Past Seminars › Seminar Series

Junior Geometry and Topology Seminar (past)

Thu, 27/10/2011
12:00 		Heinrich Hartmann Junior Geometry and Topology Seminar Add to calendar SR2
Stability conditions on K3 surfaces
We will explain Bridgelands results on the stabiltiy manifold of a K3 surface. As an application we will define the stringy Kaehler moduli space of a K3 surface and comment on the mirror symmetry picture.
Thu, 20/10/2011
12:00 		Tom Sutherland Junior Geometry and Topology Seminar Add to calendar SR2
Stability conditions, rational elliptic surfaces and Painleve equations
We will describe the space of Bridgeland stability conditions of the derived category of some CY3 algebras of quivers drawn on the Riemann sphere. We give a biholomorphic map from the upper-half plane to the space of stability conditions lifting the period map of a meromorphic differential on a 1-dimensional family of elliptic curves. The map is equivariant with respect to the actions of a subgroup of $ \mathrm{PSL}(2,\mathbb Z) $ on the left by monodromy of the rational elliptic surface and on the right by autoequivalences of the derived category. The complement of a divisor in the rational elliptic surface can be identified with Hitchin's moduli space of connections on the projective line with prescribed poles of a certain order at marked points. This is the space of initial conditions of one of the Painleve equations whose solutions describe isomonodromic deformations of these connections.
Thu, 13/10/2011
12:00 		Maria Buzano Junior Geometry and Topology Seminar Add to calendar L3
Type I singularities and ancient solutions of homogeneous Ricci flow
We will present a class of compact and connected homogeneous spaces such that the Ricci flow of invariant Riemannian metrics develops type I singularities in finite time. We will describe the singular behaviours that we can get, as we approach the singular time, and the Ricci soliton that we obtain by blowing up the solution near the singularity. Finally, we will investigate the existence of ancient solutions when the isotropy representation decomposes into two inequivalent irreducible summands.
Fri, 24/06/2011
12:00 		Steven Rayan (University of Oxford) Junior Geometry and Topology Seminar Add to calendar L3
Betti numbers of twisted Higgs bundles on P^1
As with conventional Higgs bundles, calculating Betti numbers of twisted Higgs bundle moduli spaces through Morse theory requires us to study holomorphic chains. For the case when the base is P^1, we present a recursive method for constructing all the possible stable chains of a given type and degree by representing a family of chains by a quiver. We present the Betti numbers when the twists are O(1) and O(2), the latter of which coincides with the co-Higgs bundles on P^1. We offer some open questions. In doing so, we mention how these numbers have appeared elsewhere recently, namely in calculations of Mozgovoy related to conjectures coming from the physics literature (Chuang-Diaconescu-Pan).
Fri, 17/06/2011
12:00 		Benjamin Volk (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Gromov-Witten Invariants and Integrality
We will give a quick and dirty introduction to Gromov-Witten theory and discuss some integrality properties of GW invariants. We will start by briefly recalling some basic properties of the Deligne Mumford moduli space of curves. We will then try to define GW invariants using both algebraic and symplectic geometry (both definitions will be rather sloppy, but hopefully the basic idea will become visible), talk a bit about the axiomatic definition due to to Kontsevich and Manin, and discuss some applications like quantum cohomology. Finally, we will talk a bit about integrality and the Gopakumar-Vafa conjecture. Just as a word of warning: this talk is intended as an introduction to the subject and should give an overview, so we will perhaps be a bit sloppy here and there...
Fri, 10/06/2011
12:00 		Michael Groechenig (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Fundamental groups and positive characteristic
In spirit with John's talk we will discuss how topological invariants can be defined within a purely algebraic framework. After having introduced étale fundamental groups, we will discuss conjectures of Gieseker, relating those to certain "flat bundles" in finite characteristic. If time remains we will comment on the recent proof of Esnault-Sun.
Fri, 03/06/2011
12:00 		John Calabrese (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Some random facts about the Weil conjectures
I'll start by defining the zeta function and stating the Weil conjectures (which have actually been theorems for some time now). I'll then go on by saying things like "Weil cohomology", "standard conjectures" and "Betti numbers of the Grassmannian". Hopefully by the end we'll all have learned something, including me.
