Dr. Herbert Gangl Polygonal combinatorics for algebraic cycles Starting from comparably explicit objects (algebraic cycles), Bloch and Kriz have given a tentative definition of a small yet rich category of motives (mixed Tate motives), at least over a field. They also exhibited a distinguished class of cycles corresponding to polylogarithms. One can also find _multiple_ polylogarithms as algebraic cycles, and it turns out that their differential structure can be conveniently described with the help of combinatorics of polygons. This leads to a coproduct on polygons which is a variant of the Connes-Kreimer coproduct on rooted trees. University of Durham herbert_dot_gangl_at_durham_dot_ac_dot_uk 4312 JCMB Thursday, September 25, 2008 Dr. Ivan Cheltsov Exceptional log del Pezzo surfaces Exceptional log Fano varieties are defined by Shokurov for inductive approach to study singularities and log Fano varieties. They have been studied by Shokurov, Prokhorov, McKernan, Keel and Kollar. Implicitly they have been studied by Tian, Phong, Sturm, Boyer and Rubinstein. We discuss some facts about exceptional log del Pezzo surfaces. University of Edinburgh i_dot_cheltsov_at_ed_dot_ac_dot_uk 4312 JCMB Thursday, October 02, 2008 Dr. Tamas Hausel Topology of the Hitchin map and the arithmetic of the character variety We suggest that the topology of the Hitchin map reflects the arithmetic of the character variety. This is a joint project with Mark de Cataldo and Luca Migliorini. Mathematical Institute, University of Oxford hausel_at_maths_dot_ox_dot_ac_dot_uk 4312 JCMB Friday, October 10, 2008 Dr. Elizabeth Gasparim Relating algebraic and symplectic geometry via instanton counting Report on joint paper with Melissa Liu on the Nekrasov conjecture. University of Edinburgh Elizabeth_dot_Gasparim_at_ed_dot_ac_dot_uk 4312 JCMB Thursday, October 16, 2008 Dr. Alastair Craw Tilting bundles on multilinear series The multilinear series of a collection of line bundles on a (toric) variety is a multigraded version of projective space. To further the analogy, I will construct a tilting bundle on each multilinear series, generalising Beilinson's celebrated theorem to the multigraded case. Department of Mathematics, University of Glasgow craw_at_maths_dot_gla_dot_ac_dot_uk http://www.maths.gla.ac.uk/~anc/ 4312 JCMB Thursday, October 23, 2008 Dr. Alexei Gorinov Division theorems for the cohomology of discriminant complements and applications to automorphism groups. The linear group $\mathrm{GL}_{n+1}(\mathbb{C})$ acts in an natural way on the space of equations of smooth complete intersections of given dimension and multidegree in the complex projective space of dimension $n$. We show that the rational cohomology Leray spectral sequence of the corresponding quotient map degenerates in the second term. The proof uses a cohomological analogue of the linking number of curves in the 3-space (the invariant which allows one, in particular, to show a physical chain can't be undone without tearing or breaking). As a by-product of the proof, we obtain explicit expressions divisible by the order the automorphism group of any smooth projective hypersurface of given dimension and degree >2. We shall also discuss some other applications and possible generalisations of these results. University of Liverpool 21 West Mayfield on 30th October. You are asked contact Lorette on 0131 668 2148 to let her know your estimated time of arrival. gorinov_at_liv_dot_ac_dot_uk 4312 JCMB Thursday, October 30, 2008 Dr. Dmitri Panov Simply connected Calabi-Yau manifolds via hyperbolic geometry Classically Calabi-Yau manifold are algebraic manifolds with a non-vanishing holomorphic volume form, they are both symplectic and complex manifolds. A well known idea from mirror symmetry is to consider separately symplectic and complex structures. So we can introduce a notion of Symplectic Calabi Yau and Complex nonalgebraic Calabi Yau. While in real dimension 4 these manifolds are expected to be very close to algebraic Calabi Yaus, in real dimension 6 the divergence is huge (and intriguing). We will show how to construct simply connected Symplectic Calabi Yaus with b_3=0 out of 4-dimensional hyperbolic geometry and how to construct Complex non algebraic 3-folds with trivial canonical class out of hyperbolic