic Lens Spaces

Wednesday October 19, 2005 in Rawles Hall 104

Michael McCooey, Concordance of group actions on the 4-sphere

Tuesday October 25, 2005 in Rawles Hall 104

Qayum Khan, L-theory of type III virtually cyclic groups

Given an arbitrary group, the Farrell-Jones isomorphism conjecture states roughly that its L-theory is given by its virtually cyclic subgroups and how they fit together. There are three types, and those groups which surject onto the infinite dihedral group with finite kernel are what I call type III. The computation of their L-groups reduces to a computation of associated unitary nilpotent groups UNil. I will discuss some progress in this direction, following recent work of Connolly, Davis, Kozniewski, and Ranicki.

Wednesday October 26, 2005 in Rawles Hall 104

Diarmuid Crowley, Stable diffeomorphism and Kreck's monoid $l_{2q+1}$.

Thursday October 27, 2005 in Rawles Hall 104

Howard Masur, Complexes of curves in low dimensional topology and geometry

One can associate to a surface of genus g with n boundary components a number of different complexes. Among these are the curve complex, the arc complex, the pants complex, and for the boundary of a handlebody, the meridian disc complex. The combinatorics and geometry of these complexes play a role in the geometry and topology of 3 manifolds. I will discuss some of these interactions.

Wednesday November 2, 2005 in Rawles Hall 104

Bjorn Jahren, Topological classification of free involutions on S^1xS^n

The recent calculations of UNil-groups now makes a complete such classification possible. This is based on a classification of the possible homotopy types of quotients, and we show that the only homotopy types that occur are those of four linear examples. However, some of these homotopy types can be realized by infinitely many different conjugacy classes of involutions.

Wednesday November 16, 2005 in Rawles Hall 104

Allan Edmonds, The center conjecture for equifacetal simplices

Equifacetal (geometric) simplices exist in abundance and are characterized by having a vertex transitive isometry group. This implies that such a simplex has a unique center. The center conjecture proposes that the converse is true. On the one hand, the center conjecture is true in general if one adopts a sufficiently weak definition of "center". On the other hand, a more precise conjecture is true at least through dimension six for the most stringent definition of "center".

Thursday December 1, 2005 in Rawles Hall 104

Sam Lewellen

Spring 2006

Monday January 30, 2006 in Ballantine Hall 242

Radmila Sazdanovic, Knot Theory Program: LinKnot

The Mathematica-based Windows knot theory program LinKnot is the extension of the program Knot2000 (K2K) written by M.Ochiai and N.Imafuji. LinKnot is the knot theory program that works with knots and links (KLs) given in the Conway notation. Conway symbols are the input used for creating Dowker codes or P-data (the main input for K2K functions). Instead of a graphical input or Dowker codes, for the first time in a computer program you can use human-comprehensive Conway notation of KLs represented as a Mathematica string and work with links, and not only with knots. For all KLs there is no restriction on the number of crossings. The program provides also the complete data base of alternating KLs with at most 12 crossings, and non-alternating KLs with at most 11 crossings, and the data base of basic polyhedra with at most 20 crossings. By using it, it is possible to draw KLs, calculate all polynomial invariants of KLs, work with braids, reduce KLs, etc. For the first time, it is possible to compute unknotting and unlinking numbers, calculated according to Bernhard-Jablan Conjecture. For all alternating KLs you can compute minimum Dowker codes, find all non-isomorphic projections, work with the graphs of KLs, compute linking numbers, breaking and spliting numbers, signatures, and many other KL invariants. The main property of the program is a possibility to use it as a tool in experimenting with KLs, for computing properties connected with infinite classes of KLs (KL families) and make new conjectures in knot theory. For example, except the famous Nakanishi-Bleiler exapmle of the knot 514 with the unknotting number projection gap, by using LinKnot we discovered an infinite collection of such KLs.

Tuesday January 31, 2006 in Rawles Hall 104

Slavik Jablan, Basic Polyhedra

Tuesday January 31, 2006 in Ballantine Hall 221

Radmila Sazdanovic, On Properties of Chromatic Graph Cohomology

Thursday February 2, 2006 in Rawles Hall 104

Laurentiu Maxim, Higher-order Alexander invariants of plane curve complements

I will survey various Alexander-type invariants of plane curve complements, with an emphasis on obstructions on the type of groups that can arise as fundamental groups of complements to affine curves.

Wednesday February 22, 2006 in Rawles Hall 104

L. Nicolaescu, Counting Morse functions

I will describe several equivalence relations on the set of excellent Morse functions on a compact smooth manifold M (excellent = at most one critical point per level set). I will describe Arnold's count of equivalence classes in the case M= S^1, and then I will explain how to count these classes when M= S^2. This mathematical journey will take us in unexpected places.

