London Junior Geometry Seminar

Spring Term 2010

Quantization, representations and Lie groups

Compactness Results for $\Omega$-anti-self-dual instantons

Derived Categories for the Mathematician that Works

Covering Spaces, Field Extensions, and Complex Manifolds

Gravitational instantons of type ALG and periodic monopoles

Knots and Topological characterization of knots and links arising from site-specific recombination

Autumn Term 2009

Morse homology

Seifert Fibered Spaces

Cubic surfaces and del Pezzos

Heegaard Floer homology and DNA topology

A soupçon of spectral varieties and co-Higgs bundles

Ricci-flat Kahler metrics on compact Kahler manifolds

Kähler reduction and cscK metrics

Divisors and blow-ups

Homotopy Groups and Smale's Paradox

Why CP2#-CP2 is my favourite manifold

CP2#-CP2 is great for the following reason: It exhibits a Ricci-Soliton that is not Einstein and it exhibits an extremal metric that is not CSCK and it exhibits them in an explicit manner. Before you dismiss all this solitonary as being all very trendy and New Labour, CP2#-CP2 also has a good old-fashioned Einstein metric on it to appeal to its core voters.

I will discuss the construction of these metrics.

Summer Term 2009

Period 3 Implies Chaos

Drinfeld modules

Lu's Theorem for the Bergman Kernel

Spring Term 2009

Ricci Solitons

Autumn Term 2008

Kodaira Embedding and L^2 estimates for the d-bar operator.

The Derived Category of Coherent Sheaves

Classification of compact complex surfaces.

Summer Term 2008

Generalized Dehn twists.

Given a surface, and an embedded circle within it, we can perform a Dehn twist by cutting along the circle and then re-gluing the two resulting circles with an added twist. This construction generalizes to the symplectic context, and in fact we can twist around higher-dimensional Lagrangian spheres. I will gently present the background required to understand this, and explain some interesting properties of the generalized Dehn twist.

Orbits and subgroup structure.

R. W. Richardson wrote a number of papers which constructed geometric characterisations of group-theoretic notions (and/or vice-versa depending on your taste). In his 1988 paper 'Conjugacy classes of n-tuples in Lie algebras and algebraic groups.' he introduces a notion of Strong Reductivity of a reductive group G to characterise when orbits of a subgroup H in G acting by conjugacy on G^n are closed. I'll explain how I came to be aware of the notion of Strong Reductivity, motivated by group theory, and then sketch how Richardson used it to to solve the problem he was working on. Many of the concepts involved in the latter are standard in GIT.

Spring term 2008

Gromov-Witten invariants for the complex projective plane.

How many degree d curves pass through 3d-1 generic points in the complex projective plane? For d=2, we have the result that there is just one conic passing through 5 general points. Beyond this, the question becomes more difficult, and classical enumerative geometry does not provide much hope for a complete solution.

It turns out however that a fairly simple recursion relates the numbers of such curves, allowing them to be calculated. This fact was discovered by Kontsevich in the mid 90s, exploiting the relationship between enumerative geometry and Gromov-Witten invariants.

Without going into any of the analysis underpinning Gromov-Witten invariants, I will explain their relation to enumerative geometry. Then I'll demonstrate some of the axioms of Gromov-Witten theory by describing Kontsevich's calculation in as much detail as possible.

A holomorphic symplectic manifold with singular deformation space.

I will try to illustrate the classical theory of deformations in the sense of Kodaira, Spencer & Kuranishi by calculating the Kuranishi-space of a certain manifold. It will turn out to be singular which shows that non-Kaehler manifolds with trivial canonical bundle can behave very differently from Calabi-Yau manifolds.

Constructions of 4 dimensional symplectic manifolds.

I will recall symplectic fiber sum and symplectic blow up and down and will discuss some applications, For example: "Every finetly presented group can be a fundamental group of a symplectic 4-manifold"

Numerical Kaehler-Ricci and Calabi flows (Stuart Hall)

Numerical Kaehler-Ricci and Calabi Flows:

I will try to give an overview of some algorithms for evolving Kaehler metrics on projective manifolds due to Simon Donaldson. I will hint at why they are related to other known methods for "improving a metric" such as the Ricci-flow which was talked about a few weeks ago.

An introduction to stability conditions (Sven Meinhardt)

The notion of stability in algebraic geometry was introduced 40 years ago by David Mumford in order to classify vector bundles on (complex) curves. More recently physicists developed the notion of stability of B-branes in string theory. It was Tom Bridgeland who found the appropriate language to unify both approaches of stability. He was able to show that the space Stab(X) of all stability conditions (with some properties) on the derived category of a complex manifold X has itself a natural structure of a complex manifold. This leads immediately to the following questions?

1. How does the geometry of X affects the geometry of Stab(X)?

2. Can we extract some properties of the derived category of X from the geometry of Stab(X)?

3-Manifolds via surgery (Tom Coates)

We explain how to obtain any closed 3-dimensional manifold from the 3-dimensional sphere by surgery along a knot of link.

Winter term 2007

Anti-self-dual Yang-Mills connections (Henrique Sá Earp)

I will outline SKD's paper on antiselfdual Yang-Mills connections, uses the Kahler geometry of the base manifold to reduce the gradient flow associated to the original problem to a nonlinear heat equation on the space of Hermitian bundle metrics. Hopefully, time not allowing, I will skip the gruesome analysis necessary to obtain a unique and smooth solution for all time and concentrate on the interesting energy functional, defined on the space of metrics using Chern-Bott invariants (a somewhat Chern-Simons-like construction), to show that the flow indeed converges to our desired YM connection - subject to a stability assumption along some curve.

Moment maps and scalar curvatue (Stuart Hall)

Many problems in Riemannian geometry can be phrased as:

Let M be a manifold. Is it possible to put a metric g on M so that g has property X?

In this talk 'property X' will be constant scalar curvature (csc). I will talk broadly about this problem before focusing in on the case where M is a Kahler manifold (the cscK problem). The 'moment maps' mentioned in the title refer to an idea by Fujiki, Quillen and Donaldson that one can view the problem through the framework of symplectic geometry. I hope to end by explaining this idea.

An Isoperimetric Function for Bestvina-Brady Groups (Will Dison)

The higher dimensional finiteness properties of a group generalise the notions of being finitely generated and finitely presented. Bestvina-Brady groups were introduced to settle (in the negative) the long-standing question of whether the homotopical finiteness property F_n is equivalent to its homological analogue FP_n. They have since become objects of significant interest in their own right. In this talk I will outline a proof that every finitely presented Bestvina-Brady group has a quartic upper bound on its Dehn function. This bound is known to be sharp. The Dehn function of a finitely presented group measures the non-deterministic time complexity of the word problem for the group. The proof involves a new method (area-penetration pairs) for dealing with presentation with infinitely many relators.

An introduction to geometric invariant theory (Jacopo Stoppa)

Group actions on schemes, quotients, linearisations, stability, Hilbert-Mumford criterion.

