Titles and Abstracts of Morning Lectures
Morning lectures
Tuesday, 19 April
Bennett Lecture Theatre 2
Minhyong Kim, University College London 09:10-10:05
Title: Homotopy theory and Diophantine geometry
Abstract: It is well-known that enormous amount of progress in 20C number theory was achieved through the incorporation of homological techniques in the study of Diophantine problems. It led, in particular, to the proof of the Weil conjectures, and also played key roles in the proof of the Mordell conjecture and Fermat's last theorem. On the other hand, it is perhaps less know that the use of homotopy theory was already advocated as early as the 1930's by Andre Weil and motivated his study of the moduli of vector bundles. We will survey recent progress on the use of homotopical ideas in Diophantine geometry, in particular, the ideas surrounding Grothendieck's anabelian geometry.
Colva Roney-Dougal, University of St Andrews 10:10-11:30
Title: Generation of finite groups
Abstract: Let $G$ be a finite group. Then by $d(G)$ we denote the smallest number of elements necessary to generate $G$. The talk will start by surveying some results on upper bounds for $d(G)$. For example, we could be given information about the structure of $G$, or told that $G$ can be represented as a matrix group in dimension $n$, or as a permutation group on $n$ points. I will go on to present some recent joint work with Derek Holt on bounds for $d(G)$, and on the number of random elements needed to generate $G$ with bounded probability.
Rattray Lecture Theatre
Jim Shank, University of Kent 09:15-10:05
Title: Modular Invariant Theory
Abstract: Let S:=F[x_1,...,x_n] denote the ring of polynomials in n variables with coefficients in the field F. A subgroup G of the general linear group GL_n(F) acts on the span of the x_i by linear transformations. This action extends to an action on S by ring automorphisms. The subring consisting of those polynomials fixed by this action, S^G, is known as the ring of invariants. Invariant Theorists are interested in the structure of S^G, in relating properties of G and S^G, and in algorithms for constructing generators for S^G. The subject has a rich history with natural links to representation theory, algebraic geometry and algebraic topology. I will discuss a variety of results and examples paying particular attention to the case when G is finite and the characteristic of F is a prime number dividing the order of G, i.e., modular invariant theory.
Dominic Joyce, University of Oxford 10:10-11:00
Title: Derived differential geometry
Abstract: There are many important areas in both differential and algebraic geometry which involve forming moduli spaces of geometric objects and then "counting" the points in the moduli space to get a number (or homology class, etc.). Examples include Donaldson invariants counting moduli spaces of instantons on 4-manifolds, Gromov-Witten invariants counting J-holomorphic curves in a complex or symplectic manifold, Donaldson-Thomas invariants, Lagrangian Floer cohomology, Fukaya categories, Symplectic Field Theory, ... In each case, the moduli space M may be a complicated and very singular topological space, but one treats it as if it is a compact oriented manifold of known dimension k. When k=0 we would like to "count" the points with signs to get an integer; for k>0 we want to form a fundamental cycle [M] in a homology group H_k(M,Z). Techniques for doing the "counting" when M is not a manifold are called virtual cycle constructions, and there are many different versions. A toy model for the problem of "counting" points in moduli spaces is the intersection of submanifolds. Suppose X,Y are compact oriented submanifolds of an oriented manifold Z, with dim X + dim Y = dim Z. If X,Y intersect transversely then X \cap Y is finitely many points, with signs, and the number counted with signs is the intersection number [X] \cap [Y] in the homology of Z. However, if X,Y do not intersect transversely then M = X \cap Y may be a complicated topological space, possibly infinite, and it is not obvious how to "count" points in M with multiplicity, without perturbing X,Y, to obtain the topological intersection [X] \cap [Y]. I will describe a new class of geometric spaces I call "d-manifolds". Manifolds are examples of d-manifolds -- that is, the category of manifolds embeds as a subcategory of the 2-category of d-manifolds -- but d-manifolds also include many spaces one would regard classically as singular or obstructed. A d-manifold has a virtual dimension, an integer, which may be negative. Almost all the main ideas of differential geometry have analogues for d-manifolds -- submersions, immersions, embeddings, submanifolds, orientations, transverse fibre products, and so on -- but the derived versions are often stronger. There are also good