This page is maintained by Alex Kasprzyk. It was last updated on 23rd November 2005.
This page contains abstracts from previous Calf seminars. They are listed alphabetically by speaker. For a listing by date, please return to the main Calf page.
Stacks are a more general object than schemes. Their definition is very abstract; they are in fact categories, and only after some work can their geometric structure be understood. We will start part I by defining categories fibred in groupoids, with emphasis on examples and moduli interpretations. From there we will move on to Deligne-Mumford stacks, and define the etale topology on these.
In part II we review the definition of categories fibered in groupoids. We then go on to look at how these may be assigned geometric properties, via representable morphisms, and define Deligne-Mumford stacks.
We propose an interpretation of the topological framing of a knotas a generating function for a Lagrangian submanifold of a symplectic manifold; the setting is Brylinski's space of knots (embedding of S^1 in R^3) and Maslov theory for Lagrangian submanifold. Examples and applications to existence of linear bundles with prescribed curvature will be given.
Del Pezzo surfaces provide beatiful examples of rational surfaces. Whilst their geometry is classical, they are still somewhat myserious from the point of view of a number theorist. A basic property of such surfaces is that there are infinitely many rational points on the surface provided there is at least one. I will discuss two basic questions:
For given a vector space V with a symplectic form we define a subvariety in P(V)to be legendrian if its affine cone has a Lagrangian tangent space at each smooth point. We can prove that the ideal defining a legendrian subvariety is a Lie subalgebra of the ring of functions on V with Poisson bracket. Specially interesting is the case where the ideal is generated by quadratic functions - then we can restrict our considerations to a finite dimensional Lie algebra which happens to be isomorphic to the symplectic algebra of V. Next we prove that the subgroup of Sp(V) corresponding to the subvariety act transitively on smooth points of the subvariety (in particular, if it is smooth then it is homogeneous).
The next goal is fully classify legendrian subvarieties generated by quadrics. There are not to many examples: twisted cubic in P^3, product P^1 times Q_{n-2} in P^{2n-1} (where Q_{n-2} is a smooth quadric in P^{n-1} and the embedding is Segre embedding) and four more exceptional examples. This list appears in a paper of Landsberg and Manivel and also in Mukai "Legendre varieties and simple Lie algebras". The conjecture is that these are all possible smooth legendrian subvarieties generated by quadrics. Moreover no singular example is known - so possibly the assumption of smoothness is not necessary.
For more details see the preprint math.AG/0503528.
For complex manifolds the "ddbar" lemma implies that the massey products vanish and this is how it is proved that Kaehler manifolds are formal. For symplectic manifolds satisfying the Lefschetz property Merkulov proved a similar lemma using symplectic operators analogous to d and dbar. The question that arises is "do Massey products vanish for such manifolds?" I intend to give an example of symplectic manifold satisfying the Lefschetz property with nonvanishing Massey products.
Notes for this talk are available.
I'll introduce generalized complex structures and go through their basic properties, as determined by Gualtieri in his thesis. I'll show how symplectic and complex structures fit into this generalized setting. Then I'll present some results about existence of such structures on manifolds that do not admit either complex or symplectic structures. I'll also go through a classification of generalized structures on 6-nilmanifolds and give results about their moduli space.
Notes for this talk are available.
Let V be a representation of a finite group G. Then the symmetric (S) and exterior (L) algebras of V are Koszul dual (over k), in the sense that S \otimes L^* is a bigraded algebra with a natural differential ofdegree (1,-1) which is exact (except in degree (0,0)). The exactness ofthe differential gives a well-known recurrence for the symmetric powers of a representation in terms of tensor products of exterior and symmetric powers. In particular, this gives a recurrence on the McKay matrices of these representations.
In order to see how the matrix recurrence arises in a more direct way, we should consider the following: Given a left kG-module W, define a twisted bimodule structure on Twist(W) = W \otimes kG, where the right action is (right) multiplication in kG and the left action is both the left action on W and (left) multiplication in kG. The McKay matrix of W now coincides with its decomposition matrix in terms of the irreducible kG-bimodules. Furthermore, it can be shown that Twist(S) and Twist(L) are Koszul dual rings (over kG). Hence, the recurrence on the McKay matrices reflects the fact that the differential respects the grading induced by the projectors onto the irreducible kG-bimodules.