Toric symplectic manifolds (Dominic Wright)

I will talk about the bijective correspondence between toric symplectic manifolds and the image of their moment maps, which are convex polyhedral sets. The technical terms involved will be explained, including: symplectic manifolds, Hamiltonian torus actions and moment maps.

Summer term 2007

The tautological ring of the moduli space of curves (Jacopo Stoppa)

I will report on Ravi Vakil's lectures at GAeL, where he presented some of the beautiful geometry and combinatorics related to the a subring of the Chow ring of M(g,n), called "tautological ring".

The 2nd relative homotopy group (Will Dison)

I will begin by briefly recapping the definition of the relative homotopy groups of a space. For n>2 these groups are all abelian, but the 2nd relative homotopy group has a surprising rich structure which is captured by the notion of a crossed module. After defining such crossed modules I will describe another, purely algebraic situation, in which they arise, namely the set of Peiffer sequences associated to a group presentation. During the remainder of the talk I will develop the notion of 'pictures' which will be used to show how these two manifestations of crossed modules are related.

First Steps in Yang-Mills (James Otterson)

We will go over the Yang-Mills Lagrangian and spend some time on the electromagnetic case showing that its Euler-Lagrange equations is Maxwell's equations. If there is enough time, we will tinker with the Yang-Mills lagrangian in order to obtain lagrangians that yield topological invariants.

Topological conformal fielf theories (Ed Segal)

A quantum field theory can be interpreted (roughly) as a functor which sends cobordisms of manifolds to linear maps between Hilbert spaces. In general this should depend on a choice of metric on the cobordism, however many physically interesting quantities turn out to depend only on weaker data such as the topology or the conformal class. Motivated by this, I'll explain the definition of a topological and topological conformal field theory, then give the (very easy) proof that a two-dimensional topological field theory is a Frobenius algebra. I'll then show how to generalise this to the topological conformal case.

Compact 4-orbifolds with torus actions (Dominic Wright)

The space of orbits on a compact 4-orbifold admitting a torus action is a polygon with fixed points as vertices. After defining orbifolds, I aim to describe the space of orbits, and outline how the orbifold can be reconstructed with combinatorial data prescribed on this polygon. Finally I may also discuss families of anti-self-dual metrics on these orbifolds.

Manifolds of non-negative sectional curvature (Stuart Hall)

I will briefly recap the definition of sectional curvature and explain some of the restrictions one has on the topology of a manifold with prescribed sectional curvature. After this I will show how to construct some simple families of manifolds that admit metrics of non-negative sectional curvature and discuss some of the open problems in this area.

Spring term 2007

The Mass-Tansportation Approach to Sharp Sobolev Inequalities (Jacopo Stoppa)

If you do differential geometry you need Sobolev inequalities. These in turn have some geometric content. Recently Villani et al. found a new way of proving many of these inequalities in sharp form via the existence of a transport map between two suitable measures. I think this is a beautiful trick and I hope to explain why.

Balanced Metrics and Bergman Kernels (Julien Keller)

We will introduce the notion of balanced metrics for smooth projective manifolds and explain how it is related to the "quest" for constant scalar curvature metrics. We will also speak of the relationship with the Kahler-Ricci flow and some examples will be given. The talk is intended for a large audience.

Kovalev's G2 Manifolds (Henrique Sá Earp)

Alexei Kovalev developped an idea of SKD as to how to construct Riemannian 7-manifolds with holonomy group G2 by gluing together two asymptotically cylindrical 7-manifolds with holonomy SU(3), each obtained by crossing a Calabi-Yau 3-fold with a circle. These non-compact CY are obtained under certain conditions by cutting a divisor from a Fano 3-fold, which amounts to a non-compact version of the Calabi conjecture. The adequate gluing that results in the desired torsion-free G2-structure are then obtained by "stretching the neck" of the noncompact bits and a suitable perturbation of the metric.
I will explain what all that means, including a few generalities about G2 and a "qualitative" outline of the construction. I will expect algebraic-geometric contributions from the audience as my explicit example starts from CP3 minus a quartic hence its geometry is complex.

Contact Geometry and Lagrangians (James Otterson)

We will try to present an introduction to contact geometry as a geometry of differential equations that allows one to consider more symmetries than in gauge theory. After an introduction to basic contact geometry, we will talk about contact transformations of lagrangians, the Poincare-Cartan forms with their accompanying Euler-Lagrange equations and Noether's theorem.

Autumn term 2006

Moduli Spaces and Representable Functors (Ed Segal)

A fundamental notion in most areas of geometry is the idea of a moduli space, a space that naturally parametrizes some set of objects of interest. I'll show how a little categeory theory can explain various aspects of this idea, and I'll try to include as many examples as I can.

Moment Maps and Reduction (James Otterson)

We start by proving Noether's theorem and giving some examples. Then we generalize it to momentum maps and talk about reduction (we will basicaly go over Smale's paper: Topology and Mechanics I, Invent. Mat. 1970).

Curvature and Topology (Stuart Hall)

I will rapidly recall the definitions of curvature from Riemannian Geometry and then survey the various ways that curvature tells you something about the topology of the manifold. A good start is the Gauss-Bonnet theorem. The Bonnet-Myers theorem is another particularly nice example of this so I will sketch the (variational) proof of this. In the final part I will talk about the Bochner theorems and how this gives traffic in the other direction i.e. restrictions on admissible metrics.

Summer term 2006

Elementary Chern-Simons Theory (Henrique Sá Earp)

I really mean elementary. I will define the Chern-Simons funtion for 3-manifolds in more than one way as in SKD's blue book (on Floer Homology), justify its integer periods and discuss the nice fact that its critical set is precisely the moduli space of flat connections. I will point out how to generalise CS ideas for higher dimensions under appropriate conditions. Time allowing in the end I might also show a nice illustration in the case of G2-intantons on the 7-torus, proposed by SKD to one of his students.

Minitwistors (Dominic Wright)

I aim to introduce the minitwistor correspondence between a class of 3-manifold and their spaces of geodesics. This requires reviewing the twistor correspondence, which relates anti-self-dual conformal Riemannian 4-manifolds to complex 3-manifolds, or twistors. Following a similar procedure, I will outline how an Einstein-Weyl structure on the 3-manifold gives the integrability condition necessary for the space of geodesics, or minitwistor, to become a complex surface. I will then attempt to complete the picture, by drawing on both these correspondences to examine how Einstein-Weyl 3-manifolds can arise as quotients of ASD 4-manifolds by Killing fields.

Introduction to Foliation Theory (James Otterson)

We will start by defining (singular) distributions. Since the often quoted Frobenius theorems for singular distributions are known to be wrong, we will state our own version of it. We will then explain why control theorists think that every distribution is integrable. The remaining part of the talk will consist of a crash course on foliation theory. We will introduce leaf spaces, uniform transversality, holonomy groups and Reeb stability theorems. After these we will state a theorem of Novikov (which can be thought of as an extention of the hairy ball theorem) and Schweitzer's counter-example of Seifert's conjecture (which can be thought of as the non-existence of hairy ball theorems in certain situations). In the unlikely event that there is some time left, we will introduce Lie groupoids and Lie algebroids, and show that every Lie algebroid generates a singular foliation.