knots in S^3. This is a joint work with Joel Fine. Imperial College, London 21 West Mayfield on 6th and 7th November. Please contact Lorette on 668 2148 to let her know your estimated time of arrival. d_dot_panov_at_imperial_dot_ac_dot_uk 4312 JCMB Thursday, November 06, 2008 Dr. Claire Voisin Hodge structures on cohomology algebras and geometry We study restrictions on cohomology algebras of Kaehler compact manifolds, not depending on the h^{p,q} numbers or the symplectic structure. To illustrate the effectiveness of these restrictions, we give a number of examples of compact symplectic manifolds satisfying the Lefschetz property but not having the cohomology algebra of a compact Kaehler manifold. We also prove the stability of these restrictions under products. IHES, Directrice de recherche - CNRS 12 and 13 November at 21 Mayfield. Please contact Lorette on 0131 668 2148 to let her know your estimated time of arrival. voisin_at_ihes_dot_fr 4312 JCMB Thursday, November 13, 2008 Dr. Mario Micallef Harmonic maps between complete, noncompact negatively curved manifolds. The construction of a harmonic extension of a map between the ideal boundaries of two complete noncompact manifolds of strictly negative curvature involves two parts: (i) the construction of an extension of the boundary map to the whole space with bounds on the energy density and the tension fields, (ii) deformation of the extension in (i) to a harmonic map. I will describe recent progress due to Fotiadis, Markovic and myself concerning step (ii) which enables one to allow the boundary map to be considerably more singular than previously allowed by Li-Tam and others. I will also describe a proof by Fotiadis, different from the Teichmuller proof of Markovic, of the existence of a harmonic diffeomorphism of the hyperbolic plane which extends a symmetric map of the ideal circle. University of Warwick TBA M_dot_J_dot_Micallef_at_warwick_dot_ac_dot_uk 4312 JCMB Thursday, November 20, 2008 Dr. John Bolton The moduli space of harmonic 2-spheres in round spheres I would like to report on recent work by Luis Fernandez and myself on the above moduli space M. Firstly, Fernandez has proved a conjecture of myself and Lyndon Woodward concerning the dimension of M. Secondly, in the case when the target is the unit 4-sphere, Fernandez and myself have considered the regularity of M, and we have also investigated elements of M possessing some symmetry. Durham University none needed john_dot_bolton_at_durham_dot_ac_dot_uk 4312 JCMB Thursday, November 27, 2008 Dr. Yang-Hui He Quivers and Plethystics We present some recent progress in the study of quiver theories in the context of string and gauge theory. In particular, we study gauge theories which arise from D-branes on Calabi-Yau spaces and advocate a ``Plethystic Programme'' which provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace operators and (3) the counting of multi-trace operators. Mathematically, fascinating and intricate inter-relations between quiver gauge theories, algebraic geometry and combinatorics exhibit themselves in the form of plethystics and syzygies. Mathematical Institute, University of Oxford Yang-Hui_dot_He_at_maths_dot_ox_dot_ac_dot_uk 4312 JCMB Thursday, December 04, 2008 Prof. Gabriel Paternain Transparent connections over negatively curved surfaces A unitary connection is said to be transparent if its parallel transport along every closed geodesic is the identity. We show that transparent connections over a closed negatively curved surface can be understood in terms of Backlund transformations and that the trivial connection is locally unique. The proof also uses the non-abelian Livsic theorem and a suitable Fourier analysis on the unit tangent bundle of the surface. Cambridge University g_dot_p_dot_paternain_at_dpmms_dot_cam_dot_ac_dot_uk 4312 JCMB Thursday, January 15, 2009 Dr. Eugenie Hunsicker Pseudodifferential operators for locally symmetric spaces This talk presents the first part of an ongoing collaboration between the speaker and D. Grieser on the construction of a pseudodifferential operator calculus for locally symmetric spaces and structurally similar space. The work to date is on generalizations of Q-rank 1 spaces, that is, manifolds with boundary where the boundary has a multiple fibration