Topology

In this talk we discuss a new perspective on Quillen's devissage theorem. Originally, Quillen proved devissage for algebraic $K$-theory of abelian categories. The theorem showed that given a full abelian subcategory $\mathcal{A}$ of an abelian category $\mathcal{B}$, $K(\mathcal{A})\simeq K(\mathcal{B})$ if every object of $\mathcal{B}$ has a finite filtration with quotients lying in $\mathcal{A}$. This allows us, for example, to relate the $K$-theory of torsion $\mathbf{Z}$-modules to the $K$-theories of $\mathbf{F}_p$-modules for all $p$. Generalizations of this theorem to more general contexts for $K$-theory, such as Walhdausen categories, have been notoriously difficult; although some such theorems exist they are generally much more complicated to state and prove than Quillen's original. In this talk we show how to translate Quillen's algebraic approach to a geometric context. This translation allows us to construct a devissage theorem in geometry, and prove it using Quillen's original insights.

Block sum of matrices define a group homomorphism $GL_n(R) \times GL_m(R) \to GL_{n+m}(R)$, which can be used to make the direct sum of $H_s(BGL_t(R);k)$ over all $s, t$ into a bigraded-commutative ring. A similar product may be defined on homology of mapping class groups of surfaces with one boundary component, as well as in many other examples of interest. These products have manifestations on various levels, for example there is a product on the level of spaces making the disjoint union of $BGL_n(R)$ into a homotopy commutative topological monoid. I will discuss how it, and other concrete examples, may be built by iterated cell attachments in the category of topological monoids, or better yet $E_2$ algebras, and what may be learned by this viewpoint. This is all joint work with Alexander Kupers and Oscar Randal-Williams.

This is the first lecture in a two part series: part 2

In this first talk I will explain how the machinery of the Weil Conjectures can be used to transfer information back and forth between the topology of a complex algebraic variety and its F_q points. A sample question: How many $F_q$-points does a random smooth cubic surface have? This was recently answered by Ronno Das using his (purely topological) computation of the cohomology of the universal smooth, complex cubic surface. This is part of a much larger circle of fascinating problems, most completely open.

This is the first lecture in a two part series: part 2

Following work of Röndigs-Spitzweck-Østvær and others on the stable A1-homotopy groups of the sphere spectrum, it has become possible to carry out calculations of the n-th A1-homotopy group of BGLn for small values of n. This group is notable, because it lies just outside the range where the homotopy groups of BGLn recover algebraic K-theory of fields. This group captures some information about rank-n vector bundles on schemes that is lost upon passage to algebraic K-theory. Furthermore, this group relates to a conjecture of Suslin from 1984 about the image of a map from algebraic K-theory to Milnor K-theory in degree n. This conjecture says that the image of the map consists of multiples of (n-1)!. The conjecture was previously known for the cases n=1, n=2 (Matsumoto's theorem) and n=3, where it follows from Milnor's conjecture on quadratic forms. I will establish the conjecture in the case n=4 (up to a problem with 2-torsion) and n=5 (in full). This is joint work with Aravind Asok and Jean Fasel.

Arithmetic statistics asks about the distribution of objects arising “randomly” in number-theoretic context. How many A_5-extensions of Q are there whose discriminant is a random squarefree integer? How big is the Selmer group fo a random quadratic twist of an elliptic curve over Q? How likely is the the maximal extension of Q unramified away from a random set of 3 primes to be an extension of infinite degree? Traditionally we ask these questions over Q or other number fields, but they make sense, and are expected to have broadly similar answers, when we replace Q with the function field of a curve over a finite field F_q. Having done this, one suddenly finds oneself studying the geometric properties of relevant moduli spaces over F_q, and under ideal circumstances, the topology of the corresponding moduli spaces over the complex numbers. Under even more ideal circumstances, results in topology can be used to derive theorems in arithmetic statistics over function fields. I’ll give an overview of this story, including some recent results in which the arrow goes the other way and analytic number theory can be used to prove results about the geometry of moduli spaces.

This is the first lecture in a two part series: part 2.