CALF SEMINAR ABSTRACTS
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.
Mohammad Akhtar (Imperial College) This talk is an introduction to the theory of mutations. We will discuss two closely related viewpoints on the subject: the algebraic approach interprets mutations as birational transformations, and is concerned with their action on Laurent polynomials. The combinatorial approach sees mutations as operations on lattice polytopes, and allows one to construct deformations of the corresponding toric varieties. We will explain the role played by algebraic mutations in the program to classify Fano 4-folds. We also discuss recent results concerning combinatorial mutations of weighted projective surfaces. The contents of this talk are joint work with Tom Coates, Alexander Kasprzyk and Sergey Galkin. Elizabeth Baldwin (University of Oxford)Introduction to Deligne-Mumford Stacks; Parts I & II. 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.
Moduli of stable maps as a GIT quotient.
Oliver E. Anderson (University of Liverpool)
Gergely Berczi (University of Budapest)
Fabio Bernasconi (Imperial College)
Alberto Besana (University of Milan)
Matt Booth (University of Edinburgh)
Pawel Borowka (University of Bath)
An easy exercise or an open problem?.
Non-simple abelian varieties
Nathan Broomhead (University of Bath)
The Dimer Model and Calabi-Yau Algebras.
Tim Browning (University of Oxford)
Jaroslaw Buczynski (University of Warsaw)
Legendrian varieties. For more details see the preprint math.AG/0503528.
Vittoria Bussi (Oxford) Deformations of singularities and the intersection form. The nonsingular level manifolds of a miniversal deformation of a singularity carry an intersection pairing in homology which can be thought of geometrically as intersection of cycles. By a procedure of Givental' and Varchenko it is possible to use a nondegenerate intersection pairing to furnish the base space of the deformation with a closed 2-form. This is a symplectic form if the base space is even dimensional. The symplectic form identifies a Lagrangian submanifold in the discriminant of the deformation over which level sets share the same degeneracy type. I will explain this construction, give examples of the computation of these symplectic forms and discuss the relationship between the coefficients of the form and the equations of the Lagrangian submanifold.
Francesca Carocci (Imperial College)
Gil Cavalcanti (University of Oxford) Notes for this talk are available.
Examples of generalized complex structures. Notes for this talk are available.
Andrew Chan (Warwick) Gröbner bases have several nice properties that mean that certain problems in algebraic geometry can be reduced to the construction of a Gröbner basis. For example Gröbner bases allows us to easily determine whether a polynomial lives in some ideal, find the solutions to systems of polynomial equations, as well as having applications in robotics. In this talk I shall introduce Gröbner bases and see the problems that arise when trying to adapt this theory to polynomial rings over fields with valuations. We shall discuss how these Gröbner bases are interesting to algebraic geometers and how they have important applications to tropical geometry.
Emily Cliff (Oxford)
Giulio Codogni (Cambridge)
Alex Collins (University of Bath)
Gaia Comaschi (Université des Sciences et Technologies de Lille 1) Koszul Duality and Twisted Group Algebras. 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.
An introduction to Derived Categories.
McKay Matrices, CFT Graphs, and Koszul Duality (Part I).
Stephen Coughlan (University of Warwick)
Dougal Davis (LSGNT)
Ruadhaí Dervan (Cambridge)
Carmelo Di Natale (Cambridge) Tilting, derived categories and non-commutative algebras. We can describe the derived categories of coherent sheaves on certain simple spaces by a method called tilting. This gives an equivalence of the derived category with another category built from a certain non-commutative algebra. We will work this out in some simple cases.
The McKay Correpsondence.
Vivien Easson (University of Oxford) Notes for this talk are available.
Vladimir Eremichev (University of Warwick)
Daniel Evans (University of Liverpool) Andrea Fanelli (Imperial College London) Lifting Theorems in Birational Geometry In this talk, I will try to convince you of how important lifting pluri-canonical sections is. Two main approaches can be used: the algebraic one, based on vanishing and injectivity theorems, and the analytic one, which relies on Ohsawa-Takegoshi type L^2-extension theorems. Bypassing as much as possible the birational mumbo jumbo, I will eventually discuss the Dlt Extension Conjecture proposed by Demailly, Hacon and Păun. Enrico Fatighenti (University of Warwick) Hodge Theory via deformations of affine cones Hodge Theory and Deformation Theory are known to be closely related: many example of this phenomenon occurs in the literature, such as the theory of Variation of Hodge Structure or the Griffiths Residues Calculus. In this talk we show in particular how part of the Hodge Theory of a smooth projective variety X with canonical bundle either ample, antiample or trivial can be reconstructed by looking at some specific graded component of the infinitesimal deformations module of its affine cone A. In an attempt of a global reconstruction theorem we then move to the study of the Derived deformations of the (punctured) affine cone, showing how to find amongst them the missing Hodge spaces. Aeran Fleming (University of Liverpool) Kähler packings of projective, complex manifolds. In this talk I will introduce the notion of Kähler packings and explore their connections to multipoint Seshadri constants and Nagata's conjecture. I will then briefly present a general strategy to explicitly construct Kähler packings on projective, complex manifolds and if time permits discuss some examples of blow ups of the complex projective plane. Joel Fine (Imperial College London)Constant scalar curvature Kahler metrics on fibred complex surfaces. 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.