We also discuss the related theory of almost-Koszul rings and their connection with periodic recurrences.
The derived category is a powerful homological tool and is the correct way of understanding derived functors such as Tor and Ext. Many important papers in algebraic geometry and mathematical physics are now being written in the language of derived categories, most notably Bridgeland, King and Reid's "Mukai implies McKay: the McKay correspondence as an equivalence of derived categories".
We begin by discussing abelian categories, which satisfy precisely the axioms needed to define homology. Next we encounter a problem for which homology isn't quite powerful enough to find a solution - this leads naturally to the definition of the derived category of an abelian category. The algebraic structure of the derived category is that of a triangulated category and we finish by trying to understand the interplay between abelian, derived and triangulated categories.
To a finite subgroup of SL(n) we associate a graph. We explore the possibility of classifying such graphs, and representation theory highlights a recurrence relation for which these graphs exhibit unusual behaviour. We reduce the qualitative behaviour under this recurrence to a quantitative test, which the rational conformal field theory graphs also appear to satisfy.
In subsequent talks we discuss how this test may betray the existence of a pair of Koszul or almost-Koszul dual algebras associated to the path algebra of the graph.
In the study of 3-dimensional manifolds, two of the most useful structures to have are hyperbolic geometric structures and essential surfaces lying in the manifold. Each of these relates to a different kind of representation of the fundamental group. A series of papers by Marc Culler and Peter Shalen has examined the interaction between these, using some algebraic geometry of the character variety to provide the connection. Culler-Shalen theory continues to provide new insights and deep theorems in 3-manifold topology. I intend to give an accessible introduction to the various ideas involved.
Notes for this talk are available.
I will spend half the talk motivating the search for constant scalar curvature Kahler metrics. In particular I will explain why these special metrics should be of use in studying "the majority of" smooth algebraic varieties (i.e. stably polarised ones). In the other half of the talk I will explain how to use an analytic technique called an adiabatic limit to prove the existence of constant scalar curvature Kahler metrics on a special type of complex surface.
This talk is based on the preprint math.DG/0401275.
Slides for this talk are available.
We consider fully effective orientation-preserving smooth actions of a given finite group G on smooth, closed, oriented 3-manifolds. We investigate the relations that necessarily hold between the numbers of fixed points of various non-cyclic subgroups. All such relations are in fact equations mod 2, and the number of independent equations yields information concerning low-dimensional equivariant cobordism groups. We determine all the equations for non-cyclic subgroups G of SO(3).
This talk is based on the preprint math.AT/0301159.
The classifying space BG of an algebraic group G can be approximated by algebraic varieties and therefore has a well-defined Chow ring CH^*BG, which is useful in the study of varieties acted on by G. Conjecturally this is the same as another ring defined topologically, namely using complex cobordism. These rings come equipped with a natural map to the ordinary cohomology ring. After explaining this in some detail I will give some examples of computations, using tools like the Steenrod algebra or the Morava K-theories.
Topological quantum field theories (TQFTs) have proven an interesting tool in topology, providing invariants of 3-manifolds; to every (three-dimensional) TQFT there is a "quantum invariant". But how does one construct a TQFT? One solution is through using categories with extra structure, such as a tensor product. In this talk I plan to define both TQFTs and an important class of category (braided categories) that can be used in the construction of TQFTs. I will also give a few examples that demonstrate how familiar braided categories really are.
In this talk we study geometric properties of singular del Pezzo surfaces with log terminal singularities of index less then or equal to 2. We study their (in some way) canonical embedding and use it to describe them with equations in some weighted projective space.
The aim of the talk is to describe linear systems of an irreducible curve on an Enriques surface, and the maps associated with these linear systems. We ask when a map is a morphism, what is the degree of this morphism, and describe the eventual singularities by looking at the image.
