# 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.

• Monday May 16th: Charlotte Kirchhoff-Lukat (DAMTP) Dorfman brackets in the context of double vector bundles

Dorfman brackets constitute a natural non-antisymmetric generalisation of Lie algebroids which still satisfy a type of Jacobi identity. Certain examples play an important role in string theory, where they encode infinitesimal gauge transformations or generalised diffeomorphisms.

We (CKL in collaboration with M. Jotz-Lean) aim to understand the construction principles of Dorfman brackets on vector bundles of the form $TM+E^*$ ($E \to M$ a general vector bundle).

In this talk I will show that such brackets can be viewed as restrictions of the Courant-Dorfman bracket on the standard VB-Courant algebroid $TE+T^*E$, and discuss how internal symmetries of Dorfman brackets can be described in this context.

### Previous talks

• Monday April 25th: Mathieu Stienon (Penn State)

Algebraic exponential maps

Exponential maps arise naturally in the contexts of Lie theory and connections on smooth manifolds. We will explain how exponential maps can be understood algebraically, how these maps can be extended to graded manifolds and how this problem leads naturally to Dolgushev-Fedosov resolutions.

• Monday April 18th: Madeleine Jotz Lean

Split Lie 2-algebroids and matched pairs of 2-representations

A matched pair of Lie algebroid representations is equivalent to two seemingly different objects; the bicrossproduct Lie algebroid and the double Lie algebroid of the matched pair. By looking at matched pairs of 2-representations and their bicrossproduct Lie 2-algebroids, the talk will explain how the double of a matched pair can be seen as the geometrisation of its bicrossproduct.

• Monday March 14: Cornelia Vizman (Timisoara)

Dual pairs of Poisson maps (joint work with Francois Gay-Balmaz)

After giving some examples of dual pairs of momentum maps (symplectic groupoids included), I will review the symplectic leaf correspondence in dual pairs. Then I will pass to dual pairs of momentum maps associated to mutually completely orthogonal symplectic actions of two Lie groups. In this case, symplectic reduction for one group gives coadjoint orbits for the other group.

In the end I will present dual pairs related to fluid dynamics, associated to mutually completely orthogonal actions of two diffeomorphism groups. I will use them to extract information about some coadjoint orbits of diffeomorphism groups.

• Tuesday March 8th, 4pm LT B (Note unusual day, time and place): Yvette Kosmann-Schwarzbach (Ecole polytechnique, Palaiseau)

Multiplicativity conditions for tensor fields and forms on Lie groupoids

We shall recall how the multiplicativity property of the Poisson bivectors of Poisson-Lie group theory was re-formulated in terms of morphisms of Lie groupoids by K. Mackenzie in 1992, and we shall survey the many generalizations that ensued, the multiplicativity properties of multivectors, forms and Dirac structures on Lie groupoids, as well as their infinitesimal counterparts on Lie algebroids.

The visit of Professor Kosmann-Schwarzbach is supported by the LMS under a Scheme 2 grant.

• Monday February 29th: The talk by Marie-Amelie Lawn (Imperial) is postponed to a date to be announced.

• Monday February 22nd: James Waldron (Newcastle)

Symmetries of differentiable stacks

Differentiable stacks are generalisations of smooth manifolds; they include orbifolds, quotient stacks, and the classifying stacks of Lie groups. I will talk about a 'stacky' generalisation of the fact that the space of vector fields on a smooth manifold carries a natural Lie algebra structure. Building on work of Hepworth, I will explain how vector fields on a differentiable stack form a Lie 2-algebra, and describe some examples. This is joint work with Cristian Ortiz.

If there is time, I will also talk about a stacky generalisation of the relationship between vector fields and diffeomorphims. I will describe the automorphism 2-group of a differentiable stack, and explain how this 2-group can be seen as the integration' of the Lie 2-algebra above.

• Monday December 7th: David Applebaum (Sheffield)

The Ubiquitous Heat Kernel and the Hardy-Littlewood-Sobolev Inequality

In the first part of the talk, I will introduce the heat kernel on a manifold and discuss some of its properties. In the second half I will show how the heat kernel can be used to prove a generalisation of the classical Hardy-Littlewood-Sobolev inequality to Riemannian manifolds, and even more general metric measure spaces. Based on joint work with Rodrigo Banuelos (Purdue).

• Monday November 30th: Eli Hawkins (York)

Compatibility between Metric and Poisson Tensors

If a compact Riemannian manifold can be smoothly deformed to a noncommutative geometry, then the deformation is characterized by a Poisson tensor that is locally a sum of wedge products of commuting Killing vectors. I will discuss elements of my proof. The full story involves the cotangent Lie algebroid, a contravariant connection, a higher curvature, symplectic realizations, foliations, cotangent geodesics, the Gauss-Codazzi equation, Kaluza-Klein geometry, and the secret life of Killing vectors.