Peter Frenkel (Budapest University of Technology & Economics) This talk is based on the preprint math.AT/0301159.
Pierre Guillot (University of Cambridge)
Eloïse Hamilton (University of Oxford)
Umar Hayat (University of Warwick) Thomas Hawes (University of Oxford) GIT for non-reductive groups Geometric invariant theory (GIT) is concerned with the question of constructing quotients of algebraic group actions within the category of varieties. This problem turns out to be sensitive to the kind of group being considered. When a reductive group G acts on a projective variety X, Mumford showed how to find an open subset X^s of X (depending on a linearisation of the action) that admits an honest orbit space variety X^s/G. Moreover, this admits a canonical compactification X//G, obtained by taking Proj of the finitely generated ring of invariant sections of the linearisation. This rather nice picture breaks down when the group G is not reductive, since there is the possibility of non-finitely generated rings of invariants. This talk will look at work being done to describe a similar Mumford-style picture for non-reductive group actions. After reviewing Mumford's result for reductive groups, we will look at the work done by Doran and Kirwan on GIT for unipotent group actions, which provide the key for formulating GIT for general algebraic groups. We will finish by looking at work in progress on how to extend the ideas of Doran and Kirwan to the case where the group is not unipotent. David Holmes (University of Warwick)Jacobians of hyperelliptic curves. Jacobians of curves are the natural higher-dimensional analogues of elliptic curves, and many of the familiar properties of elliptic curves carry over. In particular, the Mordell Weil theorem (that the group of rational points over a number field is finitely generated) holds on any Jacobian, and the proof is again based on a theory of heights. After giving basic definitions, we will look at how to use this to find an algorithm to compute the torsion part of the Mordell-Weil group of the Jacobian of a hyperelliptic curve, giving a method to explicitly construct the Jacobian and exploring why this isn't enough.
Julian Holstein (University of Cambridge)
Vicky Hoskins (University of Oxford)
Daniel Hoyt (University of Cardiff) Anton Isopoussu (Cambridge) K-stability, convex cones and fibrations Test configurations are a basic object in the study of canonical metrics and K-stability. We introduce two ideas into the theory. We extend the convex structure on the ample cone to the set of test configurations. The asymptotics of a filtration are described by a convex transform on the Okounkov body of a polarisation. We describe how these convex transforms change under a convex combination of test configurations. We also discuss the K-stability of varieties which have a natural projection to a base variety. Our construction appears to unify several known examples into a single framework where we can roughly classify degenerations of fibrations into three different types: degenerations of the cocycle, degenerations of the general fibre and degenerations of the base.
Seung-Jo Jung (Warwick)
Anne-Sophie Kaloghiros (Cambridge)
Grzegorz Kapustka and Michal Kapustka (Jagiellonian University, Krakow)
Grzegorz Kapustka (Jagiellonian University, Krakow)
Michal Kapustka (Jagiellonian University, Krakow)
Alexander Kasprzyk (University of Bath) Notes for the first talk are available.
Recognising toric Fano singularities.
What little I know about Fake Weighted Projective Space.
Jonathan Kirby (University of Oxford)
Weronika Krych (University of Warsaw) Roberto Laface (Leibniz Universität Hannover) Decompositions of singular Abelian surfaces Inspired by a work of Ma, in which he counts the number of decompositions of abelian surfaces by lattice-theoretical tools, we explicitly find all such decompositions in the case of singular abelian surfaces. This is done by computing the transcendental lattice of products of isogenous elliptic curves with complex multiplication, generalizing a technique of Shioda and Mitani, and by studying the action of a certain class group act on the factors of a given decomposition. Incidentally, our construction provides us with an alternative and simpler formula for the number of decompositions, which is obtained via an enumeration argument. Also, we give an application of this result to singular K3 surfaces.
Marco Lo Giudice (University of Bath, and University of Milan)
Introduction to schemes. Detailed notes on scheme theory are available. Artin level
algebras.
Cormac Long (University of Southampton)
Andrew MacPherson (Imperial College London)
A non-archimedean analogue of the SYZ conjecture
Diletta Martinelli (Imperial College London)
Mirko Mauri (LSGNT)
Francesco Meazzini (Sapienza Università di Roma)
Caitlin McAuley (University of Sheffield)
Carl McTague (University of Cambridge)
Ben Morley (University of Cambridge) Graded Riemann spheres. 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.