The aim of this talk is to describe linear systems on K3 surfaces. We are mostly concerned with their base points (or components), the morphism associated with them and its image. We also try to introduce the notion of Seshadri constants and we show some examples of linear systems on some K3 surface where we can compute them.
Toric varieties form an important class of algebraic varieties whose particular strength lies in methods of construction via combinatorial data. Understanding this construction has led to the development of a rich dictionary allowing combinatorial statements to be translated into algebraic statements, and vice versa.
In the first talk the basic details of the combinatorial approach to constructing toric varieties are given. The construction is motivated by specific examples from which the more general methods can be deduced.
The second talk will concentrate on the torus action on the variety. We will discuss the orbit closure and introduce the "star" construction. Finally, we shall apply what we have learnt to toric surfaces, analysing the singularities and seeing how they are resolved.
Notes for the first talk are available.
The talk will be in two parts. In the first I will explain what model theory is, and how it can be thought of as a generalization of the study of zeros of polynomials (aka Algebraic Geometry). In the second I will explain how simple geometric ideas crop up naturally in model theory. The aim is to give an overview of the ideas rather than any technicalities, and no familiarity with logic will be assumed.
We define false weighted projective spaces as toric varieties with fan constructed from vectors v_0,...,v_n in lattice Z^n with sum_{i=0}^n (a_i * v_i)=0 for some integers a_i. The only difference of this fan and the one of weighted projective space is that the v_i's do not span the lattice. False weighted projective space are quotients of P(a_0,...,a_n) by the action of a finite group. We distinguish false ones by introducing the fundamental group in codimension 1 and proving it is non-trivial exactly for false ones. False projective spaces are quotients of P^n and they are orbifolds. We conjecture a generalization of Mori theorem characterizing P^n as the only projective varieties with ample tangent bundle.
We will introduce projective geometry in the language of schemes. Starting from the projective spectrum of a graded ring we will explain some basic properties of projective varieties.
People tend to think about "schemes" to be synonymous with "Algebraic Geometry" but this is not quite true. As a result learning the machinery can be really frustrating, as our geometric intuition doesn't seem to fit into the picture. Actually the theory of schemes is far more general than Algebraic Geometry, and many concepts arising in the geometric context make sense only for a particular kind of schemes usually called "algebraic schemes". I will define algebraic varieties from this point of view, avoiding too much abstract nonsense and retaining the geometric point of view in evidence.
Detailed notes on scheme theory are available.
Artin level algebras are zero dimensional graded algebras, they are a generalization of Gorenstein algebras. I will describe their Hilbert function and their graded minimal resolution.
We give necessary conditions for the {3,5,3} Coxeter group to surject onto PSL(2,p^n). We also look at some of the manifolds arising from the low index normal subgroups of this group.
Riemann spheres are extremely useful in the study of two-dimensional conformal field theories. One can ask what is the corresponding structure to look at if one wishes to study a superconformal field theory. One way of introducing anti-commuting co-ordinates is to consider the sheaf functions on the Riemann sphere, and extend them by anti-commuting variables. This can be more useful than a superspace formalism, since there is still a notion of a "patching function" on intersections of "co-ordinate patches".
This talk is based on the preprint hep-th/0309243.
An introduction to the Penrose twistor corresponce will be presented. We will begin by discussing the correspondence between conformal four-manifolds and appropriate complex three-manifolds. In particular, our discussion will include the usual Penrose transform in this case. We will then discuss the various generalisations of the correspondence to other dimensions and geometric structures. We will conclude by describing some of the applications of twistor theory to gauge theory (monopoles and instantons), existence of complex structures and deformations of hypercomplex structures.
This talk will be the continuation of my talk in the Calf seminar in Liverpool (January 2005). I will spend a few minutes talking about reflection groups in integral hyperbolic lattices, and use this machinery to define the geometric realisation of a generalised Cartan matrix. There will be a short introduction to infinite-dimensional Lie algebras, based on the theory that we will give for generalised Cartan matrices.
Notes for this talk are available.
