Past Junior Geometry and Topology Seminar

The spectrum of the integers is an affine scheme which number theorists would like to complete to a projective scheme, adding a point at infinity. We will list some reasons for wanting to do this, then gather some hints about what properties the completed object might have. In particular it seems that the desired object can only exist in some setting extending traditional algebraic geometry. We will then present the proposals of Durov and Shai Haran for such extended settings and the compactifications they construct. We will explain the close relationship between both and, if time remains, relate them to a third compactification in a third setting, proposed by Toen and Vaquie.

The talk is about the homotopy type of configuration spaces. Once upon a time there was a conjecture that it is a homotopy invariant of closed manifolds. I will discuss the strong evidence supporting this claim, together with its recent disproof by a counterexample. Then I will talk about the corrected version of the original conjecture.

A -polarised K3 surface admits an embedding into weighted projective space defined by its polarisation. Let X be a family of such surfaces, then one can construct a projective model W of X such that the map from X to W realises this embedding on the general fibre. This talk considers what happens to W when we allow the fibres of the family X to degenerate.

I will explain what a perfect obstruction theory is, and how it gives rise to a "virtual" fundamental class of the right expected dimension, even when the dimension of the moduli space is wrong. These virtual fundamental classes are one of the main preoccupations of "modern" moduli theory, being the central object of study in Gromov-Witten and Donaldson-Thomas theory. The purpose of the talk is to remove the black-box status of these objects. If there is time I will do some cheer-leading for dg-schemes, and try to convince the audience that virtual fundamental classes are most happily defined to live in the dg-world.

The AJ conjecture relates two different knot invariants, namely the coloured Jones polynomial and the A-polynomial. The approach we will use will be that of 2+1 dimensional Topological Quantum Field Theory. Indeed, the coloured Jones polynomial is constructed in Reshetikhin and Turaev's formulation of a TQFT using quantum groups. The A-polynomial is defined by a subvariety of the moduli space of flat SL(2,C) connections of a torus.  Geometric quantization on this moduli space also gives a TQFT, and the correspondence between these provides a framework where the knot invariants can be compared. In the talk I will sketch the above constructions and show how we can do explicit calculations for simple knots. This is work in progress joint with J. E. Andersen.

In this talk I will outline the two constructions of the Brauer group Br($X$) of a scheme $X$, namely via etale cohomology and Azumaya algebras and briefly describe how one may compute this group using the Hochschild-Serre spectral sequence. In the early '70s Manin observed that one can use the Brauer group of a projective variety $X/k$ to define an obstruction to the existence of rational points on $X$. I will discuss this arithmetic application and time permitting, outline an example for $X$ a K3 surface.

A parabolic bundle on a marked curve is a vector bundle with extra structure (a flag) in each of the fibres over the marked points, together with data corresponding to a choice of stability condition Parabolic bundles are natural generalisations of vector bundles when the base comes with a marking (for example, they partially generalise the Narasimhan-Seshadri correspondence between representations of the fundamental group and semistable vector bundles), but they also play an important role in the study of pure sheaves on nodal curves (which are needed to compactify moduli of vector bundles on stable curves). Consider the following moduli problem: pairs $(C,E)$ of smooth marked curves $C$

and semistable parabolic bundles $E\rightarrow C$. I will sketch a construction of projective moduli spaces which compactify the above moduli problem over the space of stable curves. I'll discuss further questions of interest, including strategies for understanding the cohomology of these moduli spaces, generalisations of the construction to higher-dimensional base schemes, and possible connections with Torelli theorems for parabolic vector bundles on marked curves.