notions of d-manifolds with boundary and d-manifolds with corners, and orbifold versions of all this, d-orbifolds. Almost any moduli space which is used to define some kind of counting invariant will have a d-manifold or d-orbifold structure. Any intersection of submanifolds in a manifold has a d-manifold structure. Virtual cycle constructions also work for d-manifolds and d-orbifolds. Thus, much of Gromov-Witten theory, Donaldson-Thomas theory, Lagrangian Floer cohomology, Symplectic Field Theory, ... can be rewritten in this language. D-manifolds and d-orbifolds are related to other classes of spaces people already study. My main sources of inspiration were the Kuranishi spaces of Fukaya and Ono, which are geometric structures on moduli spaces of J-holomorphic curves in symplectic geometry, the derived algebraic geometry programme of Jacob Lurie, Bertrand Toen, ... , and the derived manifolds of David Spivak, a student of Jacob Lurie. The Fukaya-Ono definition of Kuranishi spaces has problems; I claim that the (morally and aesthetically) "right" definition of Kuranishi spaces is that they are d-orbifolds with corners. One difference between my theory and those of Lurie and Spivak is that mine is simpler -- Lurie's framework is frighteningly complex and involves infinity-categories (simplicial categories), homotopy sheaves, localization, ... In contrast, my d-manifolds form a 2-category rather than an infinity-category, are built using only fairly standard techniques from algebraic geometry, and do not involve localizing by a class of morphisms. D-manifolds are essentially a 2-category truncation of Spivak's derived manifolds, but I believe this truncation preserves the geometrically important information. All this is work in progress. I am writing a book on it; you can download a preliminary version from my webpage at http://people.maths.ox.ac.uk/~joyce/
Wednesday, 20 April
Bennett Lecture Theatre 2
Nadia Sidorova, University College London 09:15-10:05
Title: Localisation and ageing in the Parabolic Anderson model
Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation on the d-dimensional integer lattice with random potential. It describes the mass transport through a random field of sinks and sources and is actively studied in mathematical physics. We discuss, for a class of i.i.d. potentials, the intermittency effect occurring in the model, which manifests itself in increasing localisation and randomisation of the solution. We also discuss the ageing behaviour of the model as the time increases.
Tom Ward, University of East Anglia 10:10-11:30
Title: Group automorphisms from a dynamical point of view
Abstract: This will be a survey of the problems and phenomena that arise in attempting to describe the space of all compact group automorphisms modulo natural dynamical equivalences. This natural question turns out to involve diverse problems, and when the same question is asked for more general group actions entirely new rigidity phenomena arise.
Rattray Lecture Theatre
Martin Hyland, King’s College, University of Cambridge 09:15-10:05
Title: What is a linear theory?
Abstract: Intuitively a linear theory is one given by equations between expressions in which same list of variables occurs, with each appearing just once. One immediate consequence of this restriction is that the theory makes sense in linear algebra with the operations being taken to be multilinear. That understanding leads to the theory of operads. Another understanding less evident in the literature comes from the practice in basic group theory of multiplying sets of elements pointwise. The rough idea then is that any structure of a given kind should have its power set also a structure of the given kind in some natural way. For example the power set of a monoid is naturally a monoid. While operads give structures of this kind, there are many more. I shall discuss this wider class of theories, give examples and propose a characterization.
John Greenlees, University of Sheffield 10:10-11:00
Title: Homotopy invariant commutative algebra: rings, groups and spaces
Abstract: A valuable way of studying geometric objects is to look at their (commutative) ring of functions. We may then look at the original geometric objects through its ring of functions. For example looking for regular rings, complete intersections or Gorenstein rings gives interesting perspectives. Extending the idea of what constitutes the 'ring of functions' (and indeed, 'commutative') allows us to use this familiar idea in new contexts. To be robust, and reflect the appropriate geometry, we want to formulate the commutative algebra in a homotopy invariant way. As well as bringing out new structures of spaces (compact Lie groups at a prime, Gorenstein duality for groups, ...), this has benefits to commutative algebra itself.