The Futaki Invariant and Odd Symplectic Grassmannians (Jacopo Stoppa)

For any compact Fano mainfold M, the Futaki invariant f is a character defined on the Lie algebra of holomorphic vector fields on M which gives an obstruction to being Kaehler-Einstein. It is sometimes possible to compute f using a localization formula by Futaki and others. I will explain this and finally introduce a class of Fano odd symplectic grassmannians. In this case the localization formula reduces to an elementary calculation with Schubert cycles, showing the non-existence of a KE metric. This also follows from the global structure of the automorphism group, but this is much harder to work out (one would need Bott-Borel-Weil plus several vanishing results).

The Derived Category of P^1 (Ed Segal)

The derived category of projective space turns out to have a surprisingly simple description. In this talk I'll review the definitions of derived categories and functors, then sketch the construction of the Koszul resolution of the diagonal on P^1 and show why it leads to a description of the derived category.

Spring term 2006

An Introduction to Twistor Geometry (Joel Fine)

I will attempt to explain how twistors give a complex geometric description of a type of conformal Riemanian four-manifolds, namely those with anti-self-dual curvature. It turns out that for such conformal classes, other Riemannian geometric objects (e.g. instantons, Kahler representatives etc.) then have holomorphic interpretations on twistor space. This dictionary between Riemannian anti-self-dual geometry and a certain type of complex geoemtry is Penrose' famous twistor correspondence. And I happen to think its pretty neat.

The Thurston Boundary of Teichmüller Space (Henry Wilton)

"Very interesting" - Alessio Corti

Derived Categories and Derived Functors (Ed Segal)

This talk will be a misguided attempt to convince people that derived categories are interesting and useful.

Automatic Groups (Will Dison)

Automatic groups are a class of groups defined by having a collection of normal forms which can be recognised by a simple algorithmic procedure. Their theory lies at the juncture of geometry, algorithms and combinatorial group theory. As I will explain, the ability of a group to support an automatic structure is closely connected to its geometry, and in particular to non-positive curvature. Many natural classes of groups commonly studied in geometric group theory are examples of automatic groups, for example hyperbolic groups, small cancellation groups, coxeter groups, mapping class groups, braid groups and most 3-manifold groups. Despite the fact that this diverse collection of examples arise from very different geometric situations, automatic groups share a wide variety of interesting properties.

Autumn term 2005

Frobenius structures related to Coxeter groups (Misha Feigin)

To any finite Coxeter group one can relate a Frobenius manifold structure on the space of orbits of this Coxeter group acting in R^n. On the other hand, to a Coxeter group it corresponds an elementary logarithmic solution of the generalised Witten-Dijkgraaf-Verlinde-Verlinde equations, this is another way to construct most of the ingredients of a Frobenius manifold. There is a simple duality relation between the two structures.

The logarithmic solutions can be defined for a more general class of V-systems. Many V-systems can be obtained through the restrictions in the Coxeter systems. Under duality these solutions correspond to the structures on the discriminants of the Frobenius manifolds for Coxeter groups.

Subgroup Separability from a Topological Point of View (Henry Wilton)

Roughly, a group G is subgroup separable if every finitely generated subgroup can be distinguished in the finite quotients of G. This translates into the topological condition that any finite subcomplex of a covering space embeds in some finite cover. Scott and Stallings used this reinterpretation to provide easy topological proofs of M. Hall's result that free groups are subgroup separable. Scott went on to show the importance of subgroup separability in the theory of 3-manifolds, in particular its connection with the Virtually Haken Conjecture.

This talk will be a leisurely stroll through some of the topological ways of studying subgroup separability. I'll start with the concept and its topological reformulation, explain the importance of subgroup separability in the theory of 3-manifolds, and give Stallings' classic proof of Hall's theorem. If I have time I'll finish with some of my own work on subgroup separability for limit groups.

Introduction to G2-Instantons (Henrique Sá Earp)

I will try to motivate the study of gauge theory in dimension seven by showing how some notions of four-dimensional geometry can be reproduced rather authentically on G2-manifolds. In particular, I will sketch the construction of the moduli space of instantons (anti-self-dual gauge-equivalence classes of connections) for interesting (G=SO(3), SU(2), say) G-bundles on such manifolds, exploiting the analogy with SKD's construction in dimension four as far as possible - and solving the specific small difficulties that arise in the 7-dimensional case. I will avoid complications such as singularities due to reducible connections and analytical technicalities such as things with "Sobolev" in their name.

Operads and the Moduli of Curves (Ed Segal)

This talk will be an (attempted) explanation of Kevin Costello's paper The A-infinity operad and the moduli space of curves. I'll go through the definition of A-infinity algebras and show how the universal structure (operad) describing them relates to moduli spaces of riemann surfaces with boundary. We'll see that up to homotopy equivalence, these moduli spaces have a simple combinatorial description.

An Introduction to the Bass Conjecture

The statement of the Bass conjecture is algebraic, but it is extremely useful in topological arguments: it is often just what is needed to let you conclude that a homological invariant vanishes when you want it to. I'll explain the background, state the conjecture, summarize Bass's results, prove some of the useful consequences, and then look at Eckmann's strategy for establishing that a group satisfies the conjecture using a generalized Chern character in cyclic homology. As you might imagine, this is slightly technical in places, but I'll try to draw lots of pictures.

Summer term 2005

Geometrization I: An introduction to Thurston's geometrization conjecture (Henry Wilton)

Thurston's Geometrization Conjecture says that any 3-manifold has a natural decomposition into geometric pieces. This talk aims to make these statements precise, first by explaining the natural decomposition in question, achieved by cutting along essential spheres and tori, then by introducing the notion of a geometric structure. I will also say some more about Thurston's classification of 3-dimensional geometries.

Near-symplectic topology of 4-manifolds (Tim Perutz)

Every 4-manifold satisfying an elementary condition on its cohomology carries a closed 2-form which is non-degenerate away from an embedded circle Z, and zero on Z. These 'near-symplectic forms' have topological counterparts, 'broken pencils', related to them as Lefschetz pencils are to symplectic forms. I'll outline a purely symplectic construction of an invariant of broken pencils. This invariant is sometimes—I expect always—equal to the Seiberg–Witten invariant of the underlying four-manifold.

I'll do my best to keep things accessible and non-technical.

Geometrization II: Hamilton's and Perelman's approach via Ricci flow (Mark Haskins)

This is the sequel to Henry's recent talk on Thurston's Geometrization Conjecture. We will introduce the Ricci flow, some of its basic analytic and geometric properties and discuss a programme to prove Geometrization using the Ricci flow.

We will review some of the fundamental steps in this programme that were proven by Hamilton between 1982 and 1999, and point out some of the crucial missing parts. In a series of 3 preprints in 2002–3 Grisha Perelman made a major breakthrough in the program to prove geometrization via Ricci flow. We will attempt to describe some of Perelman's breakthroughs (in a not too technical way).

