Differential Geometry Seminar

Spring 2016

Meetings in spring 2016 will be on Monday afternoons, starting at 2.00, for one hour, unless otherwise stated.

The venue is Hicks J 11 unless otherwise stated.

Previous talks

Spring 2015

Autumn 2014

Spring 2014

  • Wednesday June 4: 4pm to 5pm -- note unusual day and time; this talk is for one hour only

    Elizaveta Vishnyakova (Luxembourg) Quadratic symmetric $n$-ary superalgebras via a "derived bracket'' construction

    An $n$-ary symmetric superalgebra is called quadratic if its multiplication is invariant with respect to a non-degenerate skew-symmetric form. Here one has to be careful with "skew-symmetric'' since we work in superspace!

    In superspace "skew-symmetric'' means that our form is skew-symmetric on the even part and symmetric on the odd part. Therefore, this class contains for example all Lie algebras with non-degenerate symmetric form, their $n$-ary generalizations and all commutative algebras with non-degenerate skew-symmetric form. The main observation here is that we can obtain all these algebras using a ``derived bracket'' approach from Poisson geometry.

    Our talk is devoted to a discussion of quadratic symmetric $n$-ary superalgebras and other applications of "derived brackets''.

  • Monday May 19: 2pm to 3pm -- note unusual time; this talk is for one hour only

    Brent Pym (Oxford) Categorified isomonodromic deformations via Lie groupoids

    Given a meromorphic connection on a Riemann surface, one can seek deformations of the connection in which the locations of the poles are varied but the monodromy and Stokes data are held fixed. It is well known that this "isomonodromy" condition actually characterizes the deformation up to isomorphism, suggesting that it should be implemented by a functor. I will describe joint work in progress with Marco Gualtieri, in which we construct this functor as an instance of Morita equivalence between Lie groupoids. These Morita equivalences are themselves the solutions of an isomonodromic deformation problem, for which the initial condition is the meromorphic projective connection provided by the uniformization theorem.

  • Monday May 12: John Rawnsley (Warwick) Constructing central extensions of the real symplectic group

    Let $(V,\Omega)$ be a finite dimensional real symplectic vector space. Its real symplectic group \(Sp(V,\Omega)\) is the group of linear transformations of \(V\) which preserve \(\Omega\). The maximal compact subgroup of \(Sp(V,\Omega)\) is isomorphic to a unitary group which can be constructed by taking a positive compatible complex structure \(J\) on \(V\). This means \(J \in Sp(V,\Omega)\), \(J^2 = -1\) and \(\Omega(v,Jv) > 0\) for all \(v\ne0\) in \(V\) and the unitary subgroup is the centraliser of \(J\).

    It follows that the fundamental group of \(Sp(V,\Omega)\) is the integers. We are interested in two connected covering groups, the universal cover and the double cover. The latter, the metaplectic group, is the analogue of the spin group of a symmetric bilinear form. Unlike the spin case neither of these larger groups is linear making them awkward to work with. Similarly the automorphism group of the canonical commutation relations is a central extension of the symplectic group by a circle, the symplectic analogue of \(Spin^c\), and is used in geometric quantisation to construct half-forms.

    We show how the above choice of a positive compatible \(J\) gives a means to construct all these groups as well as a compatible construction of the Lagrangian Grassmannian useful in calculations with Maslov indices.

  • Monday May 5: Bank Holiday

  • Monday April 28:

    Dr Vishnyakova has been unable to obtain a visa in time and the seminar is postponed to a date to be fixed.

    Elizaveta Vishnyakova (Luxembourg) On n-ary analogues of Lie (super)algebras

    A different reading of the standard Jacobi identity leads to various generalizations of the notion of Lie superalgebra for n-ary case. The most popular n-ary analogues were suggested by V. T. Filippov, P. Michor, A. Vinogradov, M. Vinogradov and others. For instance, A. Vinogradov and M. Vinogradov introduced a two parameter series of n-ary Lie superalgebras. The interesting fact here is that this series contains also commutative associative algebras.

    We will discuss the following: this theory in the context of quadratic n-ary Lie superalgebras using a "derived bracket" approach from Poisson Geometry; classification of simple n-ary Lie algebras and a decomposition of such algebras into elementary pieces.

  • Monday March 31: Elizabeth Mansfield (Kent) Moving frames and the calculus of variations

    The modern group-based definition of a moving frame allows for a symbolic invariant calculus that applies equally to discrete and smooth systems. In this talk I will present an introduction to these ideas and give an application to variational problems with a Lie group symmetry. Noether's Theorem guarantees such systems have conservation laws - typically conservation of energy and linear and angular momentum. I will show how the use of a moving frame adds information about the structure of the set of extremals and the conservation laws, in some cases yielding a complete solution.