Fri, 27/05/2011
12:00 		Shane Kelly (Universite Paris 13) Junior Geometry and Topology Seminar Add to calendar SR1
Derived categories of coherent sheaves and motives
The derived category of a variety has (relatively) recently come into play as an invariant of the variety, useful as a tool for classification. As the derived category contains cohomological information about the variety, it is perhaps a natural question to ask how close the derived category is to the motive of a variety. We will begin by briefly recalling Grothendieck's category of Chow motives of smooth projective varieties, recall the definition of Fourier-Mukai transforms, and state some theorems and examples. We will then discuss some conjectures of Orlov http://arxiv.org/abs/math/0512620, the most general of which is: does an equivalence of derived categories imply an isomorphism of motives?
Fri, 20/05/2011
12:00 		Laura Schaposnik (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Spectral data for principal Higgs bundles
In this talk I shall present some ongoing work on principal G-Higgs bundles, for G a simple Lie group. In particular, we will consider two non-compact real forms of GL(p+q,C) and SL(p+q,C), namely U(p,q) and SU(p,q). By means of the spectral data that principal Higgs bundles carry for these non-compact real forms, we shall give a new description of the moduli space of principal U(p,q) and SU(p,q)-Higgs bundles. As an application of our method, we will count the connected components of these moduli spaces.
Thu, 12/05/2011
13:00 		David Hume (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Topological Borovoi's theorem
Mikhail Borovoi's theorem states that any simply connected compact semisimple Lie group can be understood (as a group) as an amalgam of its rank 1 and rank 2 subgroups. Here we present a recent extension of this, which allows us to understand the same objects as a colimit of their rank 1 and rank 2 subgroups under a final group topology in the category of Lie groups. Loosely speaking, we obtain not only the group structure uniquely by understanding all rank 1 and rank 2 subgroups, but also the topology. The talk will race through the elements of Lie theory, buildings and category theory needed for this proof, to leave the audience with the underlying structure of the proof. Little prior knowledge will be assumed, but many details will be left out.
Thu, 05/05/2011
13:00 		Moritz Rodenhausen (University of Bonn) Junior Geometry and Topology Seminar Add to calendar SR1
Normal Forms, Factorability and Cohomology of HV-groups
A factorability structure on a group G is a specification of normal forms of group elements as words over a fixed generating set. There is a chain complex computing the (co)homology of G. In contrast to the well-known bar resolution, there are much less generators in each dimension of the chain complex. Although it is often difficult to understand the differential, there are examples where the differential is particularly simple, allowing computations by hand. This leads to the cohomology ring of hv-groups, which I define at the end of the talk in terms of so called "horizontal" and "vertical" generators.
Thu, 10/03/2011
13:00 		Camilo Arias Abad (University of Zurich) Junior Geometry and Topology Seminar Add to calendar SR1
The $ A_\infty $ de Rham theorem and higher holonomies
I will explain how Chen's iterated integrals can be used to construct an $ A_\infty $-version of de Rham's theorem (originally due to Gugenheim). I will then explain how to use this result to construct generalized holonomies and integrate homotopy representations in Lie theory.
Thu, 24/02/2011
13:00 		Dirk Schlueter (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Yang-Mills theory, Tamagawa numbers and matrix divisors
The topology of the moduli space of stable bundles (of coprime rank and degree) on a smooth curve can be understood from different points of view. Atiyah and Bott calculated the Betti numbers by gauge-theoretic methods (using equivariant Morse theory for the Yang-Mills functional), arriving at the same inductive formula which had been obtained previously by Harder and Narasimhan using arithmetic techniques. An intermediate interpretation (algebro-geometric in nature but dealing with infinite-dimensional parameter spaces as in the gauge theory picture) comes from thinking about vector bundles in terms of matrix divisors, generalising the Abel-Jacobi map to higher rank bundles. I'll sketch these different approaches, emphasising their parallels, and in the end I'll speculate about how (some of) these methods could be made to work when the underlying curve acquires nodal singularities.