The MNOP conjectures (Ed Segal)

We are a given a complex manifold, and we wish to study curves (i.e. compact Riemann surfaces) embedded holomorphically in it. For fixed genus and homology class, there will generically be a finite number of such curves, so we should be able to count them and produce an invariant of the manifold. However there are two different ways of doing this: we either count the holomorphic maps from a curve to our manifold, giving us Gromov–Witten invariants, or we count embedded curves (subschemes), giving us Donaldson–Thomas invariants. The MNOP conjectures give a rather mystical formula relating the two answers.

My talk will cover some of the same ground as the G&T seminars given by Pandharipande and Okounkov last term, but at a more basic level.

The h-principle (Joel Fine)

The existence of certain objects in differential topology have obstructions lying in homotopy groups. For example, if two immersions of S^1 in R^2 are to be homotopic through immersions (isotopic), they must have the same winding number. An object is said to satisfy the h-principle if this homotopy obstruction is the only obstruction to its existence. I will, hopefully, clarify this statement and explain the simplest method for proving the h-principle holds. As a consequence, this will show that two immersions of S^1 in R^2 are isotopic if and only if they have the same winding number, whilst any two immersions of S^2 in R^3 are isotopic. In particular, you can "turn S^2 inside out in R^3" by a path of immersions. I have a nice video of this which you can watch if you promise to pay attention to the talk.

Algebraic K-theory after Quillen (Michael Tweedale)

I'll start from the h-cobordism theorem, which, as everyone probably knows, implies the Poincaré conjecture in dimensions ≥ 5. For manifolds M that are not simply connected, we need to replace h by s (homotopy by simple homotopy). These simple homotopies were studied by Whitehead, and the obstruction for a homotopy to be simple turns out to lie in a quotient of a certain algebraic K group associated to the group ring Z[pi_1(M)]. I'll briefly say how this goes, then look at another topological obstruction, Wall's finiteness obstruction, which lives in a different K group. The links between the K groups will serve as motivation for putting things into a slightly more general context, namely the K theory of exact categories. I'll introduce Quillen's Q-construction, which gives a very elegant definition of all the K groups, and try to say something sketchy about how this relates to the more familiar + construction in the modules case. Finally I'll mention some applications, for example higher Chow groups.

Toric geometry III: the dual picture (Richard Thomas)

I have given lots of symplectic toric lectures, this one will assume nothing and do everything dually, via algebraic geometry. But I promise even group theorists shoild be able to follow.

Quasi-isometric rigidity of lattices in Sol (Will Dison)

I'll start out by giving the definition of a quasi-isometry and of the Lie group Sol. I'll then state a theorem about the quasi-isometric rigidity of Lattices in Sol. Along the way we'll learn a little about the geometry of Sol, including the existance of embedded hyperbolic planes and the structure of its boundary at infinity.

Winter term 2005

Generalised G2-manifolds (Fred Witt)

The talk is based on the article math.dg/0411642. We define new Riemannian structures on 7-manifolds by a differential form of mixed degree which is the critical point of a (possibly constrained) variational problem over a fixed cohomology class. The unconstrained critical points generalise the notion of a manifold of holonomy G2, while the constrained ones give rise to a new geometry without a classical counterpart. We characterise these structures by the means of spinors and show the integrability conditions to be equivalent to equations arising in supergravity and superstring theory of type IIA/B (see also hep-th/0412280.) Similar generalisations are also possible for classical Spin(7)- and SU(3)-structures. Finally, we will explain the device of T-duality as a construction method of examples.

How not to prove the Poincaré conjecture (Henry Wilton)

John Stallings' paper is the classic example of how questions in 3-manifolds topology can be rephrased in purely algebraic terms, as well as a disarmingly frank discussion of the pitfalls that mathematicians encounter. He plays topology and algebra off against each other to get a deceptively simple-looking reformulation of the Poincare Conjecture. I'll explain this reformulation and his (incorrect) proof. The essential ingredients are Heegard decompositions and Dehn's lemma.

Toric geometry I: Symplectic toric geometry (Richard Thomas)

I'll start from the definition of a symplectic manifold and work up to show how to give compact examples in a painfully simple way from the data of an integral polytope. It'll all be very easy and geometric.

Classifying spaces for proper actions and Bredon cohomology (Michael Tweedale)

I'll start with a gentle introduction to algebraic and geometric dimensions of groups for the benefit of those who haven't heard me talk about it ad nauseam already. If the group has torsion, these numbers will be infinite, but we might still get useful invariants by considering a classifying space for proper, rather than free, actions. I'll explain all this, then talk about what Bredon cohomology is, and why it provides the right algebraic counterpart to this revised notion of geometric dimension.

An introduction to twistors in four-dimensional Riemannian geometry (Joel Fine)

Penrose's twistor construction associates to every Riemannian four-manifold a six-manifold with an almost complex structure, called twistor space. Integrability of twistor space is equivalent to a condition on the curvature of the four-manifold, namely anti-self-duality. This sets up a one-to-one correspondence between ASD four-manifolds and a certain class of complex three-folds. I will explain this correspondence and, perhaps, mention how the holomorphic geometry of the three-folds can be used to solve PDEs on the Riemannian four-manifolds.

Extremal metrics and stability (Gábor Székelyhidi)

I will explain what the words in the title mean, and how moment maps can be used to relate them

Dehn functions and isoperimetric inequalities (Will Dison)

The Dehn function gives a least upper bound for the number of relators of a presentation of a group that need to be applied to a word to demonstrate it as being equal to the identity. It thus gives a measure of the complexity of the word problem for the group. Despite its algebraic definition the Dehn function has many links to geomety. These include relationships between hyperbolic groups and linear Dehn functions, and between groups which act on non-positively curved spaces and quadratic Dehn functions. I might also mention the Filling Theorem which relates isoperimetric inequalities for closed Riemannian manifolds to the Dehn functions of their fundamental groups.

Symplectic reduction (Michael Plummer)

The talk will introduce the idea of symplectic reduction in its simplest form. In doing so we will need to first look at symplectic group actions and their associated momentum maps.

Crystalline cohomology and crystals (Shu Sasaki)

Crystalline cohomology originated from the observation that ℓ-adic cohomology groups of a smooth projective (connected) variety over an algebraically closed field of characteristic p = ℓ are "miserable" in comparison to p ≠ ℓ case. Roughly speaking, Grothendieck's idea (outlined in his lectures at IHES in 1966) was to lift varieties to characteristic zero and then take the de Rham cohomology to obtain "nice" (p-adic) cohomology. However, still some questions remained. Most notably: Is it always the case that one can lift varieties? To remedy this situation, we needed more sophisticated and subtle theory. The answer was…the theory of crystalline cohomology! In my talk, I'd like to explain why this crystalline cohomology is the "right" one and if time permits, I would hope to talk about things like F-crystals to illustrate how mind-boggling this theory can sometimes be. I shall start from very basics such as Grothendieck topology so don't be scared of what I've just said above.