structure. The general philosophy is that of the Melrose b-calculus. The talk will focus on geometric motivations, the philosophy of the construction and statements of the main results. Loughborough University E_dot_Hunsicker_at_lboro_dot_ac_dot_uk 4312 JCMB Thursday, January 29, 2009 Prof. Andrew Ranicki Michel Kervaire's work in surgery and knot theory Michael Kervaire (1927-2007) was a French topologist who worked in Switzerland and the United States. Kervaire made major contributions to our understanding of the topology of high-dimensional manifolds, using homotopy theory and quadratic forms. The talk is (an abbreviated) dress rehearsal for the 2 talks I am giving at the Kervaire memorial symposium in Geneva February 10-13 (http://www.unige.ch/math/folks/kashaev/symposiumMK.html) My talk will cover Kervaire's work in the period 1955-1965, dealing with the following highlights from his oeuvre: 1. Curvatura integra and the Kervaire semicharacteristic 2. The J-homomorphism 3. The classification (with Milnor) of exotic spheres 4. The Kervaire PL manifold without differentiable structure 5. High-dimensional knot theory. These topics have a common feature: the use of stably trivialized vector bundles in topology and quadratic forms in algebra. University of Edinburgh A_dot_Ranicki_at_ed_dot_ac_dot_uk 5215 JCMB (note unusual location) Thursday, February 05, 2009 Dr. Leo Butler Exotic tori and complete integrability In hamiltonian mechanics, a key role is played by completely integrable systems. A priori, the existence of such a system is a smooth invariant of the configuration space. This talk will demonstrate that this invariant is non-trivial. We will show that, if a smooth manifold homeomorphic to the n-torus is the configuration space of a completely integrable convex hamiltonian, then it is actually diffeomorphic to the standard n-torus. On the other hand, we will exhibit topological 7-manifolds each of whose 28 smooth structures is the configuration space of a completely integrable convex hamiltonian. A key ingredient is Viterbo's work on vanishing Maslov cycles. University of Edinburgh l_dot_butler_at_ed_dot_ac_dot_uk 4312 JCMB Thursday, February 12, 2009 Dr. Gilles Carron QALE metrics on the Hilbert Scheme of n points on C2. We show that the Hyperkahler metric introduced by H. Nakajima on the Hilbert Scheme of n points on C2 is QALE (Quasi Asymptotically Locally Euclidean). University of Nantes Gilles_dot_Carron_at_math_dot_univ-nantes_dot_fr 5215 JCMB (note unusual date and location) Wednesay, February 25, 2009 Dr. Mikhail Feigin Special representations of Cherednik algebras and quantum integrable systems We investigate radical ideals in the polynomial ring which are submodules of the polynomial representation of the rational Cherednik algebra associated with a Coxeter group. These ideals are supported on certain parabolic strata that is on the orbits of special intersections of mirrors of the Coxeter group. Dunkl operators can be restricted to these strata which leads to known and new generalisations of integrable systems of Calogero-Moser type. Similar construction exists for the double affine Hecke algebras which leads to the generalized Ruijsenaars-Macdonald systems - joint work with A. Silantyev. University of Glasgow m_dot_feigin_at_maths_dot_gla_dot_ac_dot_uk 4312 JCMB Thursday, February 26, 2009 Dr. Chris Athorne A new look at \wp-functions We use some elementary representation theory to develop new machinery for deriving and classifying differential identities satisfied by \wp-functions of plane algebraic curves. University of Glasgow c_dot_athorne_at_maths_dot_gla_dot_ac_dot_uk 4312 JCMB Thursday, March 05, 2009 Prof. Tom Bridgeland Donaldson-Thomas curve-counting invariants. Donaldson-Thomas invariants are numbers that count curves on a three-dimensional Calabi-Yau variety. Conjecturally they are closely related to Gromov-Witten invariants. I will start from the beginning and explain a little bit about how such invariants are defined. It will be useful to know what a Hilbert scheme is. Then I will state a recent result which shows that the generating functions for such invariants have certain rationality properties. University of Sheffield T_dot_Bridgeland_at_shef_dot_ac_dot_uk 4312 JCMB Thursday, March 12, 2009 Dr. Susan