Forthcoming events in this series


Thu, 13 Jun 2013

15:00 - 16:00
SR1

TBA

Cancelled
Thu, 30 May 2013

12:00 - 13:00
SR1

Basic introduction to few aspects of Derived Algebraic Geometry

Vittoria Bussi
Abstract

This talk is not a detailed and precise exposition on DAG, but it is conceived more as a kind of advertisement on this theory and some of its interesting new features one should contemplate and try to understand, as they might reveal interesting new insights also on classical objects. We select some of the several motivations for introducing it (non-representability of moduli problem and non-naturality of the obstruction theory), and then we will go through the homotopy theory of simplicial commutative algebras and their cotangent complex. We will introduce the category of derived schemes and we will describe their relation with classical schemes. A good amount of time will be dedicated to examples.

Thu, 23 May 2013

15:00 - 16:00
SR1

"Geometry and topology in dimension five"

Rafael Torres
(Oxford)
Abstract

"Among the first successes of the h-cobordism theorem was the classification of simply connected closed 5-manifolds. Dimension five is sufficiently large to be able to implement the tools of surgery theory, yet low enough to allow an explicit classification of the manifolds. These traits make dimension five interesting in terms of existence results of geometric structures, like Riemannian metrics of positive Ricci/nonnegative sectional/positive sectional curvature, Einstein metrics, contact structures, Sasakian structures, among others. The talk will be a limited survey of the five-dimensional symbiosis between topology and geometry"

Thu, 16 May 2013

15:00 - 16:00
SR1

A gentle introduction to Kirby calculus

Robert Kropholler
Abstract

I will be taking us on a journey through low dimensional topology, starting in 2 dimensions motivating handles decompositions in a dimension that we can visualize, moving onto to a brief of note of what this means in 3 dimensions and then moving onto the wild world of 4 manifolds. I will be showing a way in which we can actually try and view a 4 manifold before moving onto a way of manipulating these diagrams to give diffeomorphic 4 manifolds. Hopefully, I will have time to go into some ways in which Kirby calculus has been used to show that certain potential exotic 4 spheres are not exotic and some results on stable diffeomorphims of 4 manifolds.

Thu, 09 May 2013

15:00 - 16:00
SR1

A brief survey on Ricci flow

Alejandro Betancourt
Abstract

Based on ideas from Eells and Sampson, the Ricci flow was introduced by R. Hamilton in 1982 to try to prove Thurston's Geometrization Conjecture (a path which turned out to be successful). In this talk we will introduce the Ricci flow equation and view it as a modified heat flow. Using this we will prove the basic results on existence and uniqueness, and gain some insight into the evolution of various geometric quantities under Ricci flow. With this results we will proceed to define Perelman's $\mathcal{F}$ and $\mathcal{W}$ entropy functionals to view the Ricci flow as a gradient flow. If time permits we will briefly sketch some results from Cheeger and Gromov's compactness theory, which, along with the entropy functionals, alow us to blow up singularities.This is meant to be an introductory talk so I will try to develop as much geometric intuition as possible and stay away from technical calculations.

Thu, 02 May 2013

15:00 - 16:00
SR1

Complex projective structures and dynamics in moduli space

Subhojoy Gupta
(Aarhus University)
Abstract

We shall introduce complex projective structures on a surface, and discuss a new result that relates grafting, which are certain geometric deformations of these structures, to the Teichmuller geodesic flow in the moduli space of Riemann surfaces. A consequence is that for any Fuchsian representation of a surface-group, the set of projective structures with that as holonomy, is dense in moduli space.

Thu, 07 Mar 2013

15:00 - 16:00
SR1

An introduction to Orbifold Bordism

Benjamin Volk
Abstract

This talk will give a quick and dirty introduction to orbifold bordism. We will start by briefly recalling some basic properties and definitions of orbifolds and sketch (very roughly) how orbifolds can be defined in the language of $C^\infty$-stacks due to Joyce (after introducing these). We will then review classical bordism theory for manifolds (in some nonstandard way) and discuss which definitions and results generalize to the orbifold case. A word of warning: this talk is intended to be an introduction and wants to give an overview over the subject, so it is likely that we will be sloppy here and there.

Thu, 28 Feb 2013

15:00 - 16:00
SR1

From Riches to RAAGs: Special Cube Complexes and the Virtual Haken Theorem (Part 1)

Henry Bradford
Abstract

In this first of two talks, I shall introduce the Virtual Haken Conjecture and the major players involved in the proof announced by Ian Agol last year. These are the special cube complexes studied by Dani Wise and his collaborators, with a large supporting cast including the not-inconsiderable presence of Perelman’s Geometrization Theorem and the Surface Subgroup Theorem of Kahn and Markovic. I shall sketch how the VHC follows from Agol’s result that, in spite of the name, specialness is entirely generic among non-positively curved cube complexes.

