In the talk we will define higher K-groups, and explain some of their relations to number theory

# Past Junior Geometry and Topology Seminar

A big problem in Riemannian geometry is the search for a "best possible" Riemannian metric on a given compact smooth manifold. When the manifold is complex, one very nice metric we could look for is a Kahler-Einstein metric. For compact Kahler manifolds with non-positive first chern class, these were proven to always exist by Aubin and Yau in the 70's. However, the case of positive first chern class is much more delicate, and there are non-trivial obstructions to existence. It wasn't until this decade that a complete abstract characterisation of Kahler-Einstein metrics became available, in the form of K-stability. This is a purely algebro-geometric stability condition, whose equivalence to the existence of a Kahler-Einstein metric in the Fano case is analogous to the Hitchin-Kobayashi correspondence for vector bundles. In this talk, I will cover the definition of K-stability, its relation to Kahler-Einstein metrics, and (time permitting) give some examples of how K-stability is verified or disproved in practice.

G. Dimitrov and L. Katzarkov introduced in their paper from 2016 the counting of non-commutative curves and their (semi-)stability using T. Bridgeland's stability conditions on triangulated categories. To some degree one could think of this as the non-commutative analog of Gromov-Witten theory. However, its full meaning has not yet been fully discovered. For example there seems to be a relation to proving Markov's conjecture.

For the talk, I will go over the definitions of stability conditions, non-commutative curves and their counting. After developing some tools relying on working with exceptional collections, I will consider the derived category of representations on the acyclic triangular quiver and will talk about the explicit computation of the invariants for this example.

Recently, Joyce constructed a Ringel-Hall style graded vertex algebra on the homology of moduli stacks of objects in certain categories of algebro-geometric and representation-theoretic origin. The construction is most natural for 2n-Calabi-Yau categories. We present this construction and explain the geometric reason why it exists. If time permits, we will explain how to compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety and to identify it with a lattice-type vertex algebra.

The homotopy bordism category hCob_d has as objects closed (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. This is a simplified version of the topologically enriched bordism category Cob_d whose classifying space B(Cob_d) been completely determined by Galatius-Madsen-Tillmann-Weiss in 2006. In comparison, little is known about the classifying space B(hCob_d).

In the first part of the talk I will give an introduction to bordism categories and their classifying spaces. In the second part I will identify B(hCob_1) showing, in particular, that the rational cohomology ring of hCob_1 is polynomial on classes \kappa_i in degrees 2i+2 for all i>=1. The seemingly simpler category hCob_1 hence has a more complicated classifying space than Cob_1.

Springer theory is an important branch of geometric representation theory. It is a beautiful interplay between combinatorics, geometry and representation theory.

It started with Springer correspondence, which yields geometric construction of irreducible representations of symmetric groups, and Ginzburg's construction of universal enveloping algebra U(sl_n).

Here I will present a view of two-row Springer theory of type A (thus looking at nilpotent elements with two Jordan blocks) from a scope of a symplectic topologist (hence the title), that yields connections between symplectic-topological invariants and link invariants (Floer homology and Khovanov homology) and connections to representation theory (Fukaya category and parabolic category O), thus summarising results by Abouzaid,

Seidel, Smith and Mak on the subject.

Arakelov geometry studies schemes X over ℤ, together with the Hermitian complex geometry of X(ℂ).

Most notably, it has been used to give a proof of Mordell's conjecture (Faltings's Theorem) by Paul Vojta; curves of genus greater than 1 have at most finitely many rational points.

In this talk, we'll introduce some of the ideas behind Arakelov theory, and show how many results in Araklev theory are analogous—with additional structure—to classic results such as intersection theory and Riemann Roch.

A cohomology class on the diffeomorphism group Diff(M) of a manifold M

can be thought of as a characteristic class for smooth M-bundles.

I will survey a technique for producing examples of such classes,

and then explain how the signature (of 4-manifolds) provides an

obstruction to this technique in dimension 3.

I will define Miller-Morita-Mumford classes and explain how we can

think of them as coming from classes on the cobordism category.

Madsen and Weiss showed that for a surface S of genus g all cohomology

classes

of the mapping class group MCG(S) (of degree < 2(g-2)/3) are MMM-classes.

This technique has been successfully ported to higher even dimensions d= 2n,

but it cannot possibly work in odd dimensions:

a theorem of Ebert says that for d=3 all MMM-classes are trivial.

In the second part of my talk I will sketch a new proof of (a part of)

Ebert's theorem.

I first recall the definition of the signature sign(W) of a 4 manifold W,

and some of its properties, such as additivity with respect to gluing.

Using the signature and an idea from the world of 1-2-3-TQFTs,

I then go on to define a 'central extension' of the three dimensional

cobordism category.

This central extension corresponds to a 2-cocycle on the 3d cobordism

category,

and we will see that the construction implies that the associated MMM-class

has to vanish on all 3-dimensional manifold bundles.

Topological quantum field theories (TQFTs) are an extensively studied scheme for constructing invariants of manifolds, inspired by physics. In this talk, we will discuss a particular flavour of TQFT, where we equip our manifolds with principal bundles for some finite group. After introducing TQFTs and this particular flavour, I will discuss games one can play with these TQFTs, and a possible strategy for classifying equivariant TQFTs in three dimensions.