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Series: Geometry Topology Seminar

I will describe the results of a joint project with Mike Mandell on the algebraic K-theory of the sphere spectrum, focusing on recent work that describes the fiber of the cyclotomic trace using a spectral lift of Tate-Poitou duality.

Series: Geometry Topology Seminar

In this talk, first, I'll briefly go over my proof of the conjecture that there are only afinite number of hyperbolic 3-manifolds of bounded volume and trace field degree. Then I'lldiscuss some conjectural pictures to quantitative results and applications to other similarproblems.

Series: Geometry Topology Seminar

The Ptolemy variety is an invariant of a triangulated 3-manifoldM. It detects SL(2,C)-representations of pi_1(M) in the sense that everypoint in the Ptolemy variety canonically determines a representation (up toconjugation). It is closely related to Thurston's gluing equation varietyfor PSL(2,C)-representations. Unfortunately, both the gluing equationvariety and the Ptolemy variety depend on the triangulation and may missseveral components of representations. We discuss the basic properties ofthese varieties, how to compute invariants such as trace fields and complexvolume, and how to obtain a variety, which is independent of thetriangulation.

Series: Geometry Topology Seminar

The hyperelliptic Torelli group of a genus g reference surface S_g is the subgroup of the mapping class group whose elements both commute with a fixed hyperelliptic involution of S_g and act trivially on the integral homology of S_g . This group is an important object in geometric topology and group theory, and also in algebraic geometry, where it appears as the fundamental group of the moduli space of genus g hyperelliptic curves with a homology framing. In this talk, we summarize what is known about the (infinite) topology of these moduli spaces, describe a few open problems, and report on some recent partial progress.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

The Grothendieck ring of varieties is defined to be the free abelian group generated by k-varieties, modulo the relation that for any closed subvariety Y of a variety X, we impose the relation that [X] = [Y] + [X \ Y]; the ring structure is defined by [X][Y] = [X x Y]. Last December two longstanding questions about the Grothendieck ring of varieties were answered: 1. If two varieties X and Y are piecewise isomorphic then they are equal in the Grothendieck ring; does the converse hold? 2. Is the class of the affine line a zero divisor? Both questions were answered by Borisov, who constructed an element in the kernel of multiplication by the affine line; coincidentally, the proof also constructed two varieties whose classes in the Grothendieck ring are the same but which are not piecewise isomorphic. In this talk we will investigate these questions further by constructing a topological analog of the Grothendieck ring and analyzing its higher homotopy groups. Using this extra structure we will sketch a proof that Borisov's coincidence is not a coincidence at all: that any element in the annihilator of the Lefschetz motive can be represented by a difference of varieties which are equal in the Grothendieck ring but not piecewise isomorphic.

Series: Geometry Topology Seminar

The Birman-Hilden theorem relates the mapping class groups of two orientable surfaces S and X, given a regular branched covering map p from S to X. Explicitly, it provides an isomorphism between the group of mapping classes of S that have p-equivariant representatives (mod the deck group of the covering map), and the group of mapping classes of X that have representatives that lift to homeomorphisms of S. We will translate these notions into the realm of automorphisms of free group, and prove that an obvious analogue of the Birman-Hilden theorem holds there. To indicate the proof of this, we shall explore in detail several key examples, and we shall describe some group-theoretic applications of the theorem. This is joint work with Rebecca Winarski, John Calabrese, and Tyrone Ghaswala

Series: Geometry Topology Seminar

Understanding the stable homotopy groups of spheres is one of the great challenges of algebraic topology. They form a ring which, despite its simple definition, carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the chromatic layers. This point of view organizes the homotopy groups into periodic families and reveals patterns. There are many structural conjectures about the chromatic filtration. I will talk about one of these conjectures, the \emph{chromatic splitting conjecture}, which concerns the gluing data between the different layers of the chromatic filtration.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

The theory of étale cohomology provides a bridge between two seemingly unrelated subjects: the homology of braid groups (topology) and the number of points on algebraic varieties over finite fields (arithmetic). Using this bridge, we study two problems, one from topology and one from arithmetic. First, we compute the homology of the braid groups with coefficients in the Burau representation. Then, we apply the topological result to calculate the expected number of points on a random superelliptic curve over finite fields.