Ciaran Meachan (University of Edinburgh)
Oliver Nash (University of Oxford)
Igor Netay (HSE, Moscow)
On A-infinity algebras of highest weight orbits
Alvaro Nolla de Celis (University of Warwick) Claudio Onorati (University of Bath) Moduli spaces of generalised Kummer varieties are not connected Using the recent computation of the monodromy group of irreducible holomorphic symplectic (IHS) manifolds deformation equivalent to generalised Kummer varieties, we count the number of connected components of the moduli space of both marked and polarised such manifolds. After recalling basic facts about IHS manifolds, their moduli spaces and parallel transport operators, we show how to construct a monodromy invariant which translates this problem in a combinatorial one and eventually solve this last problem. John Christian Ottem (University of Cambridge) Ample subschemes We discuss how various notions of positivity of vector bundles is related to the geometry of subschemes. Asymptotic cohomological functions.Asymptotic cohomological functions were introduced by Demailly and Küronya to measure the growth rate of the cohomology of high tensor powers of a line bundle L. These functions generalize the volume function of a line bundle and capture a lot of the positivity properties of L. In this talk I will review some recent results on them by Demailly, Küronya and Matsumura and explain how they compare with other notions of weaker positivity of a line bundle.
Kyriakos Papadopoulos (University of Liverpool) Notes for this talk are available.
Reflection groups of integral hyperbolic lattices.
Andrea Petracci (Imperial College London) Labelled Seeds and Mutation Groups This talk will introduce labelled seeds, whose definition is a modification of that of seeds of a cluster algebra. Under this new definition, the cluster algebra itself will be unchanged, but the set of labelled seeds will form a homogeneous space for a a group of mutations and permutations. We will study the automorphism group of this space, and conclude that for certain mutation classes, the orbits of this automorphism group consist of seeds with "the same cluster combinatorics", in the sense that their quivers are all related by opposing some connected components. Knowledge of cluster algebras will not be assumed, and indeed one goal is to provide an introduction to the subject, albeit in a slightly esoteric way.
Ice Quivers with Potential and Internally 3CY Algebras.
Thomas Prince (Cambridge)
Qiu Yu (University of Bath)
Lisema Rammea (University of Bath)
Construction of Non-General Type surfaces in P^4_w.
Nils Henry Rasmussen (University of Bergen)
Jorgen Rennemo (Imperial College)
Sönke Rollenske (Imperial College London)
Taro Sano (University of Warwick)
Deformations of weak Fano manifolds
Shu Sasaki (Imperial College London)
Danny Scarponi (Oxford/Tolouse)
Ed Segal (Imperial College London)
Crepant resolutions and quiver algebras Superpotential algebras from three-fold singularities. The orbifold X = C^3 / Z_3 is a simple but interesting example of a (non-compact) Calabi-Yau threefold. Physicists predict that type II string theory on X reduces in the low-energy limit to a gauge theory, which is described by a quiver and a superpotential. We'll discuss how these objects arise mathematically.
Lars Sektnan (Imperial College)
Yuhi Sekiya (University of Nagoya)
Michael Selig (Warwick University) We use the following well-known graded ring construction: given a polarised variety (X,D), under certain assumptions the graded ring R(X,D) = ⊕ n≥0H0(X,nD) gives an embedding XProj(R(X,D)) ∈ wℙ. It is well known that the numerical data of (X,D) is encoded in the Hilbert series PX(t) := ∑ n≥0h0(X,nD)tn. We aim to break down the Hilbert series into terms associated to the orbifold loci of X. The talk should be fairly introductory. I will explain the ideas behind the work from scratch, exhibit some results in 3-D and explain some ideas for the 4-D case.
Orbifold Riemann-Roch and Hilbert Series.
Kenneth Shackleton (University of Southampton) This talk is based on the preprint math.GT/0412078.
Alexander Shannon (University of Cambridge)
Geometry without geometry.
Dirk Schlueter (University of Oxford)
YongJoo Shin (Sogang University)
James Smith (University of Warwick) Notes for the second part of this talk are available.
K3s as quotients of symmetric surfaces.
David Stern (University of Sheffield)
Vocabulary made easy.
Jacopo Stoppa (Imperial College)
Andrew Strangeway (Imperial College) Stability conditions for the one-arrow quiver. Stability conditions are needed in order to construct nice moduli spaces, the classical example being vector bundles over a curve. Spaces of stability conditions of Calabi-Yau threefolds are also important in studying mirror symmetry which is a duality for Calabi-Yau threefolds arising in string theory. In this talk we will give an introduction to stability conditions in algebraic geometry and then study the space of stability conditions of a particularly simple CY3 category described by the one-arrow quiver Affine cubic surfaces and cluster varieties In this talk we will consider affine cubic surfaces obtained as the complement of three lines in a cubic surface where it intersects a tritangent plane. We will interpret certain families of these affine cubic surfaces as moduli spaces of local systems on the punctured Riemann sphere. We will see how to draw quivers on the sphere so that the associated cluster variety is related to the total space of these families.
Rosemary Taylor (University of Warwick)
Elisa Tenni (University of Warwick)
Alan Thompson (University of Oxford)
Models for Threefolds Fibred by K3 surfaces of Degree Two.
Andrey Trepalin (HSE, Moscow)
Jorge Vitoria (University of Warwick)
Anna Lena Winstel (TU Kaiserlautern)
John Wunderle (University of Liverpool)
Jacobians of hyperelliptic curves.
Christian Wuthrich (University of Cambridge)
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CALF SEMINAR
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