I will briefly describe a differential geometric construction of Hitchin's projectively flat connection in the Verlinde bundle, over Teichm\"uller space, formed by the Hilbert spaces arising from geometric quantization of the moduli space of flat connections on a Riemann surface. We will work on a general symplectic manifold sharing certain properties with the moduli space. Toeplitz operators enter the picture when quantizing classical observables, but they are also closely connected with the notion of deformation quantization. Furthermore, through an intimate relationship between Toeplitz operators, the Hitchin connection manifests itself in the world of deformation quantization as a connection on formal functions. As we shall see, this formal Hitchin connection can be used to construct a deformation quantization, which is independent of the Kähler polarization used for quantization. In the presence of a symmetry group, this deformation quantization can (under certain cohomological conditions) be constructed invariantly. The talk presents joint work with J. E. Andersen.

I will present a new proof of Mumford's conjecture on the rational cohomology of moduli spaces of curves, which is substantially different from those given by Madsen--Weiss and Galatius--Madsen--Tillmann--Weiss: in particular, it makes no use of Harer--Ivanov stability for the homology of mapping class groups, which played a decisive role in the previously known proofs. This talk represents joint work with Soren Galatius.

This talk concerns the relationships between Donaldson-Thomas, Pandharipande-Thomas, and Szendroi invariants established via analysis of the geometry of wall crossing phenomena of suitably general moduli spaces. I aim to give a reasonably detailed account of the simplest example, the conifold, where in fact all of the major ideas can be easily seen.

I will go over my paper (arXiv:0902.2135v1) which explains how semi-flat Calabi-Yau / G$_2$ manifolds can be constructed from minimal 3-submanifolds in a signature (3,3) vector space.

I will briefly discuss the construction of the moduli spaces of (semi)stable bundles on a given curve. The main aim of the talk will be to describe various features of the geometry and topology of these moduli spaces, with emphasis on methods as much as on results. Topics may include irreducibility, cohomology, Verlinde numbers, Torelli theorems.

We prove uniqueness of the Kerr black holes within the connected, non-degenerate, analytic class of regular vacuum black holes. (This is joint work with Piotr Chrusciel. arXiv:0806.0016)

We introduce the notion of $G$-Higgs bundle from studying the representations of the fundamental group of a closed connected oriented surface $X$ in a Lie group $G$. If $G$ turns to be the isometry group of a Hermitian symmetric space, much more can be said about the moduli space of $G$-Higgs bundles, but this also implies dealing with exceptional cases. We will try to face all these subjects intuitively and historically, when possible!

In this talk I will discuss some elementary notions of deformation theory in algebraic geometry like Schlessinger's Criterion. I will describe obstructions and deformations of sheaves in detail and will point out relations to moduli spaces of sheaves.

Fibrations are a valuable tool in the study of the geometry of higher dimensional algebraic varieties. By expressing a higher dimensional variety as a fibration by lower dimensional varieties, we can deduce much about its properties. Whilst the theory of elliptic fibrations is very well developed, fibrations by higher dimensional varieties, especially K3 surfaces, are only just beginning to be studied. In this talk I study a special case of the K3-fibration, where the general fibres admit a <2>-polarisation and the base of the fibration is a nonsingular curve.

I will give a survey-type introduction to manifolds equipped with $G_2$ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk is the second of two parts.

I will give a survey-type introduction to manifolds equipped with $G_2$ structures, emphasizing the similarities and differences with Riemannian manifolds equipped with almost complex structures, and with oriented Riemannian 3-manifolds. Along the way I may discuss the Berger classification of Riemannian holonomy, the Calabi-Yau theorem, exceptional geometric structures arising from the algebra of the Octonions, and calibrated submanifolds. This talk will be in two parts.

Geometrically, the problem of descent asks when giving some structure on a space is the same as giving some structure on a cover of the space, plus perhaps some extra data.
In algebraic geometry, faithfully flat descent says that if $X\rightarrow Y$ is a faithfully flat morphism of schemes, then giving a sheaf on $Y$ is the same as giving a collection of sheaves on a certain simplicial resolution constructed from $X$, satisfying certain compatibility conditions. Translated to algebra, it says that if $S\rightarrow R$ is a faithfully flat morphism of rings, then giving an $S$-module is the same as giving a certain simplical module over a simplicial ring constructed from $R$. In topology, given an etale cover $X\rightarrow Y$ one can recover $Y$ (at least up to homotopy equivalence) from a simplical space constructed from $X$.

