# Upcoming Talks

The seminar will meet at 4:30 on Monday in 2-131 unless otherwise noted.

• Sep 282017

### Lars Hesselholt (Nagoya University and University of Copenhagen)

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#### Higher algebra and arithmetic

The seminar will meet at 4:30 PM in 2-190.

This talk is part of the Brandeis–Harvard–MIT–Northeastern joint Mathematics Colloquium

This talk concerns a twenty-thousand-year old mistake: The natural numbers record only the result of counting and not the process of counting. As algebra is rooted in the natural numbers, the higher algebra of Joyal and Lurie is rooted in a more basic notion of number which also records the process of counting. Long advocated by Waldhausen, the arithmetic of these more basic numbers should eliminate denominators. Notable manifestations of this vision include the Bökstedt-Hsiang-Madsen topological cyclic homology, which receives a denominator-free Chern character, and the related Bhatt-Morrow-Scholze integral $p$-adic Hodge theory, which makes it possible to exploit torsion cohomology classes in arithmetic geometry. Moreover, for schemes smooth and proper over a finite field, the analogue of de Rham cohomology in this setting naturally gives rise to the cohomological interpretation of the Hasse-Weil zeta function by regularized determinants envisioned by Deninger.

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• Oct 022017

### Craig Westerland (University of Minnesota)

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#### Fox-Neuwirth/Fuks cells, quantum shuffle algebras, and Malle's conjecture for function fields

In 2002, Malle formulated a conjecture regarding the distribution of number fields with specified Galois group. The conjecture is an enormous strengthening of the inverse Galois problem; it is known to hold for abelian Galois groups, but for very few non-abelian groups.

We may reformulate Malle's conjecture in the function field setting, where it becomes a question about the number of branched covers of the affine line (over a finite field) with specified Galois group. In joint work with Jordan Ellenberg and TriThang Tran, we have shown that the upper bound in Malle's conjecture does hold in this setting.

The techniques used involve a computation of the cohomology of the (complex points of the) Hurwitz moduli spaces of these branched covers. Surprisingly (at least to me), these cohomology computations can be rephrased in terms of the homological algebra of certain braided Hopf algebras arising in combinatorial representation theory and the classification of Hopf algebras. This relationship can be leveraged to provide the upper bound in Malle's conjecture.

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• Oct 162017

### David Ayala (Montana State University)

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#### The Bruhat stratified orthogonal group acts on higher categories by adjoints

The majority of this talk will examine the Bruhat stratified orthogonal group:

- The Bruhat cells of the general linear group assemble as a combinatorial stratification of the orthogonal group.

- Compatibility of this stratification with matrix multiplication can be articulated as an associative algebra structure on its exit-path category in a certain *Morita* category of categories.

- Articulated as so, there is an action of this Bruhat stratified orthogonal group $O(n)$ on the category of $n$-categories; this action is given by adjoining adjoints.

- This results in a continuous action of the topological group $O(n)$ on the category of n-categories with adjoints.

The last point is a ket input into a proof of the Cobordism Hypothesis using factorization homology -- this context will be discussed.

This is joint work with John Francis.

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• Oct 232017

• Nov 062017

• Nov 132017

• Dec 042017

• Dec 112017

### Julie Bergner (University of Virginia)

The mailing list for this seminar is the MIT topology google group.
Email Mike Hopkins if you want to join the list.