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.

# Past Junior Geometry and Topology Seminar

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.

The world of quantum invariants began in 1983 with the discovery of the Jones polynomial. Later on, Reshetikhin and Turaev developed an algebraic machinery that provides knot invariants. This algebraic construction leads to a sequence of quantum generalisations of this invariant, called coloured Jones polynomials. The original Jones polynomial can be defined by so called skein relations. However, unlike other classical invariants for knots like the Alexander polynomial, its relation to the topology of the complement is still a mysterious and deep question. On the topological side, R. Lawrence defined a sequence of braid group representations on the homology of coverings of configuration spaces. Then, based on her work, Bigelow gave a topological model for the Jones polynomial, as a graded intersection pairing between certain homology classes. We aim to create a bridge between these theories, which interplays between representation theory and low dimensional topology. We describe the Bigelow-Lawrence model, emphasising the construction of the homology classes. Then, we show that the sequence of coloured Jones polynomials can be seen through the same formalism, as topological intersection pairings of homology classes in coverings of the configuration space in the punctured disc.

My goal for the talk is to give a "from the ground-up" introduction to symplectic topology. We will cover the Darboux lemma, pseudo-holomorphic curves, Gromov-Witten invariants, quantum cohomology and Floer cohomology.

Gauge-theoretic invariants such as Donaldson or Seiberg–Witten invariants of 4-manifolds, Casson invariants of 3-manifolds, Donaldson–Thomas invariants of Calabi–Yau 3- and 4-folds, and putative Donaldson–Segal invariants of G_2 manifolds are defined by constructing a moduli space of solutions to an elliptic PDE as a (derived) manifold and integrating the (virtual) fundamental class against cohomology classes. For a moduli space to have a (virtual) fundamental class it must be compact, oriented, and (quasi-)smooth. We first describe a general framework for addressing orientability of gauge-theoretic moduli spaces due to Joyce–Tanaka–Upmeier. We then show that the moduli stack of perfect complexes of coherent sheaves on a Calabi–Yau 4-fold X is a homotopy-theoretic group completion of the topological realisation of the moduli stack of algebraic vector bundles on X. This allows one to extend orientations on the locus of algebraic vector bundles to the boundary of the (compact) moduli space of coherent sheaves using the universal property of homotopy-theoretic group completions. This is a necessary step in constructing Donaldson–Thomas invariants of Calabi–Yau 4-folds. This is joint work with Yalong Cao and Dominic Joyce.

Multiplicative preprojective algebras (MPAs) were originally defined by Crawley-Boevey and Shaw to encode solutions of the Deligne-Simpson problem as irreducible representations.

MPAs have recently appeared in the literature from different perspectives including Fukaya categories of plumbed cotangent bundles (Etgü and Lekili) and, similarly, microlocal sheaves

on rational curves (Bezrukavnikov and Kapronov.) After some motivation, I'll suggest a purely algebraic approach to study these algebras. Namely, I'll outline a proof that MPAs are

2-Calabi-Yau if Q contains a cycle and an inductive argument to reduce to the case of the cycle itself.

In this talk, I will sketch a geometrically flavoured proof of the

Madsen-Weiss theorem based on work by Eliashberg-Galatius-Mishachev.

In order to prove the triviality of appropriate relative bordism groups,

in a first step a variant of the wrinkling theorem shows

that one can reduce to consider fold maps (with additional structure).

In a subsequent step, a geometric version of the Harer stability

theorem is used to get rid of the folds via surgery. I will focus on

this second step.

Standard representation theory transforms groups=algebra into vector spaces = (linear) algebra. The modern approach, geometric representation theory constructs geometric objects from algebra and captures various algebraic representations through geometric gadgets/invariants on these objects. This field started with celebrated Borel-Weil-Bott and Beilinson-Bernstein theorems but equally is in rapid expansion nowadays. I will start from the very beginnings of this field and try to get to the recent developments (time permitting).