Topology seminar

Location

Mondays at 2:30 in Ford Hall 110

Talks in Fall 2019

• Sep 09 '19

  • Weiyan Chen (University of Minnesota)

  • Cohomology of the space of complex irreducible polynomials in several variables

  We will show that the space of complex irreducible polynomials of degree d in n variables satisfies two forms of homological stability: first, its cohomology stabilizes as d increases, and second, its compactly supported cohomology stabilizes as n increases. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.

• Sep 09 '19

  • TBA ()

  • Topology Seminar

  Abstract not available

• Sep 16 '19

  • Peter Webb (University of Minnesota)

  • Transfer in the homology and cohomology of categories

  The cohomology of a category has many properties that extend those that are familiar when the category is a group. Second cohomology classifies equivalence classes of category extensions, first cohomology parametrizes conjugacy classes of splittings, first homology is the abelianization of the fundamental group, and second homology has a theory that extends that of the Schur multiplier. Defining restriction and corestriction maps on the homology of categories is problematic: most attempts to do this require induction and restriction functors to be adjoint on both sides, and this typically does not happen with categories. We describe an approach to defining these maps that includes all the situations where they can be defined in group cohomology, at least when the coefficient ring is a field. The approach uses bisets for categories, the construction by Bouc and Keller of a map on Hochschild homology associated to a bimodule, and the realization by Xu of category cohomology as a summand of Hochschild cohomology.

• Sep 23 '19

  • Dongkwan Kim (University of Minnesota)

  • Homology class of Deligne-Lusztig varieties

  Since first defined by Deligne and Lusztig, a Deligne-Lusztig variety has become unavoidable when studying the representation theory of finite groups of Lie type. This is a certain subvariety of the flag variety of the corresponding reductive group, and its cohomology groups are naturally endowed with the action such finite groups, which in turn gives a decomposition of irreducible representations called Lusztig series. In this talk, I will briefly discuss the background of Deligne-Lusztig theory and provide a formula to calculate the homology class of Deligne-Lusztig varieties in the Chow group of the flag variety. If time permits, I will also discuss their analogues.

• Sep 30 '19

  • Sasha Voronov (University of Minnesota)

  • Mysterious Duality

  “Mysterious Duality” was discovered by Iqbal, Neitzke, and Vafa in 2001. They noticed that toroidal compactifications of M-theory lead to the same series of combinatorial objects as del Pezzo surfaces do, along with numerous mysterious coincidences: both toroidal compactifications and del Pezzo surfaces give rise to the exceptional series E_k; the U-duality group corresponds to the Weyl group W(E_k), arising also as a group of automorphisms of the del Pezzo surface; a collection of various M- and D-branes corresponds to a set of divisors; the brane tension is related to the “area” of the corresponding divisor, etc. The mystery is that it is not at all clear where this duality comes from. In the talk, I will present another series of mathematical objects: certain versions of multiple loop spaces of the sphere S^4, which are, on the one hand, directly connected to M-theory and its combinatorics, and, on the hand, possess the same combinatorics as the del Pezzo surfaces. This is a report on an ongoing work with Hisham Sati.

• Oct 14 '19

  • Craig Westerland (University of Minnesota)

  • Second order terms in arithmetic statistics

  The machinery of the Weil conjectures often allows us to relate the singular cohomology of the complex points of a scheme to the cardinality of its set of points over a finite field. When we apply these methods to a moduli scheme, we obtain an enumeration of the objects the moduli parameterizes. It's rare that we can actually fully compute the cohomology of these moduli spaces, but homological stability results often give a first order approximation to the homology.

  In this talk, we'll explain how to obtain second order homological computations for a class of Hurwitz moduli spaces of branched covers; these yield second order terms in enumerating the moduli over finite fields. We may interpret these as second order terms in a function field analogue of the function which counts number fields, ordered by discriminant. Our second order terms match those of Taniguchi-Thorne/Bhargava-Shankar-Tsimerman in the cubic case, and give new predictions in other Galois settings.

  This is joint (and ongoing) work with Berglund, Michel, and Tran.

• Oct 21 '19

  • Liam Keenan (University of Minnesota)

  • Topology Seminar

  TBA

• Oct 28 '19

  • Elden Elmanto (Harvard University)

  • On the K-theory of Universal Homeomorphisms

  A universal homeomorphism of schemes is a morphism of schemes (frighteningly, with no finiteness conditions) which is a homeomorphism on underlying topological spaces (!) and remains so after arbitrary pullback. In some sense, beginning with at least Grothendieck, one should view a universal homeomorphism as analogous to the topologists’ notion of a “weak equivalence.” In this talk, I will explain this philosophy, give examples, and explain two results about the behavior of algebraic K-theory under universal homeomorphisms. The first is an equicharacteristic result: if k is a field of exponential characteristic c, then homotopy K-theory is invariant under universal homeomorphisms of k-schemes, after inverting c; this is a consequence of joint work with Adeel Khan and implies the statement for usual K-theory in positive characteristics. The second is a mixed characteristic result for usual (but p-inverted) K-theory: over Z_p schemes, universal homeomorphisms are completely controlled by its “characteristic zero part.” The formulation of the theorem, and its proof, is inspired by some results in mixed characteristic birational geometry. This is joint work in progress with Akhil Mathew and Jakub Witaszek.