The birational classification of varieties is an interesting and ongoing problem in algebraic geometry. This talk aims to give an

overview of the progress made on this problem in the special case where the varieties considered are surfaces in projective space.

Hilbert schemes classify subschemes of a given projective variety / scheme. They are special cases of Quot schemes which are moduli spaces for quotients of a fixed coherent sheaf. Hilb and Quot are among the first examples of moduli spaces in algebraic geometry, and they are crucial for solving many other moduli problems. I will try to give you a flavour of the subject by sketching the construction of Hilb and Quot and by discussing the role they play in applications, in particular moduli spaces of stable curves and moduli spaces of stable sheaves.

The usual procedure to obtain uniqueness theorems for black hole space-times ("No Hair" Theorems) requires the construction of global coordinates for the domain of outer communications (intuitively: the region outside the black hole). Besides an heuristic argument by Carter and a few other failed attempts the existence of such a (global) coordinate system as been neglected, becoming a quite hairy hypothesis.

After a review of the basic aspects of causal theory and a brief discussion of the definition of black-hole we will show how to construct such coordinates focusing on the non-negativity of the "area function".

Extending work of Klyachko and Perling, we develop a combinatorial description of pure equivariant sheaves on an arbitrary nonsingular toric

variety X. This combinatorial description can be used to construct moduli spaces of stable equivariant sheaves on X using Geometric Invariant Theory (analogous to techniques used in case of equivariant vector bundles on X by Payne and Perling). We study how the moduli spaces of stable equivariant sheaves on X can be used to explicitly compute the fixed point locus of the moduli space of all stable sheaves on X, i.e. the subscheme of invariant stable sheaves on X.

For continuous maps $f: S^{2n-1} \to S^n$ one can define an integer-valued invariant, the so-called Hopf invariant. The problem of determining for which $n$ there are maps having Hopf invariant one can be related to many problems in topology and geometry, such as which spheres are parallelisable, which spheres are H-spaces (that is, have a product), and what are the division algebras over $\mathbb{R}$.

The best way to solve this problem is using complex K-theory and Adams operations. I will show how all the above problems are related, give an introduction to complex K-theory and it's operations, and show how to use it to solve this problem.

The Nielsen realisation problem asks when a collection of diffeomorphisms, which form a group up to isotopy, is isotopic to a collection of diffeomorphisms which form a group on the nose. For surfaces this problem is well-studied, I'll talk about this problem in the context of K3 surfaces.

Will discuss several constructions of the Grothendieck group in different contexts together with Wall's solution of the problem of determining homotopy types of finite CW complexes as a motivating application.

Given an algebraic variety $X$ over the finite field ${\bf F}_{q}$, it is known that the zeta function of $X$,

$$ Z(X,T):=\mbox{exp}\left( \sum_{k=1}^{\infty} \frac{#X({\bf F}_{q^{k}})T^{k}}{k} \right) $$

is a rational function of $T$. It is an ongoing topic of research to efficiently compute $Z(X,T)$ given the defining equation of $X$.

I will summarize how we can use Berthelot's rigid cohomology (sparing you the actual construction) to compute $Z(X,T)$, first done for hyperelliptic curves by Kedlaya. I will go on to describe Lauder's deformation algorithm, and the promising fibration algorithm, outlining the present drawbacks.

In this talk I will introduce hyperbolic 3-manifolds, state some major conjectures about them, and discuss some group-theoretic properties of their fundamental groups.

This talk will be about the systematic simplification of differential equations.

After giving a geometric reformulation of the concept of a differential equation using prolongations, I will show how we can prolong group actions relatively easily at the level of Lie algebras. I will then discuss group-invariant solutions.

The key example will be the heat equation.