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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.

Series: Geometry Topology Seminar

In this work, a novel approach is used to study geometric properties of the indicatrix bundle and the natural foliations on the tangent bundle of a Finsler manifold. By using this approach, one can find the necessary and sufficient conditions on the Finsler manifold (M; F) in order that its indicatrix bundle has the Sasakian structure.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

This is joint work with Saul Schleimer. Veering structures onideal triangulations of cusped manifolds were introduced by Ian Agol, whoshowed that every pseudo-Anosov mapping torus over a surface, drilled alongall singular points of the measured foliations, has an ideal triangulationwith a veering structure. Any such structure coming from Agol'sconstruction is necessarily layered, although a few non-layered structureshave been found by randomised search. We introduce veering Dehn surgery,which can be applied to certain veering triangulations, to produceveering triangulationsof a surgered manifold. As an application we find an infinite family oftransverse veering triangulations none of which are layered. Untilrecently, it was hoped that veering triangulations might be geometric,however the first counterexamples were found recently by Issa, Hodgson andme. We also apply our surgery construction to find a different infinitefamily of transverse veering triangulations, none of which are geometric.

Series: Geometry Topology Seminar

It is a well understood story that one can extract linkinvariants associated to simple Lie algebras. These invariants arecalled Reshetikhin-Turaev invariants and the famous Jones polynomialis the simplest example. Kauffman showed that the Jones polynomialcould be described very simply by replacing crossings in a knotdiagram by various smoothings. In this talk we will explainCautis-Kamnitzer-Licata's simple new approach to understanding theseinvariants using basic representation theory and the quantum Weylgroup action. Their approach is based on a version of Howe duality forexterior algebras called skew-Howe duality. Even the graphical (orskein theory) description of these invariants can be recovered in anelementary way from this data. The advantage of this approach isthat it suggests a `categorification' where knot homology theoriesarise in an elementary way from higher representation theory and thestructure of categorified quantum groups. Joint work with David Rose and Hoel Queffelec