A walk through the landscape of elliptic genera (Imma Galvez)

I will define Hirzebruch genera, that is, graded unital ring homomorphisms whose source is a ring of cobordism classes of manifolds. Classical examples are the signature, the A-hat genus, the Todd genus and the arithmetic genus. The Atiyah–Singer theorem says then that the invariants obtained are actually indices of well-known operators on the manifolds. So there is no doubt that Hirzebruch genera provide a wealth of interesting invariants.

Then I will focus on elliptic genera. These mushroomed simultaneously in the late 80s both in physics and in algebraic topology. Elliptic genera are Hirzebruch genera whose target is usually some ring of modular forms, and they come in many interesting flavours. The q-expansions of their values encode in their coefficients the indices of twisted operators on the manifold, and, following Witten, we may interpret them as indices of operators on the loop space of the original manifold. But this is only the beginning….

Autumn term 2004

A 'quasi' approach to Poisson structures on non-rotating BTZ black holes (Sébastien Racanière)

In this talk I will construct a black hole (the BTZ black hole) and then show how the theory of quasi-Poisson actions can be used to construct a Poisson structure on this black hole. Of course, no prior knowledge of black holes, Poisson or quasi-Poisson structures will be assumed.

Crossed modules, pictures, and 2-dimensional topology (Michael Tweedale)

Two-dimensional cell complexes form a remarkably rich class of objects; indeed, one can regard all of group theory as a sub-theory of 2-dimensional topology. One of the key invariants at our disposal is the second homotopy group, pi_2, and starting with the work of Whitehead and Reidemeister, there is a good classical theory of the structure of pi_2. The aim of the talk is to describe this basic structure, and explain the connection with group theory.

Cohomology of finite groups (Will Dison)

Topological definition of group homology in terms of Eilenberg–Mac Lane spaces. Algebraic definition using free resolutions. Using cohomology to count group extensions.

Elliptic operators and the Atiyah–Singer index theorem (Gábor Székelyhidi)

The index theorem relates the analytic index of an elliptic operator to topological invariants. I will try to explain why such a result might be true, starting from the index theory of Toeplitz operators. It will hopefully be very elementary so everyone can understand. I will not give the cohomological version of the index theorem, in case that's the only thing you want to know.

Algebraic geometry over groups (Henry Wilton)

Zlil Sela's solution to the Tarski problem is a tour de force of ingenious ideas and original techniques. His starting point is the development of a theory of algebraic geometry over free groups—that is, a description of sets of solutions to systems of equations in free groups. I shall explain this description in terms of Makanin–Razborov diagrams and limit groups, and show how the diagrams are constructed. The techniques used range from classical algebraic geometry, to non-standard analysis, to the theory of group actions on trees.

Spherical functions and symmetric powers of the projective plane (Misha Feigin)

I am going to talk a bit on the spherical functions, which may be thought of as the restrictions to a sphere of harmonic polynomials in R^3. Maxwell's theorem states that these functions can be uniquely obtained by differentiating 1/r along constant directions. I'll follow Arnold to explain this theorem together with its topological content that symmetric power of a projective plane is a projective space, and maybe mention related topological theorems.

Lefschetz fibrations and monodromy (Matthew Gould)

I will give the definition and examples of Lefschetz pencils and fibrations. Then move on to the monodromy of such and how this encodes the fibration structure in terms of Dehn twists.

What I think A. Connes means by a non-commutative manifold (Henrique Sá Earp)

I will explain how a "geometry" is defined by a spectral triple (A,H,D), given by an algebra A, a Hilbert space H and a "Dirac operator" D, and I will discuss how the notion of "non-commutative geometry" can be made precise in this context, following Alain Connes's book. I will provide the necessary background, sacrificing rigour for intuition whenever I feel like, so everybody can come with less ambitions but also less fears.

Closed geodesics (Ed Segal)

A closed geodesic (on a Riemannian manifold) is just a geodesic that loops round and repeats itself. I'll be discussing the proof by Gromoll and Meyer that any manifold contains infinitely many distinct closed geodesics, provided that the Betti numbers of its free loop space are unbounded. The proof goes via putting a Hilbert manifold structure onto the free loop space, then applying some Morse theory to the energy function on it.

Variational principles for diffeomorphism groups (Colin Cotter)

I will show how to make a variational principle for a general set of fluid dynamical equations without having to resort to nasty Lagrangian flow variables. This is done by figuring out what the variations look like when constrained to the Lie algebra of vector fields (i.e. the fluid velocities). If there is enough time I will show how to extend this theory to include advected scalar, vector and tensor quantities (i.e. the semi-direct product version of the theory).

Summer term 2004

Gauss–Bonnet theorem for a 2-sphere with an integral affine structure (Dmitri Panov)

A manifold with affine structure is a manifold glued from pieces of R^n by affine identifications. The only two-dimensional surface that has a non-singular affine structure is a torus. The affine structure on a surface is called integral if it has a finite number of singularities and its holonomy is contained in SL(2,Z). One can see that any regular polyhedron (exept the icosahedron) generates an integral affine structure on the sphere. These guys are interesting because of their relation to symplectic and complex geometry. I will speak about this relation and about the generalisation of the Gauss–Bonnet theorem (the sum of apexes of a 2-dimensional polyhedron is equal to its Euler characteristic) to integral affine geometry.

Topological field theories and the Alexander polynomial (Tim Perutz)

The Casson and Seiberg–Witten invariants of 3-manifolds can both be defined using gauge theory, but both can be computed, on a homology-(S^1 x S^2), from the classical Alexander polynomial. Donaldson's explanation is that both invariants come from topological field theories, in which a 3-manifold-with-boundary has an invariant in a vector space associated to its boundary. These vector spaces, built from the homology of the surface, actually determine the whole theory; but in another model one can see the role of the Alexander polynomial.

Complex dynamics and quasiconformal maps (Gábor Székelyhidi)

The talk will be about the relation between complex dynamics and quasiconformal maps, in particular the use of holomorphic motions. I will concentrate on the family of quadratic maps, and show that in a hyperbolic component of the Mandelbrot set, the Julia sets of the corresponding quadratic maps move holomorphically. This motion can in favourable cases be extended to a global quasiconformal conjugacy between the corresponding dynamical systems.

Bass–Serre theory (Henry Wilton)

Bass–Serre theory provides the right language to talk about decompositions of groups, analogously to decompositions of spaces of the type used in the the Geometrization conjecture. Amalgamated free products and HNN extensions are, by van Kampen's theorem, the group-theoretic analogies to cutting a manifold along a codimension-one submanifold. The correct generalization of these is the concept of a graph of groups, which differential topologists may like to think of as an orbigraph. By an insight of Serre all graphs of groups are developable, so to discuss the splittings of a group it suffices to consider actions on trees. I will explain what all these terms mean, outline the essentials of the theory, and prove Serre's theorem. If I have time I'll explain some applications, including a classic decomposition of SL_2(Z).

Grothendieck's Riemann–Roch Theorem (Kevin Costello)

I'll describe Grothendieck's proof of his Riemann–Roch theorem, starting with his definition of K homology and cohomology groups for varieties. In particular, I'd like to explain why the Todd class arises in this theorem.

