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There is a Pure Mathematics Colloquium on each Friday in term time, at 4pm in Room 211. After the colloquium we go for drinks in the Everyman at 6pm and then to dinner.
For confirmation and further information please contact Jonathan Woolf on (0151) 794 4052.
Vladimir Guletskii (Liverpool) "Motivic finite-dimensionality and algebraic cycles on threefolds over a field"
Let X be a non-singular projective threefold over a field of characteristic zero, and let A^2(X) be the group of algebraically trivial codimension 2 algebraic cycles on X modulo rational equivalence with coefficients in Q. Assume X is birationally equivalent to another threefold X' admitting a fibration over an integral curve C whose generic fiber X'_{\bar \eta } satisfies the following three conditions: (i) the motive M(X'_{\bar \eta }) is finite-dimensional, (ii) H^1_{et}(X_{\bar \eta },\mathbb Q _l)=0 and (iii) H^2_{et}(X_{\bar \eta },\mathbb Q_l(1)) is spanned by divisors on X_{\bar \eta }. The main result in the lecture is as follows. Provided the above three assumptions, the group A^2(X) is representable in the weak sense: there exists a curve Y and a correspondence z on Y\times X, such that z induces an epimorphism A^1(Y)\to A^2(X), where A^1(Y) is Pic ^0(Y) tensored with Q. In particular, the result holds for three-dimensional Del Pezzo fibrations over curves over C. The lecture will start with a brief introduction to Chow groups, motives and some basic conjectures on algebraic cycles.
Alastair King (Bath) "Why are moduli spaces projective varieties?"
I will talk about why spaces which parametrise certain types of objects naturally have homogeneous coordinates? Starting from Plucker coordinates on Grassmannians and determinantal semi-invariants of Kronecker modules, I will explain how a new perspective on the construction of moduli spaces of vector bundles and coherent sheaves yields an improved understanding of theta functions, the homogeneous coordinates on such spaces.
Dmitry Anosov (Moscow) "On hyperbolic diffeomorphisms of the 2-dimensional torus"
This is a survey talk on results of Zhirov, Klimenko, and Kolyutskiy. Classical theorems by Lagrange and Gauss can be reformulated in terms of dynamics on the 2-dimensional torus. We describe a new way to classify hyperbolic diffeomorphisms of the torus up to topological conjugacy. Related properties of simplest Markov partitions are discussed.
Prof. Anosov's visit is funded by an LMS Scheme 2 grant.
Simon Willerton (Sheffield) "Gerbes and topological quantum field theory"
The intention is to make the following paragraph intelligible by the end of the talk:
A 3-d topological field theory associates a vector space to each surface and linear map to a 3-d cobordism between surfaces. For example, in Chern-Simons theory the vector space of a surface is some space of sections of a certain line bundle over an associated moduli space. Evidence suggests that many 3-d TQFTs can be extended `downwards' to also get `invariants' of 1-manifolds and 0-manifolds, but the `invariants' here are not vector spaces or numbers but rather categories and 2-categories. In simple cases these can be obtained by taking `spaces of sections' of `gerbes' over suitable moduli space.
Andreas Langer (Exeter) "Introduction to de Rham-Witt cohomology"
For an algebraic variety (i.e a curve or a surface) defined over a finite field the number of rational points on the variety can be expressed via the Frobenius action on a suitable cohomology theory. If the variety is smooth and proper crystalline cohomology is such a good cohomology theory. One way to desribe it is via a complex of sheaves on the the ring of Witt vectors, the de Rham Witt complex which was invented by Bloch, Deligne and Illusie.
In my talk I will present this complex and discuss a generalization of it to a relative situation, i.e when we consider a family of varieties or a scheme over p-adic base rings. To understand relative crystalline cohomology will become important in deformation theory. For smooth nonproper varieties (for examples affines) the right cohomology to consider is rigid cohomology.
At the end I will briefly discuss work in progress on the construction of an overconvergent de Rham Witt complex which ought to compute rigid cohomology. This is joint work with Thomas Zink.
Mike Prest (Manchester) "The spectrum of a module category".
The typical module category is far too complicated to understand in any detail so it is interesting to find structures (which may well be topological or geometric) which organise some of the information they contain. I will describe one such structure: it is like the Zariski spectrum of a ring but carried one representation level up.
Nige Ray (Manchester) "The rise of toric topology"
Toric topology has recently come of age as a branch of algebraic topology, and concerns itself with well-structured actions of the compact torus T^n. It has roots in algebraic and symplectic geometry, and was initiated by Davis and Januszkiewicz's paper of 1991. Much of the appeal of the subject lies in its interdisciplinary nature, as it now draws additional inspiration from category theory, cobordism theory, combinatorics, complex and convex geometry, differential topology, and homotopy theory. I shall try to provide an overview of the subject by describing basic examples and background material from some of the sources above. My hope is to be able to enthuses a general audience, by working at an appropriate level and modifying the material according to audience reaction as the talk proceeds!
Anna Pratoussevitch (Liverpool) "Fundamental Domains in Lorentz Geometry"
We describe a construction of fundamental domains for an action of a group of isometries on a quadric in a pseudo-Euclidean space. Our main example is the Lie group $SL(2,R)$ equipped with the Lorentzian metric of constant curvature induced by the Killing form. It can be identified with a quadric in the $4$-dimensional pseudo-Euclidean space of signature $(2,2)$. A discrete subgroup $G$ of $SL(2,R)$ acts on it by left translations. We also generalise the construction to obtain fundamental domains in covering spaces of quadrics, for example in the universal cover of $SL(2,R)$. The quotients of the universal cover of $SL(2,R)$ by certain discrete subgroups can be identified with the links of quasi-homogeneous Gorenstein singularities. The quasi-homogeneous singularities of Arnold's series $E, Z, Q$ are of this type. We compute the fundamental domains for the corresponding groups. They are represented by polyhedra in Lorentz $3$-space. Each series exhibits a regular characteristic pattern of its combinatorial geometry related to classical uniform polyhedra.