Cooper In search of exactness Certain data about a finite set of distinct, reduced points in projective space can be obtained from its Hilbert function. It is well known what these Hilbert functions look like, and it is natural to try to generalize this characterization to non-reduced schemes. In particular, we consider a fat point scheme determined by a set of distinct points (called the support) and non-negative integers (called the multiplicities). In general, it is not yet known what the Hilbert functions are for fat points with fixed multiplicities as the support points vary. However, if the points are in projective 2-space and the number of support points is 8 or less, we can write down all of the possible Hilbert functions for any given set of multiplicities (due to Guardo-Harbourne and Geramita-Harbourne-Migliore). In this talk we focus on what can be said, in projective 2-space, given information about what collinearities occur among the support points. Using this information we measure how far related sequences can be from being exact on global sections. Doing so, we obtain upper and lower bounds for the Hilbert function of the fat point scheme. Moreover, we give a simple criterion for when the bounds coincide yielding a precise calculation of the Hilbert function in this case. This is joint work with B. Harbourne and Z. Teitler. University of Nebraska - Lincoln scooper4_at_math_dot_unl_dot_edu 4312 JCMB Thursday, March 19, 2009 Dr. Gavin Brown Diptych varieties and the equations of Mori flips A Mori flip is a particular class of surgery operations that can be applied to a complex 3-folds: roughly speaking, a flip excises one curve and recompactifies with another, making the line bundle of rational 'volume' forms more positive and the singularities of the 3-fold more tame in the process. Their existence was proved by Mori 20 years ago, and their classification into A,D,E types completed by Kollar-Mori soon after. I will report on work with Miles Reid that computes the equations of flips of type A, extending more recent work of Mori describing a Euclidean-style algorithm in a certain group of line bundles. Our constructions are built on a new class of nearly-symmetric spaces called 'diptych varieties'. I will explain their role and the relation to the Mori-Euclidean algorithm. University of Kent G_dot_D_dot_Brown_at_kent_dot_ac_dot_uk 4312 JCMB Thursday, March 19, 2009 Dr. Joel Fine Calabi flow and projective embeddings Let X subset CP^N be a smooth projective variety. I will describe a flow, called balancing flow, which tries to move the embedding of X until it has zero centre of mass in CP^N. I will then explain how this flow is related to Calabi flow. Calabi flow is a parabolic flow for a Kahler metric which tries to deform the metric until it has constant scalar curvature. If L->X is a Kahler manifold with ample line bundle, high powers L^k of L give embeddings X->CP^N in larger and larger projective spaces. For each k, we evolve the embedding via balancing flow and restrict the Fubini-Study metric to X to get a sequence of metric flows. Prorvided the initial embeddings are correctly chosen, this sequence of flows converges to Calabi flow. Universite Libre de Bruxelles Joel_dot_Fine_at_ulb_dot_ac_dot_be TBA Wednesay, April 01, 2009 Dr. Jens Kroeske Cartan Geometries and Invariant Multilinear Differential Operators For a differential operator in Euclidean space (or better on the sphere) the notion of being invariant under conformal transformations is well understood. Recent research has extended this notion for linear differential operators to conformal manifolds and more generally to a wide class of Cartan Geometries. The classification of such invariant linear differential operators has led to many interesting results. This talk will give a brief introduction to the concept of Cartan Geometries, discuss the general notion of invariance and motivate the study of multilinear differential operators. General classification results and the ideas behind it that go back to Penrose's Twistor Theory will be presented. ThinkTankMaths jkroeske_at_gmail_dot_com 4312 JCMB Thursday, April 16, 2009 Prof. Simon Gindikin 50 years of integral geometry Rutgers University gindikin_at_math_dot_rutgers_dot_edu 4312 JCMB Thursday, April 30, 2009