Wednesday March 1, 2006 in Rawles Hall 104

Jan Jaworowski, Equivariant Maps of Manifolds

The talk is related to the paper by Yasuhiro Hara (Y. Hara, The degree of equivariant maps, Topology and its Appl. vol 148, 2004, p. 113-1210). I take a slightly different approach from that used by Hara and prove a little more. The talk will be very elementary, certainly suitable for graduate students.

Monday March 6, 2006 in Ballantine 011

Vaughan Jones, Tangles: linear and quadratic

Wednesday March 22, 2006 in Rawles Hall 104

J. E. McClure, E-infinity Structures on Surgery Spectra

Tuesday March 28, 2006 in Rawles Hall 104

Matt Hedden, Recent work regarding smooth knot concordance invariants

In the past few years, several new invariants of smooth knot concordance have been discovered. These invariants have arisen in very different contexts: some with analytical origins and others defined combinatorially with close ties to representation theory . I'll survey these invariants and discuss generalizations which should exist. I will probably focus most attention on two concordance invariants, denoted s(K) and tau(K), associated to a knot, K. tau(K) was discovered by Ozsvath and Szabo, and independently by Rasmussen, while s(K) was discovered by Rasmussen. They are perhaps the most notable of the recently discovered invariants and were conjectured to be equal. In fact, they are not and I'll discuss the first counterexamples to this conjecture, jointly discovered with Philip Ording.

Wednesday March 29, 2006 in Rawles Hall 104

Ed Swartz, Face enumeration on manifolds

In the over 100 years since the discovery of the Euler-Poincar? formula there have been tremendous advances in the understanding of the geometry and topology of manifolds. However, the enumerative properties of triangulations remain mysterious. For instance, the minimum number of faces needed to produce a closed $d$-manifold with nonzero first Betti number was unknown until recently. We will survey what is known and sample a variety of new results and techniques.

Monday April 3, 2006 in Rawles Hall 104

Danielle O'Donnol, Intrinsically n-linked complete bipartite graphs

Tuesday April 4, 2006 in Rawles Hall 104

Zhenghan Wang, Classification of TQFTs

Wednesday April 5, 2006 in Kirkwood 200

Stavros Garoufalidis, Asymptotic expansions in quantum topology

A celebrated invariant of knots in 3-space is the Jones polynomial. Using the Jones polynomial one can construct "analytic" invariants of 3-manifolds, where "analytic" has a precise number-theoretic meaning. In the talk we will discuss the relation between quantum invariants and their better known classical analogues: such as the algebrotoplogical invariants of the knot complements and the riemannian (mostly hyperbolic) invariants of complements.

Thursday April 6, 2006 in Rawles Hall 104

Denis Auroux, Near symplectic structures and Lefschetz pencils on smooth 4-manifolds

Every compact smooth 4-manifold with nonzero b+ carries a near-symplectic structure, i.e. a closed 2-form which degenerates in a specific way along a union of circles. The study of these structures is largely motivated by Taubes' program to interprete Seiberg-Witten invariants in terms of enumerative geometry in this context. In this talk, we will describe an extension to the near-symplectic situation of Donaldson's construction of Lefschetz pencils (this is joint work with S. Donaldson and L. Katzarkov). More precisely, we will define the appropriate notion of singular Lefschetz pencil, state the existence result, and discuss some simple topological considerations and examples.

Wednesday April 26, 2006 in Rawles Hall 104

P. Kirk, Symplectic 4-manifolds with arbitrary fundamental group near the BMY line

Wednesday May 3, 2006 in Rawles Hall 104

Shmuel Weinberger, Spaces of Translations

The simplest nontrivial dynamical systems are arguably irrational rotation on the circle. In this talk, I will try to generalize their classification to other (almost as simple) translation systems.

Friday May 26, 2006 in Rawles Hall 104

Vladimir Turaev, A generalization of several classical invariants of links

We extend several classical invariants of links in the 3-sphere to links in so-called quasi-cylinders. These invariants include the linking number, the Seifert form, the Alexander module, the Alexander-Conway polynomial and the Tristram-Levine signatures. This is joint work with D. Cimazoni.

Tuesday May 30, 2006 in Rawles Hall 104

Stefan Friedl, Symplectic products and subgroup separability

A folklore conjecture says that if a product $S^1\times N^3$ is symplectic, then $N^3$ fibers over $S^1$. We will give a proof for this conjecture in the case that $N$ has vanishing Thurston norm, and we will reduce this conjecture to a conjecture on Kleinian groups. (This is joint work with Stefano Vidussi).