Catharina Stroppel (Glasgow) "From the Jones polynomial to Khovanov homology from the perspective of representation theory"
In this talk I will give an overview about the connections between the Jones polynomial, Khovanov homology and knots invariants coming from the quantum group of sl(2). All this is from a perspective of representation theory. It will turn out that all the mentioned invariants emerge naturally from the representation theory of the Lie algebras sl(n) (for various n)
Oleg Viro (Uppsala) "Tropics and subtropics of Algebraic Geometry"
Algebraic Geometry has a piecewise linear core visible in logarithmic coordinates. This core was recently named Tropical Geometry. The relation between Tropical and Algebraic Geometries is similar to that between Classical and Quantum Mechanics. Deformations similar to quantization turn tropical varieties into usual complex or real algebraic varieties. This is used as a powerful way to construct algebraic varieties with interesting properties.
Sabir Gusein-Zade (Liverpool / Steklov Institute) "What is the meaning of the expression (1 + A_1 t + A_2 t^2 + ...)^M where A_1, A_2,..., and M are varieties?"
I'll describe the meaning of the expression mentioned in the title. This gives a structure of a new sort over the, so called, Grothendieck ring of complex quasi-projective varieties. It turns out that this structure can be useful for formulation and proof of some formulae for generating series of invariants of configurations spaces (e.g. Hilbert schemes of zero-dimensional subschemes ("fat points") on a smooth quasi-projective variety). (Joint results with I.Luengo and A.Melle, Madrid).
Peter Kropholler (Glasgow) "High dimensional cohomology of groups"
Yann Rollin (Imperial) "Construction of Kaehler surfaces with constant scalar curvature"
A new construction is presented of constant scalar curvature Kaehler metrics on non-minimal ruled surfaces. The method is based on the resolution of singularities of orbifold ruled surfaces which are closely related to rank-2 parabolically stable holomorphic bundles. This rather general construction is shown also to give new examples of low genus: in particular, it is shown that CP^2 blown up at 10 suitably chosen points, admits a scalar-flat Kaehler metric; this answers a question raised by Claude LeBrun in 1986 in connection with the classification of compact self-dual 4-manifolds.
Alexander Odesskii (Manchester / Landau Institute) "Algebraic structures connected with pairs of compatible associative algebras"
We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM-structures naturally leads to affine Dynkin diagrams of A, D, E-type.
Farid Tari (Durham) "Projections in hyperbolic space"
(Joint work with Shyuichi Izumiya) There is a great deal of work on projections of surfaces in the Euclidean space R3 to planes, with important applications to differential geometry and computer vision. I will talk about recent work with Shyuichi Izumiya on an analogous study for surfaces in the hyperbolic space. I will first define what we mean by a projection in this context. The projections we deal with are given by families of maps. The singularities of the members of these families give us geometric information about the surface itself. I will mention Koendrink type theorems that give the curvature of the surface in terms of the curvatures of the contour and the normal section of the surface. I will also present some duality results concerning the bifurcation sets of the families of projections.
John Greenlees (Sheffield) "Equivariant cohomology theories with algebraic models"
Morita theory provides a method for showing that mysterious categories are categories of modules over a ring E. This is only helpful if the ring E is under good control. The talk will give some examples to show that superficial knowledge of E is not enough, and examples where Koszul duality replaces E by a ring which is rigid. The particular example I care about comes from equivariant topology, but most of the talk is not specific to that context.
Theodore Voronov (Manchester) "Operators on superspaces and generalizations of the Gelfand-Kolmogorov theorem"
Gelfand and Kolmogorov in 1939 proved that any compact Hausdorff space X is canonically embedded into the infinite-dimensional vector space C(X)^* (the dual space to the algebra of continuous functions C(X)) as an "algebraic variety" specified by an infinite number of quadratic equations of the form f(a^2)=f(a)^2. Buchstaber and Rees have recently extended this to all symmetric powers Sym^n(X) using their notion of "Frobenius n-homomorphisms". We give a simplification and a further extension of this theory, basing, rather unexpectedly, on results from super linear algebra. It is a joint work with H. Khudaverdian. See arXiv:math.RA/0612072.
Philip Boyland (Florida)
Andriy Bovykin (Liverpool) "Unprovable statements about braids, zeta-function, analytic combinatorics and diophantine approximation."
My talk will be about recent developments in the interdisciplinary study of Unprovability in Mathematics. I shall start off with a listener-friendly survey of unprovable statements from Godel till nowadays, with particular emphasis on concrete arithmetical and combinatorial unprovable assertions. This is intended to be accessible to a widest audience without any special expertise in logic. The second half of the talk will cover my recent results on braid groups (long games on braids), unprovable statements about the Riemann zeta-function, the study of analytic threshold phenomena (a passage between provability and unprovability in combinatorics) and simple unprovable statements from analysis based on simultaneous Diophantine approximation theory. Are there truly difficult mathematical questions? I expect mathematicians, philosophers and computer scientists to be equally entertained by this story.