Thu, 17/02/2011
13:00 		Ben Davison (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
The geometry and topology of chromatic polynomials
I will talk about a recent paper of Huh, who, building on a wealth of pretty geometry and topology, has given a proof of a conjecture dating back to 1968 regarding the chromatic polynomial (the polynomial that determines how many ways there are of colouring the vertices of a graph with n colours in such a way that no vertices which are joined by an edge have the same colour). I will mainly talk about the way in which a problem that is explicitly a combinatorics problem came to be encoded in algebraic geometry, and give an overview of the geometry and topology that goes into the solution. The talk should be accessible to everyone: no stacks, I promise.
Thu, 10/02/2011
13:00 		Imran Qureshi (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Graded rings and polarised varieties
Many classes of polarised projective algebraic varieties can be constructed via explicit constructions of corresponding graded rings. In this talk we will discuss two methods, namely Basket data method and Key varieties method, which are often used in such constructions. In the first method we will construct graded rings corresponding to some topological data of the polarised varieties. The second method is based on the notion of weighted flag variety, which is the weighted projective analogue of a flag variety. We will describe this notion and show how one can use their graded rings to construct interesting classes of polarised varieties.
Thu, 03/02/2011
13:00 		Victoria Hoskins (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Quotients of group actions in algebraic and symplectic geometry
Consider the action of a complex reductive group on a complex projective variety X embedded in projective space. Geometric Invariant Theory allows us to construct a 'categorical' quotient of an open subset of X, called the semistable set. If in addition X is smooth then it is a symplectic manifold and in nice cases we can construct a moment map for the action and the Marsden-Weinstein reduction gives a symplectic quotient of the group action on an open subset of X. We will discuss both of these constructions and the relationship between the GIT quotient and the Marsden-Weinstein reduction. The quotients we have discussed provide a quotient for only an open subset of X and so we then go on to discuss how we can construct quotients of certain subvarieties contained in the complement of the semistable locus.
Thu, 27/01/2011
13:00 		Martin Palmer (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Homological stability of configuration spaces
I will first introduce and motivate the notion of 'homological stability' for a sequence of spaces and maps. I will then describe a method of proving homological stability for configuration spaces of n unordered points in a manifold as n goes to infinity (and applications of this to sequences of braid groups). This method also generalises to the situation where the configuration has some additional local data: a continuous parameter attached to each point. However, the method says nothing about the case of adding global data to the configurations, and indeed such configuration spaces rarely do have homological stability. I will sketch a proof – using an entirely different method – which shows that in some cases, spaces of configurations with additional global data do have homological stability. Specifically, this holds for the simplest possible global datum for a configuration: an ordering of the points up to even permutations. As a corollary, for example, this proves homological stability for the sequence of alternating groups.
Thu, 20/01/2011
13:00 		Tom Sutherland (University of Oxford) Junior Geometry and Topology Seminar Add to calendar SR1
Stability conditions for curves
This talk will be an introduction to the space of Bridgeland stability conditions on a triangulated category, focussing on the case of the derived category of coherent sheaves on a curve. These spaces of stability conditions have their roots in physics, and have a mirror theoretic interpretation as moduli of complex structures on the mirror variety. For curves of genus g > 0, we will see that any stability condition comes from the classical notion of slope stability for torsion-free sheaves. On the projective line we can study the more complicated behaviour via a derived equivalence to the derived category of modules over the Kronecker quiver.
Wed, 08/12/2010
12:00 		Shane Kelly Junior Geometry and Topology Seminar Add to calendar SR1
A very brief introduction to stable $ \AA^1 $-homotopy theory
$ \AA^1 $-homotopy theory is the homotopy theory for smooth algebraic varieties which uses the affine line as a replacement for the unit interval. The stable $ \AA^1 $-homotopy category is a generalisation of the topological stable homotopy category, and in particular, gives a setting where algebraic cohomology theories such as motivic cohomology, and homotopy invariant algebraic $ K $-theory can be represented. We give a brief overview of some aspects of the construction and some properties of both the topological stable homotopy category and the new $ \AA^1 $-stable homotopy category.

For any corrections/updates to these listings, please contact seminars [-at-] maths [dot] ox [dot] ac [dot] uk.

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