Department of Mathematics
Topology Seminar


Spring 2013
Time and place: Wednedays 4:30-5:30 in Rec 108
Organizer: Ralph Kaufmann. Contact organizer.


09/02/15 Arnold Yim (Purdue). Homological Properties of Determinantal Arrangements.
            Abstract:  An important aspect of a divisor $Y$ on a complex analytic manifold $X$ is its singular locus. By studying the vector fields on $X$ that are tangent to $Y$, one can understand just how complex the singular locus is. In particular, if the collection of these vector fields form a free module, then the singular locus is simple and we call the divisor a \emph{free divisor}.
    Free divisors show up naturally in many different settings. For example, many of the classically arising hyperplane arrangements (such as braid arrangements and Coxeter arrangements) are free. Moreover, we can get a lot of information about the topology of the complement of an arrangement if we know that it is free. Though much is known for hyperplane arrangements, things become more difficult when we consider arrangements of more general hypersurfaces. In this talk, we explore freeness for determinantal arrangements (arrangements defined by minors of a generic matrix), and the topology of the complement of such arrangements.

09/02/15 Dan Li (Purdue). Topological insulators and K-theory.
        Abstract: Topological insulators are a hot topic in mathematical physics these days.
The behaviour of so-called Majorana zero modes gives rise to a Z/2Z-valued topological invariant, which characterizes time reversal invariant topological insulators.
    I will talk about the topological Z/2Z invariant in the framework of index theory and K-theory. 

09/23/15 Jeremy Miller (Purdue). Localization and homological stability
        Abstract: Traditionally, homological stability concerns sequences of spaces with maps between them that induce isomorphisms on homology in a range tending to infinity. I will talk about homological stability phenomena in situations where there are no natural maps between the spaces. The prototypical example of this phenomenon is configuration spaces of particles in a closed manifold. In this and other situations, the homological stability patterns depend heavily on what coefficient ring one considers.

09/30/15 Ben Ward (Simons Center, Stony Brook)  Operads of the baroque era.
        Abstract:  The failure of an associative algebra to be commutative is controlled by an instance of the Koszul dual structure: the commutator Lie bracket.  This has been well understood since the Renaissance period of the 1990s. I will discuss an E_2 analog of this fact.

10/07/15 Stephan Stolz (Notre Dame): Functorial field theories from factorization algebras
        Abstract: There are various, quite different mathematical approaches to quantum field theories, among them functorial field theories in the sense of Atiyah and Segal and the  factorization algebras of quantum observables constructed by Costello and Gwilliam.
        In the talk, I will describe a construction that produces a twisted functorial field theory  from a factorization algebra, thus relating these two approaches. This is joint work with Bill Dwyer and Peter Teichner.

10/14/15 Lauretiu Maxim (Wisconsin): Equivariant invariants of external and symmetric products of quasi-projective varieties
        Abstract: I will start by revisiting formulae for the generating series of genera of symmetric products (with suitable coefficients), which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical results in the literature as special cases. Important specializations of these results include generating series for extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures of symmetric products of complex projective varieties. In the second part of the talk, I will describe a generating series formula for equivariant invariants of external products, which includes all of the above-mentioned results as special cases. This is joint work with Joerg Schuermann.

10/28/15 Alexander Kupers (Stanford). H-principles using delooping
        Abstract: H-principles are about reducing geometric problems to homotopy-theoretic ones. Gromov gave a general criterion for h-principles to hold on open manifolds and I will explain how to extend this to closed manifolds. As consequences we will deduce Vassiliev's h-principle on smooth maps with moderate singularities and the contractibility of the space of framed functions.

