## Seminar Categories

## Current Series

Mon Sep 09 |
## Topology Seminar2:30pm - Ford Hall 110Cohomology of the space of complex irreducible polynomials in several variables Weiyan Chen, University of Minnesota 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. |

Mon Sep 16 |
## Topology Seminar2:30pm - Ford Hall 110Transfer in the homology and cohomology of categories Peter Webb, University of Minnesota 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. |

Mon Sep 23 |
## Topology Seminar2:30pm - Ford Hall 110Homology class of Deligne-Lusztig varieties Dongkwan Kim, University of Minnesota 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. |

Mon Sep 30 |
## Topology Seminar2:30pm - Ford Hall 110Mysterious Duality Sasha Voronov, University of Minnesota 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. |

Mon Oct 14 |
## Topology Seminar2:30pm - Ford Hall 110Second order terms in arithmetic statistics Craig Westerland, University of Minnesota 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. |

Mon Oct 21 |
## Topology Seminar2:30pm - Ford Hall 110Descent properties of topological Hochschild homology Liam Keenan, University of Minnesota Algebraic K-theory is an extremely rich but notoriously difficult invariant to compute. In order to make calculations tractible, topological Hochschild homology and topological cyclic homology were introduced, along with the Dennis and cyclotomic trace maps. A natural question to consider is whether or not these invariants are sheaves for various topologies arising in algebraic geometry. In fact, it turns out that topological Hochschild homology is a sheaf for the fpqc topology on connective commutative ring spectra. In this talk, I plan to introduce the language necessary and sketch the argument of this result. |

Mon Oct 28 |
## Topology Seminar2:30pm - Ford Hall 110Compactifying the étale topos Elden Elmanto, Harvard University The speaker has long feared the technicalities and intricacies of equivariant stable homotopy theory. Fortunately, beginning with the work of Glasman, major simplification on the foundations of the subject has been made (cf. the work of Ayala-Mazel-Gee-Rozenblyum, Nikolaus-Scholze and the Barwick school). We offer another perspective (that the speaker has a chance of understanding) on equivariant stable homotopy theory, at least for the group C_2, via algebraic geometry. We view it as a way to remedy an infamous annoyance: the 2-étale cohomological dimension of the field of real numbers is infinite. We do this by identifying the genuine C_2-spectra with a category of motives based on Real algebraic geometry ala Scheiderer. This is joint work with Jay Shah. |

Mon Nov 04 |
## Topology Seminar2:30pm - Ford Hall 110Topology Seminar TBA |

Mon Nov 11 |
## Topology Seminar2:30pm - Ford Hall 110Cochain models for the unit group of a differential graded algebra Tyler Lawson, University of Minnesota Abstract not available. |

Mon Nov 18 |
## Topology Seminar2:30pm - Ford Hall 110Topology Seminar TBA |

Mon Nov 25 |
## Topology Seminar2:30pm - Ford Hall 110Topology Seminar TBA |

Mon Dec 02 |
## Topology Seminar2:30pm - Ford Hall 110Topology and Arithmetic Statistics Weiyan Chen, University of Minnesota Topology studies the shape of spaces. Arithmetic statistics studies the behavior of random algebraic objects such as integers and polynomials. I will talk about a circle of ideas connecting these two seemingly unrelated areas. To illuminate the connection, I will focus on three concrete examples: (1) the Burau representation of the braid groups, (2) analytic number theory for effective 0-cycles on a variety, and (3) cohomology of the space of multivariate irreducible polynomials. These projects are parts of a broader research program, with numerous contributions by topologists, algebraic geometers, and number theorists in the past decade, and lead to many future directions yet to be explored. |

Mon Dec 09 |
## Topology Seminar2:30pm - Ford Hall 110Topology Seminar TBA |

Mon Jan 27 |
## Topology Seminar2:30pm - VinH 364Smooth 4-manifolds and the geometry of 3-manifolds Matthew Stoffregen, MIT One of the interests of low-dimensional topologists is |

Mon Mar 02 |
## Topology Seminar2:30pm - VinH 364Topology Seminar - TBA George Shabat, Russian State University for the Humanities TBA |

Mon Mar 16 |
## Topology Seminar2:30pm - VinH 364Topology Seminar - TBA Marcy Robertson, University of Melbourne TBA |