## Seminar Categories

## Current Series

Mon Oct 01 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: Grothendieck-Teichmuller group and braided Hopf algebras Craig Westerland, University of Minnesota The primitives in a classical Hopf algebra form a Lie algebra (in fact, a Lie subalgebra of the Hopf algebra). For a braided Hopf algebra, this is no longer the case. Consequently, all of the structure theorems for Hopf algebras (e.g., the Milnor-Moore and Poincaré-Birkhoff-Witt theorems) break down in this setting. This is a report on ongoing work in which we construct an operad (a braided form of the Lie operad) which governs the algebraic structure of the primitives in a braided Hopf algebra. We can interpret this operad in terms of the homology of covering spaces of the 2-dimensional little disks operad. This gives rise to an action of Drinfeld's Grothendieck-Teichmuller group on this operad which may be related to Drinfeld's original definition of this group. |

Mon Oct 08 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: A chromatic approach to tmf cooperations Paul van Koughnett, Purdue The topological modular forms spectrum, tmf, is a cohomology theory constructed from elliptic curves that captures information about stable homotopy at chromatic heights less than or equal to 2. We present a description of the height 1 part of the algebra of tmf cooperations, and interpret it in terms of modular forms. This is joint work with Dominic Culver. |

Mon Oct 15 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: Rigidity of warped cones and coarse geometry of expanders Wouter van Limbeek, UIC Finitely generated subgroups of compact Lie groups give rise to expander graphs via a warped cone construction. We study the dependence of the coarse geometry of such expander graphs on the original subgroup and establish a dynamical analogue of quasi-isometric rigidity theorems in geometric group theory: Namely, the coarse geometry of the warped cone determines the subgroup up to commensurability, unless the group has abelian factors. This is joint work with David Fisher and Thang Nguyen. |

Mon Oct 22 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: Finitely generated sequences of linear subspace arrangements Nir Gadish, University of Chicago Hyperplane arrangements are a classical meeting point of topology, combinatorics and representation theory. Generalizing to arrangements of linear subspaces of arbitrary codimension, the theory becomes much more complicated. However, a crucial observation is that many natural sequences of arrangements seem to be defined using a finite amount of data. In this talk I will describe a notion of 'finitely generation' for collections of arrangements, unifying the treatment of known examples. Such collections turn out to exhibit strong forms of stability, both in their combinatorics and in their cohomology representation. This structure makes the appearance of representation stability transparent and opens the door to generalizations. |

Mon Oct 29 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: Symmetric monoidal categories and \Gamma-categories Amit Sharma, Kent State University A \Gamma-category is a functor from the category of finite based sets and basepoint preserving functions \Gamma^op to Cat. We construct a model category structure on the category of \Gamma-categories, which is symmetric monoidal closed to the Day convolution product. The fibrant objects in this model category structure are those References: |

Mon Nov 05 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: An isovariant Elmendorf's theorem Sarah Yeakel, University of Maryland An isovariant map is an equivariant map which preserves isotropy groups. Isovariant maps show up in equivariant surgery theory and in other settings when homotopy theory is applied to geometry. For a finite group G, we consider the category of G-spaces with morphisms given by isovariant maps. We will discuss a cofibrantly generated model structure on this category, along with isovariant versions of Elmendorf's theorem and a theorem of Klein and Williams about homotoping a map off a submanifold. |

Mon Nov 12 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: Localizing the E_2 page of the Adams spectral sequence Eva Belmont, Northwestern The Adams spectral sequence is one of the central tools for calculating the stable homotopy groups of spheres, one of the motivating problems in stable homotopy theory. In this talk, I will discuss an approach for computing the Adams E_2 page for the sphere at p = 3 in an infinite region, by computing its localization by the non-nilpotent element b_{10}. This approach relies on computing an analogue of the Adams spectral sequence in Palmieri's stable category of comodules, which can be regarded as an algebraic analogue of stable homotopy theory. This computation fits in the framework of chromatic homotopy theory in the stable category of comodules. |

Mon Nov 19 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: The spectrum of units of a height 2 theory Jeremy Hahn, Harvard University The space BSU admits two infinite loop space structures, one arising from addition of vector bundles and the other from tensor product. A surprising fact, due to Adams and Priddy, is that these two infinite loop spaces become equivalent after p-completion. I will explain how the Adams-Priddy theorem allows for an identification of sl_1(ku_p), the spectrum of units of p-complete complex K-theory. I will then describe work, joint with Andrew Senger, that attempts to similarly understand the spectrum of units of the 2-completion of tmf_1(3). |

Mon Nov 26 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: Cohomology of arithmetic groups and characteristic classes of manifold bundles Bena Tshishiku, Harvard University A basic problem in the study of fiber bundles is to compute the ring H*(BDiff(M)) of characteristic classes of bundles with fiber a smooth manifold M. When M is a surface, this problem has ties to algebraic topology, geometric group theory, and algebraic geometry. We have a good understanding of the cohomology in the "stable range", but this accounts for a small percentage of the total cohomology, and little is known beyond that. In this talk I describe some new characteristic classes (in the case dim M >>0) that come from the unstable cohomology of arithmetic groups. |

Fri Dec 07 |
## Topology Seminar3:30pm - 301 Vincent HallTopology Seminar: Power operations in normed motivic spectra Tom Bachmann, MIT In joint work with M. Hoyois, we established (the beginnings of) a theory of "normed motivic spectra". These are motivic spectra with some extra structure, enhancing the standard notion of a motivic E_oo-ring spectrum (this is similar to the notion of G-commutative ring spectra in equivariant stable homotopy theory). It was clear from the beginning that the homotopy groups of such normed spectra afford interesting *power operations*. In ongoing joint work with E. Elmanto and J. Heller, we attempt to establish a theory of these operations and exploit them calculationally. I will report on this, and more specifically on our proof of a weak motivic analog of the following classical result of Würgler: any (homotopy) ring spectrum with 2=0 is generalized Eilenberg-MacLane. |

Mon Dec 10 |
## Topology Seminar3:30pm - Vincent Hall 301Topology Seminar: The geometric average size of Selmer groups over function fields Aaron Landesman, Stanford We use topological methods to investigate the average size of n-Selmer groups of elliptic curves over F_q(t). |