Ends of groups and compact Kähler manifolds (Michael Tweedale)

Using a hard theorem of Taubes one can show that any finitely presented group occurs as the fundamental group of a complex, symplectic closed manifold. However, demanding that Γ be the fundamental group of a Kähler manifold places very strong restrictions on Γ. These restrictions have been clarified by research in the last fifteen years, but they still could not be described as well-understood. After a selection of basic results, I'll try to describe one of the early landmarks in the theory, essentially due to Gromov: Kähler groups are, in a strong sense, indecomposable—for example, they don't split over finite subgroups. The tools involved are nothing if not varied: analysis to construct a holomorphic map to a curve inducing an isomorphism on certain L^2-cohomology groups, a classical algebraic invariant (the "ends" of a group), and a smattering of Hodge theory.

Winter term 2004

L^2 Cohomology and Gromov's Kähler hyperbolicity (Michael Tweedale)

I'll define L^2-cohomology for a smooth manifold, and explain briefly how this can be formulated more algebraically in terms of von Neumann algebras and so extended to arbitrary CW-complexes. I'll show that the L^2-Betti numbers are obstructions for a closed manifold to fibre over the circle, and then it's on to L^2-Hodge theory. Finally, I'll describe a rather eccentric definition of a "Kähler hyperbolic" (closed) manifold due to Gromov, and indicate how he used L^2 methods to prove that such things are projective algebraic varieties.

Automorphisms of surfaces (Henry Wilton)

In the wake of the work of Nielson and Thurston, surface automorphisms are more-or-less completely understood. I'll start by explaining what geodesic laminations are, then say what they've got to do with aperiodic automorphisms. The next step is the slightly knotty business of passing from laminations to foliations, and finally I'll show that aperiodic irreducible automorphisms are in fact pseudo-Anosov.

Compact Kähler manifolds not homotopy equivalent to projective manifolds (Tim Perutz)

These improbable beasts have recently been constructed by Claire Voisin; her examples (cunningly blown up tori, in dimension 4 and higher) can be distinguished from projective algebraic manifolds on the basis of their cohomology ring alone. This proves spectacularly false a widely-believed conjecture that compact Kähler manifolds should be deformations of projective ones. Voisin's construction is direct, and only uses standard tools from the complex geometer's kit—the Albanese torus, weight 1 Hodge structures, blow-ups (I'll review these).

Bend and Break (Richard Thomas)

I'll give a little introduction to algebraic geometry, to, I hope, make the talk accessible to all, leading to "the ample cone". I'll then explain the main technique for its study, due to Mori, which involves producing rational curves (S^2s!) under certain conditions. I'll mainly concentrate on the geometry of how the curve is produced, rather than the amazing characteristic p trick that is also required (which I'll probably only have time to mention briefly).

Honda–Tate theory (Toby Gee)

Honda–Tate theory provides a complete description of abelian varieties over finite fields, and the isogenies between them. The classification involves some rather tasty algebra (central simple algebras, Brauer groups); I'll give enough background (assuming nothing more than some idea of what varieties and finite fields are) to state the main theorem, and show how it applies to the theory of supersingular elliptic curves.

Floer homology and 4-manifold invariants (Jeremy Bryant)

I will talk about the construction of Floer homology on 3-manifolds, emphasizing how it generalises Witten's Morse theory construction of homology on a manifold, which I will review. The 4-manifold invariants generalise to manifolds with boundaries, taking values in the Floer groups. I will describe the gluing properties of the Floer groups, and approaches to the Witten conjecture. This is meant as an overview of Floer theory, keeping the main points clear, for those who have never seen it before. I will then describe the construction of a spectrum by Manolescu, yielding a simpler construction of Floer homology.

Spectral Sequences (Diego Matessi)

Spectral sequences are a useful tool in algebraic topology. For example they are useful in the computation of the cohomology of the total space of a fibre bundle (Leray's spectral sequence). I will explain the construction of the spectral sequence associated to a filtered or double complex, which leads to Leray's spectral sequence. Applications will follow, such as a quick proof of the fact that for simply-connected manifolds, the first non-zero homology and homotopy groups occur in the same dimension and are equal (Hurewicz Isomorphism Theorem).

Introduction to geometric K-theory (Joel Fine)

I will define the K-theory of vector bundles over a compact Hausdorff space, and show that it defines a cohomology theory. I will then sketch a proof of Bott periodicity, the amount of detail dictated by the amount of time available. I will assume very little prior knowledge about vector bundles, although I may well state foundational results without proofs.

Autumn 2003

Representing 4-manifolds: Kirby calculus and applications (Daniele Zuddas, Scuola Normale Superiore, Pisa)

This is a presentation of basic ideas and some advanced and recent results in 4-manifold theory; particular attention to examples is given. topics: Handlebody decompositions, focusing on dimension 3 and 4. Heegaard splittings of 3-manifolds. Montesinos's theorem and Kirby diagrams. Kirby moves for 4-manifolds. Mapping class groups of closed surfaces and Dehn twists. Surgery presentation of 3-manifolds and the Lickorish–Wallace theorem. Kirby diagrams and Kirby moves for 3-manifolds. Some applications of Kirby moves. Lefschetz fibrations and branched covers. Open book decompositions of 3-manifolds and contact structures. Kirby diagrams of Lefschetz fibrations. Ribbon surfaces in the 4-ball. 4-manifolds with boundary as branched covers of the 4-ball and as Lefschetz fibrations on the 2-ball The Hilden–Montesinos theorem. Symmetrization of branch links in the 3-sphere and universal links. The 4-dimensional case. Rudolph's theorem about positive braids. Eliashberg's theorem and Stein surfaces with boundary as branched covers of the 4-ball. Stein-fillable 3-manifolds. Open problems.

The Adams spectral sequence and the Steenrod algebra (Michael Tweedale)

I'll start with some gentle motivation for cohomology operations, and cite the Steenrod squares as an example. This leads to the definition of the Steenrod algebra. Then a brief excursion into the Realm of Algebra to gather up basic definitions and properties of Hopf algebras, which lets us say something about the structure of the Steenrod algebra.

We then start afresh with a rapid history of stable homotopy groups of spheres. I'll try to explain where the Adams spectral sequence comes from, and the ideas that underlie it as far as I understand them. I'll explain how the Steenrod algebra enters into the picture in a crucial way, and if time permits I might attempt a few simple calculations.

Realising homology classes by submanifolds (Tim Perutz)

In this talk I'll be indulging in some really old topology (Thom, 1954) in the hope that it will be unfamiliar to others as it was to me. Problem: Which of the homology classes of a smooth, closed, oriented manifold can be realised as smoothly embedded submanifolds? The Pontrjagin–Thom construction gives an effective criterion, which leads to results both positive and negative. Sample: there's always a submanifold representing some multiple of a chosen class; on the 10-manifold Sp(2), the generator of H_7=Z cannot be realised. The proofs involve cohomology operations and obstruction theory.

