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Seminars During Academic Year 2007-08
Fall 2007
- Thursday August 2, 2007 in Rawles Hall 316
- Kate Kearney, Eight Faces of the Poincare Homology Sphere
The Poincare Homology Sphere is a 3-manifold that arises in many different ways. This talk will discuss eight such ways, as presented in a paper with the same title as this talk, written by Kirby and Scharlemann.
- Wednesday August 29, 2007 in Rawles Hall 104
- Matt Hedden, On a conjecture regarding knots in the three-sphere on which surgery yields lens spaces
- Thursday August 30, 2007 in Rawles Hall 104
- Nadya Shirokova, The categorification of Vassiliev theory
Many recent theories, e.g. Khovanov, Ozsvath-Szabo homologies, can be considered as categorifications of classical invariants. We introduce a program of classification of such theories. We consider them as constructible sheaves on the spaces of manifolds, extend them to the singular locus, introduce the definition of the theory of finite type and construct our first examples. The resulting theory can be considered as the categorification of Vassiliev theory.
- Wednesday September 12, 2007 in Rawles Hall 104
- Pierre Py, Joint Topology-Dynamics Seminar: Quasi-morphisms and Hamiltonian diffeomorphisms
A quasi-morphism on a group $\Gamma$ is a map $\phi : \Gamma \to \mathbb{R}$ such that there exist a constant $C$ with $\vert \phi(xy)-\phi(x)-\phi(y) \vert \le C$, for all $x,y \in \Gamma$. Recently, Gambaudo and Ghys, as well as Entov and Polterovich discovered that many groups of Hamiltonian diffeomorphisms admit non-trivial quasi-morphisms, although they do not admit any non-trivial homomorphism to $\mathbb{R}$. I would like to survey and compare all these constructions.
- Wednesday September 26, 2007 in Rawles Hall 104
- Kevin Pilgrim, An algebraic formulation of Thurston's characterization of rational functions
- Wednesday October 3, 2007 in Rawles Hall 104
- Kari Ragnarsson, Completion of G-spectra and stable maps between classifying spaces
For a finite group G, completion at the augmentation ideal I(G) of the Burnside ring A(G) features in cohomological completion theorems such as the Atiyah–Segal completion theorem and Carlsson's theorem (formerly the Segal Conjecture). Completion of G-spectra at ideals of the Burnside ring was introduced by Greenlees–May to obtain topological versions of these theorems, letting the completion take place at the level of spectra rather than cohomology groups. When G is a p-group, May–McClure showed that I(G)-adic completion coincides with p-completion (modulo some basepoint issues) but in general I(G)-adic completion seems very complicated.
In this talk I will describe how, for the purposes of I(G)-completion, one may replace a G-spectrum E by a related G-spectrum E' whose isotropy groups are all groups of prime power order. The I(G)-adic completion of the latter is often more approachable, admitting a description as a colimit of completions at primes. As an application I will give a simplified description of the function spectrum F(BG,BK) for a finite group and a compact Lie group K.
- Wednesday October 10, 2007 in Rawles Hall 104
- Mihai Staic, Hopf Algebras, Braids, and Knot Invariants
- Thursday October 11, 2007 in Rawles Hall 104
- Sergei Chmutov, Thistlethwaite's theorem for virtual links
Regions of a link diagram can be colored in black and white in a checkerboard manner. Putting a vertex in each black region and connecting two vertices by an edge if the corresponding regions share a crossing yields a planar graph. In 1987 Thistlethwaite proved that the Jones polynomial of the link can be obtained by a specialization of the Tutte polynomial of this planar graph. I will explain a generalization of this theorem to virtual links. In this case the graph will be a ribbon graph, which means that it will be embedded into a (higher genus, possibly non-oriented) surface. For such graphs we use a generalization of the Tutte polynomial discovered recently by B. Bollobas and O. Riordan. This is a joint work with Jeremy Voltz.
- Wednesday October 31, 2007 in Rawles Hall 104
- Nathan Geer, Renormalized quantum invariants
In this talk I will discuss a renormalization of the Reshetikhin-Turaev quantum invariants, by "fake quantum dimensions." In the case of simple Lie algebras these "fake quantum dimensions" are proportional to the genuine quantum dimensions. More interestingly I will discuss two examples where the genuine quantum dimensions vanish but the "fake quantum dimensions" are non-zero and lead to non-trivial link invariants. The first of these examples is a class of invariants arising from Lie superalgebras defined by Patureau-Mirand and myself. These invariants are multivariable and generalize the multivariable Alexander polynomial. The second example, is a hierarchy of invariants arising from nilpotent representations of quantized sl(2) at a root of unity. These invariants contain Kashaev's quantum dilogarithm invariants of links. This is joint work with Bertrand Patureau-Mirand and Vladimir Turaev.
- Thursday November 1, 2007 in Rawles Hall 104
- Chris Douglas, Higher Clifford Algebras
Real K-Theory is 8-periodic. This periodicity can be seen algebraically from the periodicity of Clifford algebras — I will discuss how the notion of Clifford algebra arises naturally from thinking about how to define K-theory. It turns out that Clifford algebras form a 2-category, and the generator of that 2-category has order 8. We are interested in the analogous story for elliptic cohomology — I will describe the geometric approach to elliptic cohomology introduced by Segal, Stolz, and Teichner, and explain the role of what we call "higher Clifford algebras". These objects ought to form a 3-category — we introduce such a 3-category and show that its generator has order at least 24. This is joint work with Arthur Bartels and Andre Henriques.