Thu, 21 Feb 2013

15:00 - 16:00
SR1

Supersymmetry and Morse Theory

Thomas Wasserman
Abstract

Morse theory gives an estimate of the dimensions of the cohomology groups of a manifold in terms of the critical points of a function.
One can do better and compute the cohomology in terms of this function using the so-called Witten complex.
Already implicit in work of Smale in the fifties, it was rediscovered by Witten in the eighties using techniques from (supersymmetric) quantum field theories.
I will explain Witten's (heuristic) arguments and describe the Witten complex.

Thu, 07 Feb 2013

15:00 - 16:00
SR1

What a Higgs bundle is - and why you should care.

Jakob Blaavand
Abstract

This talk is a basic introduction to the wonderful world of Higgs bundles on a Riemann Surface, and their moduli space. We will only survey the basics of the theory focusing on the rich geometry of the moduli space of Higgs bundles, and the relation to moduli space of vector bundles. In the end we consider small applications of Higgs bundles. As this talk will be very basic we won't go into any new developments of the theory, but just mention the areas in which Higgs bundles are used today.

Thu, 31 Jan 2013

15:00 - 16:00
SR1

Introduction to Stacks by way of Vector Bundles on a Curve

Tom Hawes
Abstract

The aim of this talk is to introduce the notion of a stack, by considering in some detail the example of the the stack of vector bundles on a curve. One of the key areas of modern geometry is the study of moduli problems and associated moduli spaces, if they exist. For example, can we find a `fine moduli space' which parameterises isomorphism classes of vector bundles on a smooth curve and contains information about how such vector bundles vary in families? Quite often such a space doesn't exist in the category where we posed the original moduli problem, but we can enlarge our category and construct a `stack' which in a reasonable sense gives us the key properties of a fine moduli space we were looking for. This talk will be quite sketchy and won't even properly define a stack, but we hope to at least give some feel of how these objects are defined and why one might want to consider them.

Thu, 24 Jan 2013

15:00 - 16:00
SR1

From Borel to Yu (via Gromov): topology via geometric group theory

David Hume
Abstract

The Borel conjecture is one of the most important (and difficult) conjectures in Topology. We explain how some weaker but highly related conjectures are being tackled through the coarse geometry of finitely generated groups.

Thu, 17 Jan 2013

15:00 - 16:00
SR1

Useful geometry and modular forms

Jan Vonk
Abstract

Algebraic geometry has become the standard language for many number theorists in recent decades. In this talk, we will define modular forms and related objects in the language of modern geometers, thereby giving a geometric motivation for their study. We will ask some naive questions from a purely geometric point of view about these objects, and try to answer them using standard geometric techniques. If time permits, we will discuss some rather deep consequences in number theory of our geometric excursion, and mention open problems in geometry whose solution would have profound consequences in number theory.

Thu, 29 Nov 2012

15:00 - 16:00
SR1

Hamiltonian evolution of half-flat SU(3) structures

Thomas Madsen
(King's College London)
Abstract

This talk surveys the well known relationship between half-flat SU(3) structures on 6-manifolds M and metrics with holonomy in G_2 on Mx(a,b), focusing on the case in which M=S3xS3 with solutions invariant by SO(4).

Thu, 22 Nov 2012

15:00 - 16:00
SR1

Teichmüller Curves in TQFT

Shehryar Sikander
(Aarhus University)
Abstract

In this talk we show how Teichmüller curves can be used to compute

quantum invariants of certain Pseudo-Anasov mapping tori. This involves

computing monodromy of the Hitchin connection along closed geodesics of

the Teichmüller curve using iterated integrals. We will mainly focus on

the well known Teichmüller curve generated by a pair of regular

pentagons. This is joint work with J. E. Andersen.

Thu, 15 Nov 2012

12:00 - 13:00
SR1

Witten--Reshetikhin--Turaev invariants of mapping tori via skein theory

Søren Fuglede Jørgensen
(Aarhus University)
Abstract
Quantum representations are finite-dimensional projective representations of the mapping class group of a compact oriented surface that arise from the study of Chern--Simons theory; a 3-dimensional quantum field theory. The input to Chern--Simons theory is a compact, connected and simply connected Lie group $G$ (and in my talks, the relevant groups are $G = SU(N)$) and a natural number $k$ called the level. In these talks, I will discuss the representations from two very different and disjoint viewpoints. Part I: Quantum representations and their asymptotics The characters of the representations are directly related to the so-called quantum SU(N)-invariants of 3-manifolds that physically correspond to the Chern--Simons partition function of the 3-manifold under scrutiny. In this talk I will give a definition of the quantum representation using the geometric quantization of the moduli space of flat $SU(N)$-manifolds, where Hitchin's projectively flat connection over Teichmüller space plays a key role. I will give examples of the large level asymptotic behaviour of the characters of the representations and discuss a general conjecture, known as the Asymptotic Expansion Conjecture, for the asymptotics. Whereas I will likely be somewhat hand-wavy about the details of the construction, I hope to introduce the main objects going into it -- some prior knowledge of the geometry of moduli spaces of flat connections will be an advantage but not necessarily necessary. Part II: Quantum representations and their algebraic properties In this part, I will redefine the quantum representations for $G = SU(2)$ making no mention of flat connections at all, instead appealing to a purely combinatorial construction using the knot theory of the Jones polynomial. Using these, I will discuss some of the properties of the representations, their strengths and their shortcomings. One of their main properties, conjectured by Vladimir Turaev and proved by Jørgen Ellegaard Andersen, is that the collection of the representations forms an infinite-dimensional faithful representation. As it is still an open question whether or not mapping class groups admit faithful finite-dimensional representations, it becomes natural to consider the kernels of the individual representations. Furthermore, I will hopefully discuss Andersen's proof that mapping class groups of closed surfaces do not have Kazhdan's Property (T), which makes essential use of quantum representations.
Thu, 08 Nov 2012