  • Monday March 24: Cesare Tronci (Surrey) The geometry of collisionless kinetic theories

    Kinetic theories of multiparticle systems are dynamical continuum models governing the evolution of a probability density function on phase space. These theories are well known to possess a Lie-Poisson structure on the Poisson algebra of Hamiltonian functions. Recently, the statistical method of moments has been shown to possesses momentum map features conferring moments the same Lie algebra structure as the symbols of differential operators. The geometry underlying these structures involves coadjoint orbits on the group of strict contactomorphisms (aka quantomorphisms) of the prequantization bundle. This talk reviews recent progress on these topics and shows how certain moment closures produce integrable systems such as the Camassa-Holm equation on the diffeomorphism group and Bloch-Iserles system on the Jacobi group.

  • Monday March 17: Madeleine Jotz Lean, Dorfman connections and Courant algebroids.

    I will explain how linear connections are useful for the study of tangent spaces of vector bundles. I will then define Dorfman connections and explain how they arise naturally in the study of Courant algebroids, before explaining how a certain class of Dorfman connections can be used to describe the standard Courant algebroid over a vector bundle. (The standard Courant algebroid over a manifold is the direct sum of its tangent bundle with its cotangent bundle, equipped with a bracket extending the Lie bracket of vector fields on the tangent side.)

    This talk should be particularly accessible to postgraduate students.

  • Monday March 10: No seminar.

  • Monday March 3: James Montaldi (Manchester), Topology of spaces of symmetric loops

    There is a classical result that the set of connected components of the loop space of a manifold is in 1-1 correspondence with the conjugacy classes of the fundamental group of the manifold. Motivated by the study of symmetries of planar choreographies, I will describe the extension to symmetric loops (periodic orbits) of this result. Joint work with my PhD student Katie Steckles.

  • Monday February 24: Silvia Sabatini (Lisbon). Chern numbers of compact symplectic manifolds with a circle action

    In this talk I will discuss some problems related to the classification of some equivariant topological invariants of compact symplectic manifolds with a symplectic circle action.

    In particular I will present some recent results of my work currently in progress, as well as those obtained with L. Godinho in arXiv:1206.3195 [math.SG] and A. Pelayo in arXiv:1307.6766 [math.SG], involving the Chern numbers of the manifold, and show how these can be used to: (a) classify the equivariant cohomology ring and Chern classes when the action is Hamiltonian, and (b) give a lower bound on the number of fixed points when the action is not Hamiltonian and the first Chern class of the manifold vanishes.

  • Monday February 17: No Differential Geometry seminar.

    Kirill's informal talk on quotients of Lie groupoids and Lie algebroids will take place on another occasion. (This is the global theory behind the infinitesimal ideal systems of Madeleine's talk in the PM Colloquium.)

  • Monday February 10: Theodore Voronov (Manchester). On volumes of classical supermanifolds.

    As has been established in recent decades, constructions of algebra and geometry possess a natural self-consistent extension (some view it as "closure") in which the usual notion of commutativity is replaced by its \({\mathbb Z}_2\)-graded analog. This is known as superalgebra and supergeometry and has its origins in unification of bosons and fermions in quantum field theory. In particular, supermanifolds are the corresponding generalizations of ordinary manifolds (differentiable or complex-analytic). The concept of volume in the super world may show unexpected features. For example, the total volume of a supermanifold may vanish. This is due to the fact that the underlying "Berezin integral" is not of the usual measure-theoretic nature, so considerations based in positivity are not applicable. A striking example is the superanalog of the unitary group, the unitary supergroup \(U(n|m)\), whose total "Haar measure" is zero whenever \(nm > 0\) (Berezin, 1970s).

    Recently Witten raised a question as to whether the Liouville volume of every compact symplectic supermanifold is zero. The answer is negative, as is shown by a counterexample. As such one can take the superanalog of a complex projective space endowed with the analog of the classical Fubini--Study form. The explicit formula for the volume of this complex projective superspace \({\mathbb C}P^{n|m}\) is an analytic continuation of the corresponding formula for the ordinary complex projective space (up to a factor). Conjecturally, this is the case for other classical supermanifolds (such as the unitary supergroup). In the talk, I will explain how this all works. This is very much a work in progress and there remain open questions. There may be an interesting relation with recent works on "universal formulas" in Lie algebra theory by Mkrtchyan--Veselov.

    (No preliminary knowledge of supermanifolds is assumed. All necessary notions will be introduced in the talk.)

  • Monday February 3: Kirill will talk informally about morphisms of Lie algebroids. These include Maurer-Cartan forms, the infinitesimal form of momentum maps, and sigma-models. There are characterizations in terms of Poisson structures and of super geometry.

  • Monday January 27: Madeleine Jotz Lean, The infinitesimal description of Poisson and Dirac Lie groups.

    I will first give a quick introduction to Poisson Lie groups and Dirac manifolds. Drinfel'd proved classifications of Poisson Lie groups via Lie bialgebras, and of Poisson homogeneous spaces of a Poisson group via Dirac subspaces of its Lie bialgebra. This can be understood more naturally in the context of Dirac groups and their homogeneous spaces.