11/4/15 Heather Lee (Purdue).   Homological mirror symmetry for open Riemann surfaces from pair-of-pants decompositions
        Abstract: Mirror symmetry is a duality between symplectic and complex geometries, and the homological mirror symmetry (HMS) conjecture was formulated by Kontsevich to capture this phenomenon by relating two triangulated categories. 
        In this talk, we will prove one direction of the HMS conjecture for punctured Riemann surfaces -- the wrapped Fukaya category of a punctured Riemann surface H is equivalent to the category of singularities of the toric Landau-Ginzburg mirror (X, W), where W is a holomorphic function from X to the complex plane.
        Given a Riemann surface with a pair-of-pants decomposition, we compute its wrapped Fukaya category in a suitable model by reconstructing it from those of various pairs of pants. The pieces are glued together in the sense that the restrictions of the wrapped Floer complexes from two adjacent pairs of pants to their adjoining cylindrical piece agree. The A_\infty-structures are given by those in the pairs of pants. The category of singularities of the mirror Landau-Ginzburg model can also be constructed in the same way from a Cech cover by local affine pieces that are mirrors of the pairs of pants.  In fact, HMS serves as our guide in developing this sheaf theoretic method for computing the wrapped Fukaya category.

11/11/15  Ilya Grigoriev (U of Chicago). Characteristic classes of manifold bundles
        Abstract: For every smooth fiber bundle  f: E\to B with fiber a closed, oriented manifold  M^d of dimension d and any characteristic class of vector bundles  p in H^*( BSO(d)), one can define a “generalized Miller-Morita-Mumford class” or “kappa-class” κ_p in H^*(B). We are interested in the ideal I_M of all the polynomials in the kappa classes which vanish for every bundle with fiber diffeomorphic to M, as well as the algebraic structure of the quotient  R_M = Q[κ _p]/I_M of the free polynomial algebra by this ideal. I will talk mainly about the case where the manifold is a connected sums of g copies of S^n \times S^n, with n odd. In this case, we can compute the ring R_M modulo nilpotents, and show that the Krull dimension of R_M is n-1 for all g>1. This is joint work with Søren Galatius and Oscar Randal-Williams. 

11/18/15 John Wiltshire-Gordon (U of Michigan).  Algebraic invariants of configuration space via representation theory of finite sets
Abstract: The space of n-tuples of distinct points in a smooth manifold M is called the nth configuration space of M.  As n grows, what happens to configuration space?  This attractive question continues to receive plenty of attention.  Recently, Church-Ellenberg-Farb obtained strong results on the eventual behavior of the cohomology of configuration space using the representation theory of finite sets.  I will use recent advances in this theory to prove a theorem about configuration space when M admits a nowhere-vanishing vector field.  Finally, I will use Goodwillie calculus to give a similar result for configurations of smoothly embedded circles if M has almost-complex structure.  This talk is based on joint work with Jordan Ellenberg.

12/2/15 John Harper (OSU). Derived Koszul duality of spaces and structured ring spectra
        Abstract:Consider a flavor of structured ring spectra that can be described as algebras over an operad O in spectra. A natural question to ask is when the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated Koszul dual comonad K can be modified to turn it into an equivalence of homotopy theories. In their 2012 Selecta Math. paper, Francis and Gaitsgory conjecture that replacing O-algebras with the full subcategory of homotopy pro-nilpotent O-algebras will do the trick. In joint work with Kathryn Hess, we show that every 0-connected O-algebra is homotopy pro-nilpotent. This talk will describe recent work, joint with Michael Ching, that resolves in the affirmative the 0-connected case of the
Francis-Gaitsgory conjecture. If time permits, we will also outline recent work, joint with Jake Blomquist, on derived Koszul duality for

12/9/15 Jason Lucas (Purdue). Decorated Feynman Categories.
        Abstract: Feynman categories provide us with a strong categorical framework for discussing operadic theories. A good way to understand this setup is within the framework of graph structures, which are natural examples. This allows one to capitalize on graph theoretic constructions by using them as a guide. When working in this context, for instance, it is common to label the vertices of these graphs with elements of the object being studied. Decorated Feynman categories formalize this process. We will define the notion of a decorated Feynman category and establish some of its basic properties. In addition, we will discuss some applications of decorated Feynman categories to the study of surfaces with arcs.