Quantisation and C*-algebras (Sebastien Racaniere)

Roughly speaking, a strict deformation quantisation is like a deformation quantisation where one can give values to the parameter h and make it tend to zero. More precisely, it is a family of C^* algebras parametrised by h. In this talk, after defining the relevant tools, I will explain what is the quantisation of the dual of a Lie algebra (with its natural Poisson structure). The answer is that it is more or less the convolution algebra of L^1 functions on the group itself

Stability of algebraic varieties (Julius Ross)

I will start by describing what stability means in general,and give some stability notions for projective manifolds (or projective varieties). I will assume very little algebraic geometry, and no knowledge of geometric invariant theory, despite talking ad nauseum about it in junior geometry last year.

Then I will talk a little about the conjectured correspondence between stability and the existence of Kähler metrics of constant scalar curvature. Finally I will define a notion of slope for a variety (and its subvarieties) which give interesting examples that are not stable.

Constant scalar curvature metrics on fibred Kähler surface (Joel Fine)

I will begin by briefly describing some of the reasons to be interested in constant scalar curvature Kähler metrics. Then I will go on to discuss a specific existence theorem for such metrics on a special class of compact complex surfaces. I will outline the proof (a so-called adiabatic limit). If there is time, I will describe the correct generalisation to higher dimensional complex manifolds.

Operads and the moduli space of curves (Kevin Costello)

Operads are a formalization of the notion of algebraic structure. There are operads describing associative algebras, Lie algebras , commutative algebras, Poisson algebras…. Operads also arise in geometry and string theory. I'll start by defining operads and the bar construction (which is a duality between operads). Then I'll describe the "open string" operad, which is an operad of singular chains on certain moduli spaces of decorated Riemann surfaces. The main result is that a type of bar construction, applied to the operad of associative algebras, yields the open string operad.

JSJ decomposition of groups (Henry Wilton)

In the late '70s Jaco and Shalen and, independently, Johannson proved a theorem of Waldhausen on canonical splittings of 3-manifolds along essential embedded tori. Group theorists interpret this result as a classification of the splittings of the manifold's fundamental group over free abelian subgroups of rank 2. I'll describe a construction that classifies more general splittings of arbitrary finitely-presented groups.

Hyperkähler extension of Teichmüller space (Thomas Hodge)

Results of Feix and Kaledin show that any Kähler manifold has an essentially unique Hyperkähler extension inside its cotangent bundle. Since Teichmüller space with the Weil–Petersson metric is a Kähler manifold, it is natural to try and construct its Hyperkähler extension. This has recently been done by Donaldson. I shall (try and) outline this work and also describe an attempt to explicitly identify the resulting moduli space with the classical quasi-Fuchsian moduli space of Kleinian group theory.

Compact G2 manifolds (Diego Matessi)

I will sketch Joyce's construction of compact 7 manifolds with holonomy equal to the exceptional group G2. It contains many interesting ideas from analysis and algebraic geometry: Hodge theory, resolution of singularities, Riemannian geometry. I will explain in particular the construction of the most simple G2 manifold, which was inspired by Kummer's K3 surface.

Summer term 2003

Hyperbolic spaces, R-trees and outer automorphism groups (Henry Wilton)

The notion of negative curvature can be extended to general metric spaces, and has important applications in group theory. We start with the definition of a hyperbolic metric space, and apply this to groups via the Cayley graph. A particularly important case is that of R-trees, which arise naturally as degenerations of hyperbolic metric spaces. This idea is applied to prove a theorem of Paulin's.

Winter term 2003

Morse theory on polyhedral cell complexes (Michael Tweedale)

We define polyhedral cell complexes, develop a theory there which closely parallels the familiar Morse theory for smooth manifolds, apply the results to a celebrated family of examples (certain cubical complexes), and then run out of time before we can explain why anyone wanted to do any of this in the first place.

De Rham homotopy theory (Tim Perutz)

The complex of differential forms on a manifold contains more information than meets the eye. On a simply-connected manifold it encodes all the homotopy data defined over the reals. Taking cohomology usually wastes all the information contained in the non-closed forms, but on a Kähler manifold you can read off the real homotopy groups from the cohomology ring. This is all old news—it comes from a beautiful 1975 paper of Deligne–Griffiths–Morgan–Sullivan, and I'll do no more than explain some bits of the paper.

Jet methods for symmetries of Partial Differential Equations (Gianni Manno)

In this seminar the definition of a (non linear) PDE, its symmetries and its solutions is described in the geometrical setting of Jet spaces. Jet spaces are differential manifolds of independent and dependent variables. Any differential equation can be understood as a submanifold of a suitable jet space, provided with the Cartan distribution. In this seminar we shall discuss the Lie–Backlund theorem, which permits us to classify contact transformations and then symmetries of a given equation.

Geometric theory of the measure and Heisenberg group space (Andrea Mirand)

In this seminar the notion of functions of bounded variation is described in the setting of metric measure spaces. The development of a geometric measure theory in this setting is motivated by a wide literature concerning Sobolev spaces in this abstract framework. Also the notion of sets of finite perimeter is given, and some of its fine properties are investigated, such as representation formula and locality properties. There are many examples of such spaces, such as a wide class of Riemannian manifolds, particular sub-Riemannian manifolds, the Carnot–Caratheodory spaces, and also many fractal sets. In this seminar we will focus our attention on the Heisenberg group, a particular example of a Carnot–Caratheodory space.

The ADHM Contruction of Instantons—big matrices with nasty coefficients (Jonathan Munn)

The ADHM construction for me is one of the great mysteries of gauge theory. Take certain matrices with quaternionic coefficients, bung 'em together, stir nicely and out pops a nice little instanton, or vector bundle over S^4 with ASD connection to spell it out in full. What's more, but take a simple quotient of the space of these matrices by a compact Lie group, and the resulting space is actually isomorphic to the moduli space of instantons under gauge equivalence. Which is nice.

Autumn 2002

Scalar curvature and variations of complex structure (Joel Fine)

Given a Kähler manifold, its group of symplectomorphisms acts on the space of compatible complex structures. I will attempt to explain Donaldson's calculation of the moment map for this action. I will basically be looking in detail at his original paper on the subject which appears in "Fields medallists' lectures," and hopefully managing some of the calculations in it which are left to the reader.

Gentle Introduction To Geometric Invariant Theory (Julius Ross)

I will give a very gentle introduction to geometric invariant theory and the notion of stability. My aim is to talk about the Hilbert–Mumford criterion and give an example concerning the stability in the space of sets of points on the projective line. This example is a good illustration of the connection with symplectic reduction and the moment map which is the topic for the subsequent talk by Tom.

Mirror Symmetry nach Leung, Yau und Zaslow (Tommaso Paccini)

MS is an ugly beast (might this explain the German title?), partly because everyone talks about it but no-one has yet figured out what it exactly is. Happily, LYZ have provided a very basic example (cfr. math.DG/0005118), which I will try to pass on to the audience. The seminar should be fairly simple, without much background needed. Bring girlfriends, cushions and ear-plugs.