This is an introductory talk on reflection groups of integral hyperbolic symmetric bilinear forms. Lobachevskii (hyperbolic) geometry is a strong tool in mathematics, and lots of problems which appeared in algebraic geometry have been attacked using this tool. We will divide the lecture into two parts; in the first one we will present all the preliminaries, and in the second part we will formulate Vinberg's algorithm. This algorithm permits us to find all cells of a polyhedron C of an acceptable set P(C) of orthogonal vectors to C, where C is the fundamental chamber for a subgroup W of the group W(M) (the group generated by reflections in all elements of M), where S:MxM -> Z is a given quadratic form. Hopefully, this material will be used as a basis, for a future lecture on hyperbolic Kac-Moody algebras.
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\neq 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 permitting, 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.
We give explicit bounds on the intersection number between any curve on a tight geodesic and the two ending curves. We use this to construct all tight geodesics and so conclude that distances are computable. The algorithm applies to all surfaces. The central argument makes no use of the geometric limit arguments seen in the recent work of Bowditch (2003) and Masur-Minsky (2000). From this we recover the finiteness result of Masur-Minsky for tight geodesics.
This talk is based on the preprint math.GT/0412078.
The two talks will cover some basic aspects of K3 surfaces with the following aims: First, to say what a K3 surface is and how to recognise one. Examples will be given and the difficult question of why K3s are interesting may be tackled.Second, to become familiar with some of the methods used in the study of K3 surfaces such as lattice theory and Hodge structures. Time permitting, we may look at some deeper aspects of the subject and try and build an understanding of the moduli space of K3 surfaces.
In the second talk, by looking at explicit examples, we shall illustrate some general properties of K3 surfaces. In particular, we look at variations of Hodge structure, periods and the associated Picard-Fuchs differential equation, and use these to visualise the moduli space of certain one-parameter families of K3 surfaces.
Notes for the second part of this talk are available.
We consider the action of finite subgroups of SO(4) on P^3. Recent work of W. Barth and A. Sarti provides three examples of families of K3 surfaces that arise as the quotient of invariant surfaces modulo this group action. We describe an easy way to prove this and to find more examples using graded ring methods and invariant theory. This talk will cover a basic introduction to algebraic K3 surfaces and will demonstrate the use of graded rings and weighted projective spaces to their study.
In this I will working in the context of the bounded derived categories of coherent sheaves of a fano surface Z and it's canonical bundle \omega_{Z} which is a Calabi Yau 3-fold. I will breifly state how quivers relate to t-structures and tilting them, to exceptional collections and their mutations, and if any physists are present to Seiberg duality. I will then use this to explain Tom Bridgeland's result in "T-structures on some local Calabi-Yau varieties" and if all goes well give a brief description of my current work.
The aim of this talk is to provide an alternative understanding of derived categories, focusing on using the formal definitions of a t-structure and the heart of a t-structure to get visual understanding of what a derived category is, even for those with little prior knowedge. Then depending on time and peoples interest I will use this 'picture' to give simple explainations, of things like torsion pairs, tilting with respect to torsion pairs, stability conditions (Tom Bridgelands description), etc.
We will investigate some of the geometric and number theoretic properties of curves of genus two, which admit various types of isogenies. We will look at these via covering techniques and go on to extend some of the results regarding curves with bad reduction at 2 and p, where p is some prime.
The resolution of Diophantine equations over the rationals is one with a deep history. In this talk I will consider ways to solve a general family of curves - specifically the Fermat Quartic curves (x^4+y^4=c). We present work from Flynn and Wetherell and expand upon their work. We consider a "flow diagram" approach to solving these curves and present explicit examples as well as a general method for approaching the curves. Finally we explain how these methods can be adopted to suit other diophantine equations over the rationals.
About twenty years ago, following an initial idea of Bernardi and Neron, Perrin-Riou and Schneider found a canonical p-adic height pairing on an elliptic curve defined over a number field. The associated p-adic regulator appears in the p-adic version of the conjecture of Birch and Swinnerton-Dyer, but it is still unknown if this pairing is non-degenerate except for special cases. Following the work done for the real-valued pairing, one can analyse the behaviour of the p-adic height as a point varies in a family of elliptic curves, and get so new information about this pairing.
This page is maintained by Alex Kasprzyk. It was last updated on 23rd November 2005.