15:00 - 16:00
SR1

Homology-stability for configuration spaces of submanifolds

Martin Palmer
Abstract

Fix a connected manifold-with-boundary M and a closed, connected submanifold P of its boundary. The set of all possible submanifolds of M whose components are pairwise unlinked and each isotopic to P can be given a natural topology, and splits into a disjoint union depending on the number of components of the submanifold. When P is a point this is just the usual (unordered) configuration space on M. It is a classical result, going back to Segal and McDuff, that for these spaces their homology in any fixed degree is eventually independent of the number of points of the configuration (as the number of points goes to infinity). I will talk about some very recent work on extending this result to higher-dimensional submanifolds: in the above setup, as long as P is of sufficiently large codimension in M, the homology in any fixed degree is eventually independent of the number of components. In particular I will try to give an idea of how the codimension restriction arises, and how it can be improved in some special cases.

Thu, 25 Oct 2012

15:00 - 16:00
L3

SU(3)-Structures in Heterotic Compactifications

Eirik Svanes
(Department of Physics)
Abstract

I will give an introduction to how SU(3)-structures appear in heterotic string theory and string compactifications. I will start by considering the zeroth order SU(3)-holonomy Calabi-Yau scenario, and then see how this generalizes when higher order effects are considered. If time, I will discuss some of my own work.

Thu, 18 Oct 2012

15:00 - 16:00
SR1

On Moduli of Quiver Representations

Alberto Cazzaniga
Abstract

We will go through the GIT construction of the moduli space of quiver representations. Concentrating on examples (probably the cases of Hilbert schemes of points of $\mathbb{C}^{2}$ and $\mathbb{C}^{3}$) we will try to give an idea of why this methods became relevant in modern (algebraic) geometry.

No prerequisites required, experts would probably get bored.

Thu, 11 Oct 2012

12:00 - 13:00
SR1

Nahm transforms in differential geometry

Jakob Blaavand
Abstract

This talk will discuss the notion of a Nahm transform in differential geometry, as a way of relating solutions to one differential equation on a manifold, to solutions of another differential equation on a different manifold. The guiding example is the correspondence between solutions to the Bogomolny equations on $\mathbb{R}^3$ and Nahm equations on $\mathbb{R}$. We extract the key features from this example to create a general framework.

Thu, 14 Jun 2012

12:00 - 13:00
L3

A gentle introduction to hyperbolic groups.

Dawid Kielak
Abstract

This is intended as an introductory talk about one of the most

important (and most geometric) aspects of Geometric Group Theory. No

prior knowledge of any maths will be assumed.

Thu, 07 Jun 2012

12:00 - 13:00

An Introduction to Reductive GIT

Tom Hawes
Abstract

The aim of this talk is to give an introduction to Geometric Invariant Theory (GIT) for reductive groups over the complex numbers. Roughly speaking, GIT is concerned with constructing quotients of group actions in the category of algebraic varieties. We begin by discussing what properties we should like quotient varieties to possess, highlighting so-called `good' and `geometric' quotients, and then turn to search for these quotients in the case of affine and projective varieties. Here we shall see that the construction runs most smoothly when we assume our group to be reductive (meaning it can be described as the complexification of a maximal compact subgroup). Finally, we hope to say something about the Hilbert-Mumford criterion regarding semi-stability and stability of points, illustrating it by constructing the rough moduli space of elliptic curves.

Thu, 31 May 2012

12:00 - 13:00
L3

Diffeomorphism equivariance and the scanning map

Richard Manthorpe
Abstract

Given a manifold $M$ and a basepointed labelling space $X$ the space of unordered finite configurations in $M$ with labels in $X$, $C(M;X)$ is the space of finite unordered tuples of points in $M$, each point with an associated point in $X$. The space is topologised so that particles cannot collide. Given a compact submanifold $M_0\subset M$ we define $C(M,M_0;X)$ to be the space of unordered finite configuration in which points `vanish' in $M_0$. The scanning map is a homotopy equivalence between the configuration space and a section space of a certain $\Sigma^nX$-bundle over $M$. Throughout the 70s and 80s this map has been given several unsatisfactory and convoluted definitions. A natural question to ask is whether the map is equivariant under the diffeomorphism group of the underlying manifold. However, any description of the map relies heavily on `little round $\varepsilon$-balls' and so only actions by isometry have any chance at equivariance. The goal of this talk is to give a more natural definition of the scanning map and show that diffeomorphism equivariance is an easy consequence.