Singularities of Complex Analytic Curves (Michael Tweedale)

Singularities of Complex Analytic Curves We try to understand the geometrical content of a definition by Zariski of 'equisingularity' (when should we say two singularities are 'the same'?) by developing the very pretty, very geometric, and distinctly left-field theory of infinitely-near points, and look briefly at other approaches to equisingularity.

More Geometric Invariant Theory—this time applied to the Moduli of Sheaves (Julius Ross)

After a recap of my last talk I will discuss geometric invariant theory applied to the moduli space of sheaves (via Simpson's construction). Restricting to the case of vector bundles on curves I aim to show that stability in the sense of geometric invariant theory becomes the well known notion of slope stability. If time allows I will discuss how this applies to a base of higher dimension.

Summer term 2002

Introduction to toric varieties (Julius Ross)

I will give a very gentle introduction to the correspondence between toric varieties and the combinatorial study of cones and fans. It will be a summary of the lectures given last term by Richard Thomas. If time allows I will talk about 1-parameter subgroups and show when a fan represents a compact toric variety. Although everything I say will over the complex numbers I have been led to believe that most of the theory is true over arbitrary fields.

Mean Curvature Flow for Dummies (Tommaso Pacini)

MCF of a submanifold is a difficult problem, but if the submanifold has symmetries we may reduce the number of variables, hopefully making things easier. The goal of the seminar will be to examine the simplest case, ie MCF of the orbits of isometry groups of the ambient space. The final picture constitutes, for several reasons, a good "example zero" of MCF. It is simple; it generalizes the standard example of the "shrinking sphere" in R^n; it is codimension-independent; and especially, in the orbit setting, "everything we might want to be true for MCF, is true". We will also discuss some applications.

An instanton over a product of curves; an example of an adiabatic limit (Joel Fine)

The adiabatic limit technique is used to solve PDEs on fibre bundles. If fibrewise behaviour of the PDE is understood then an approx soln is produced by stretching the base out very large. Hopefully is the local geometry beomes dominated that of the fibres. The hard part is showing that the approx soln can be perturbed to a genuine one. As an example I will prove the following: Let E —> S x S' be a holo bundle over a product of curves with a Hermitian inner prod s.t. the restricted bundles E_s —> S' are all flat and irreducible. Then E admits an instanton. (Algebraic geometers note: there is an alg-geom version of this which you can amuse yourselves trying to prove during the talk whilst I go through the analysis; all will be explained.)

Kähler & symplectic toric varieties (Aleksis Raza)

Guillemin constructed a way in which to study T^m-symmetric Kähler manifolds by looking at their representations as polytopes in R^m=Lie(T^m). So a Kähler toric manifold is a Kähler manifold M equipped with the Hamiltonian action of T^m, and its polytope P is the image of M under the moment map (which coincides with a Legendre transformation). The motivation is to do Kähler geometry in symplectic (i.e. action-angle) coordinates on P, rather than in holomorphic coordinates on M. If time permits, I shall show a new construction of scalar-flat Kähler metric on C^2 blown up at the origin, working on the polytope.

Winter term 2002

p-adic analysis & easy rigid analytic varieties (Ed Nevens)

In these talks, I'll try to give a brief overview of the theory of rigid analytic spaces, starting from the very beginning. Talk 1 will be an introduction to the p-adic numbers, largely by example and NO PRIOR KNOWLEDGE WILL BE ASSUMED. Rigid analysis takes place over a field, k, that is complete with respect to a non-archimedean absolute value. In talk 2, I'll introduce rigid analytic spaces (or varieties) which are analogues of complex analytic varieties that work over such fields. In fact, for any (reduced) scheme X of finite type /k there is a functor X—>X^an which associates a rigid analytic space to X. When studying important arithmetic objects, such as the supersingular locus of the modular curve X_1(N) (which is neither Zariski open nor closed) it has often proved useful to study the associated rigid analytic space. In fact, Buzzard and Taylor have provided powerful applications to the theory of Galois representations.

Handle Bodies and Heegaard diagrams (Joel Fine)

A diffeomorphism of a closed surface S can be used to glue together two copies of the 3-manifold whose boundary is S. This gives a closed 3-manifold Y, and this decomposition of Y is called a Heegaard splitting. I will explain why all 3-manifolds admit a Heegaard splitting and how these can be described via Heegaard diagrams. I will focus on results and examples needed for the course Brendan is giving on new invariants for 3 and 4-manifolds.

Noether's theorem (Gianluca Allemandi)

The seminar will deal about the geometric objects needed to define conserved quantities in field theories via the Noether theorem. This formalism is particularly important both from a mathematical and a physical point of view as it is intrinsically global and covariant. We introduce the concept of k-order jet bundle and of variational morphisms which are the natural framework for field theories and for the variational calculus. The definition of a Lagrangian for a field theory as a bundle morphism, will allow us to introduce the Noether theorem for a field theory which admits symmetries.

Autumn 2001

Riemann–Roch Theorem For Surfaces (Tim Perutz)

The original Riemann–Roch theorem determined, for generic divisors D on a compact Riemann surface R, the dimension of the vector space of meromorphic functions on R with poles "at worst D". Hirzebruch's generalisation is harder to understand: he expresses the holomorphic Euler characteristic of a vector bundle E on a complex manifold M as a polynomial in the Chern classes of E and of M. The aim of this talk is to bridge the gap by looking at the case of complex surfaces.

Clifford algebras, spin structures & the Dirac operator (Aleksis Raza)

My intro to clifford algebras will be purely geometrical i.e. dealing with the case of a euclidean vector space over the reals. note that, strictly speaking, no knowledge of clifford algebras is necessary to define spin structures on riemannian manifolds. but they are useful when defining the positive dirac operator as an elliptic operator translating +ve spinors (i.e. sections of +ve spin bundle) to negative spinors (i.e. sections of the -ve spin bundle) (and vice versa for its formal adjoint). bear in mind that im a 4-manifoldist, and generalization to higher dimensions is not one of my specialities.

Equivalent formulations of the Kähler condition on a complex manifold (Tom Hodge)

There are various equivalent definitions of the Kähler condition on a complex manifold. However it requires a bit of work to see why the conditions are indeed equivalent. In this talk i shall demonstrate the key result that grad w = 0 iff dw=0 for the 1,1 form w.

Sobolev spaces (James Smith)

Sobolev spaces are essentially normed vector spaces of L^p functions whose (weak) derivatives are also L^p functions. Their importance lies in the fact that they are complete (i.e. Banach spaces). I shall begin the talk by showing how they arise naturally when trying to solve PDEs, in particular I shall look at the Dirichlet problem for the Laplace equation. I shall then define the W^{k,p} spaces (where k,p are integers) and give some of their most important properties, in particular I shall look at the well known embedding theores of Sobolev and Rellich.

Hodge theory (Julius Ross)

The Hodge decomposition of the space of differential forms groups on a Riemannian manifold gives canonical representatives for each cohomology class. I intend to discuss this theory and outline the analysis used in its proof using the Sobolev theory from the previous week. Finally I shall give some applications e.g. a proof of Poincaré duality.

Back to the homepage