- Thursday November 1, 2007 in Rawles Hall 104
- Matt Mauntel, Algebraic K-Theory
- Wednesday November 28, 2007 in Rawles Hall 104
- Frank Quinn, An approach to the Farrell-Jones conjecture
The conjecture describes the algebraic K-theory of group rings via an assembly map involving only the K-theory of virtually cyclic groups. Special cases have been obtained slowly and with difficulty by Farrell-Jones, Bartels-Lueck-Reich, Farrell-Linnell, Davis-Lueck and others. We describe an approach to the general result that has led to a number of curious ideas and should at least give some injectivity results.
- Wednesday December 5, 2007 in Rawles Hall 104
- Chuck Livingston, Finite Representations of Knot Groups
- Thursday December 13, 2007 in Rawles Hall 104
- David Roshenthal, On the algebraic K-theory of groups with finite asymptotic dimension
In this joint work with Arthur Bartels, it is proved that the assembly maps in algebraic K-and L-theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for the universal space for proper actions. The result also applies to certain groups that admit only a finite dimensional model for the universal space. In particular, it applies to discrete subgroups of virtually connected Lie groups.
Spring 2008
- Wednesday January 9, 2008 in Rawles Hall 104
- Patricia Hersh, Regular cell complexes modeling Bruhat intervals
We give a new criterion for determining whether a finite CW complex is regular. This involves both combinatorial conditions on the closure poset and also topological conditions on the codimension one cell incidences. As an application, we prove a conjecture of Fomin and Shapiro answering a question of Bjorner, namely the question of finding naturally arising regular CW complexes with the Bruhat intervals as their closure posets. To this end, we confirm a conjecture of Fomin and Shapiro that certain stratified totally positive spaces known to have the proper closure posets are regular CW decompositions; this involves showing that an intriguing map of Lusztig yields the characteristic maps.
- Tuesday January 15, 2008 in Rawles Hall 104
- Allan Edmonds, The Plus Construction for Manifolds
This is a short coda to our seminar of Fall, 2007, in which I will interpret the plus construction in the category of manifolds. One nice consequence is that topological or PL homology n-spheres, n>=5, bound contractible manifolds.
- Wednesday February 13, 2008 in Rawles Hall 104
- Neza Mramor-Kosta, The degree of maps between manifolds with free group actions
The degree of a map between two manifolds with free group actions is in some cases determined by the classifying maps of the two manifolds. In particular this is true for equivariant maps, and for maps which are in certain ways close to being equivariant. We will describe some classes of such maps, and the resulting formulas for the degree.
- Wednesday February 20, 2008 in Rawles Hall 104
- Chuck Livingston, Twisted Alexander Polynomials and Knot Slicing
- Tuesday April 8, 2008 in Rawles Hall 104
- Andrew Ranicki, The geometric Hopf invariant
This is a joint project with Michael Crabb (Aberdeen). The geometric Hopf invariant of a stable map F:Sigma^kX to Sigma^kY is a Z_2-equivariant map h_{R^k}(F) which “counts the double points” of F. The homotopy class of h_{R^k}(F) is the primary obstruction to F being homotopic to the k-fold suspension Sigma^kF_0 of an unstable map F_0:X to Y. The geometric Hopf invariant has applications to double points of immersions of manifolds, and surgery obstruction theory (including the non-simply-connected case).
- Wednesday April 9, 2008 in Rawles Hall 104
- Danny Ruberman, Knot concordance and Heegaard Floer Homology of Branched covers
- Thursday April 10, 2008 in Rawles Hall 104
- Danny Ruberman, Rohlin's invariant and periodic-end Dirac operators
- Wednesday April 16, 2008 in Rawles 104
- Jamie Pommersheim, The geometry of summation
The classical Euler-Maclaurin formula expresses the sum of a function $f$ of one variable as the integral $f$ plus terms involving derivatives of $f$ at the endpoints. This formula may be viewed as expressing the sum $f$ over the integral points in a one dimensional polytope (i.e., a line segment) in terms of certain integrals over the faces of the polytope. In this talk, we explore higher dimensional analogs of Euler-Maclaurin formula, which allow one to sum functions over polytopes of any dimension. An important special case is summing the constant function $f=1$, i.e., counting the number of lattice points inside the polytope. Much progress on this problem has been made in the last 20 years using the theory of toric varieties. After discussing some of these results, we will introduce the Euler-Maclaurin formulas of Berline and Vergne, as well as joint work of the Garoufalidis and the speaker.
- Thursday April 24, 2008 in Rawles Hall 104
- Jim McClure, A convenient category of $C^\infty$ manifolds.
Joint with Gerd Laures. Gluing of smooth manifolds has always been a delicate topic because it is only well defined if one chooses “gluing data” (such as vector fields). But in fact one can expand the category of smooth manifolds with boundary in such a way that gluing is canonically defined and is a categorical pushout. Moreover, if we redefine the word “set” then gluing becomes strictly associative (not just up to isomorphism). Our category also has Cartesian products and has the same bordism groups as the usual category of smooth manifolds with boundary.
- Wednesday May 28, 2008 in Rawles Hall 104
- Charles Livingston, The concordance genus of knots