Thursday, 21 April
Bennett Lecture Theatre 2
Niels Jacob, Swansea University 09:15-10:05
Title: Metric Spaces Isometric to Hilbert Spaces, Metric Measure Spaces, and the Transition Densities of Levy Processes
Joint work with Viktoria Knopova, Sandra Landwehr and Rene Schilling
Abstract: A classical theorem of Schoenberg states that if a metric space is isometrically embedded into a Hilbert space then the square of the metric is related to a continuous negative definite function g , i.e. g(0) is non-negative and for all t > 0 the function exp(-tg) is positive definite in the sense of Bochner. Restricting our considerations to the n-dimensional Euclidean space and choosing Lebesgue measure we obtain metric measure spaces related to continuous negative definite functions. Continuous negative definite functions enter Mathematics also in a quite different context: they characterize Levy processes, i.e. processes with stationary and independent increments. In this talk we discuss how the underlying metric measure spaces give us some geometric interpretation of transition densities of Levy processes. For diffusion processes such results are meanwhile best studied, not least due to the work of the E.M.Stein school on sub-elliptic differential operators of second order. For Levy processes (and more general jump-type processes) so far nothing similar is known. Indeed there is a surprise: We have to work with two (one-parameter) families of in general different metrics. However eventually we get for a large class of Levy processes results analogous to diffusions.
Steven Hurder, University of Illinois at Chicago 10:10-11:00
Title: Invitation to Matchbox Manifolds
Abstract: We take a tour through a developing area of dynamics and geometry, where the basic objects are "Matchbox Manifolds", aka "laminations", "zero-dimensional foliated spaces", "inverse limit spaces", or "exceptional minimal sets" of foliations. There are many examples, a variety of techniques and results, and many more questions. In this talk we will give a sampling of the ideas, recent results and open problems.
Rattray Lecture Theatre
MARM, Mentoring African Research in Mathematics 09:15-10:05
An LMS-IMU-AMMSI initiative The Mentoring African Research in Mathematics (MARM) scheme supports mathematics and its teaching in the countries of sub-Saharan Africa. It is designed to counter the mathematics "brain-drain" from sub-Saharan Africa by supporting qualified mathematics professionals in situ. The scheme concentrates on the creation of joint research partnerships between UK mathematicians, their colleagues in sub-Saharan Africa, and doctoral students of those colleagues. It is an initiative of the the London Mathematical Society, the International Mathematical Union (IMU), and the African Mathematics Millennium Science Initiative (AMMSI), funded by the Nuffield Foundation and the Leverhulme Trust. MARM is five years old and has set up thirteen partnerships. So far it is only a small scale project, but the results are very encouraging. This BMC meeting will provide background information about MARM, and how it complements other schemes such as the Volunteer Lecturer Programme. It will also present first-hand accounts and experiences of MARM mentors, and will invite questions and contributions. The presentation and discussion will be led by Stephen Huggett (University of Plymouth, LMS Programme Secretary, and MARM board member), Balazs Szendroi (University of Oxford and MARM mentor) and Frank Neumann (University of Leicester, MARM mentor, and MARM board member).
Konstanze Rietsch, King’s College London 10:10-11:00
Title: On mirror symmetry for Grassmannians
Joint work with Robert Marsh
Abstract: The cohomology ring of a Grassmannian is a very well understood classical object. Both it and its quantum analogue have a beautiful combinatorial structure. I will describe how these and a related structure can be obtained from a `mirror dual' perspective, using a particular rational function (called the superpotential) on a `dual' Grassmannian. This mirror picture also has combinatorial aspects coming from the cluster algebra structure of the homogeneous coordinate ring of a Grassmannian, leading to some pretty pictures.