• Monday November 23rd: Kirill Mackenzie (Sheffield)

The outlines of double Lie theory

I will describe the main ideas of the Lie theory of double Lie groupoids. The talk is intended for a general pure mathematics audience.

(The title may appear ungrammatical but it should become clear in the course of the talk.)

• Monday November 16th: Simone Gutt (ULB, Brussels)

Symplectic analogues of spaces of constant curvature

We show how symplectic symmetric spaces with Ricci-type curvature are the symplectic analogues of space forms. The large number of their totally geodesic symplectic (or Lagrangian) submanifolds opens up the possibility of defining some Radon-type transform in a symplectic context.

• Monday October 26th: David Saunders

Cartan geometries and Lie algebroids

"A Cartan geometry is a Klein geometry with curvature”: that is, given a Klein geometry as a homogeneous space G/H where G is a Lie group and H a closed Lie subgroup, a Cartan geometry is a smooth manifold M which locally is “like G/H”. A modern approach to Cartan geometry is given in a book by Sharpe, where the structure is given by a principal H-bundle over M and a “Cartan connection”, a 1-form on M taking values in the Lie algebra of G (rather than H). In this talk I shall describe an alternative approach to Cartan geometry using, rather than a principal bundle, a fibre bundle with standard fibre G/H. The morphisms of this structure form a Lie groupoid with a distinguished Lie subgroupoid, and the geometry is given by a path connection. The corresponding infinitesimal structures are Lie algebroids and an infinitesimal connection. An advantage of this approach is that the Lie algebroids obtained in this way can be identified with certain Lie algebroids of projectable vector fields on the fibre bundle. This gives a means of relating the present approach to those classical studies of projective and conformal geometry which used methods of tensor calculus.

• Monday October 19th: Weiyi Zhang (Warwick)

Geometric structures, Gromov norm and Kodaira dimensions

Kodaira dimension provides a very successful classification scheme for complex manifolds. The notion was extended to symplectic 4-manifolds. In this talk, we will define the Kodaira dimension for 3-manifolds through Thurston’s eight geometries. It is compatible with the mapping order and other Kodaira dimensions in the sense of “additivity”. This idea could be extended to 4-dimensional geometric manifolds. Those with highest Kodaira dimension are distinguished by nonvanishing Gromov norm. Finally, we will see how it is sitting in a potential classification of 4-manifolds.

### Spring 2015

• Monday June 8th. Kirill will give the third of three talks on Poisson Lie groups and Lie bialgebras.

• Monday June 1st at 2pm in J 11: Ekaterina Shemyakova SUNY at New Paltz

Darboux transformations for differential operators on the superline

We give a full description of Darboux transformations of any order for arbitrary (nondegenerate) differential operators on the superline. We show that every Darboux transformation of such operators factorizes into elementary Darboux transformations of order one. Similar statement holds for operators on the ordinary line. (Joint work with Ted Voronov and my student Sean Hill.)

Supported by the LMS through a Scheme 2 grant.

• Monday May 25th. No seminar (Bank Holiday)

• Monday May 18th. Kirill will give the much-postponed second talk on Poisson Lie groups and Lie bialgebras.

• Monday May 11th. Cancelled.

• Monday May 4th. No seminar (Bank Holiday)

• Monday April 27th. No seminar.

• Monday April 20th at 2pm in J 11: Christian Blohmann MPI, Bonn

Removable presymplectic singularities and the local splitting of Dirac manifolds

I call a singularity of a presymplectic form removable if its graph extends to a smooth Dirac structure over the singularity. The guiding example is the symplectic form of a magnetic monopole in 2 dimensions. The question whether a given singularity of a presymplectic or Poisson structure is removable in this sense is surprisingly subtle. For example, the singularity of a monopole in 3 dimensions is not removable. I will report on recent results that give a complete understanding of the structure of removable singularities. The key tool is a (new) local splitting theorem for Dirac manifolds. Finally, I will explain how this leads to the generalized prequantization of singular symplectic manifolds.

• Monday April 13th, J 11, 2pm: Arkady Vaintrob (Oregon)

Hidden supersymmetries of generalized geometries

Generalized complex structures, introduced by Hitchin as a common generalization of complex and symplectic structures on manifolds, found many applications in differential geometry and in physics.They also have some peculiar features, such as the the extended diffeomorphism group (the so-called B-field action), D-branes (submanifolds with additional structure), and several competing notions of a generalized holomorphic map.

In my talk I will explain how to describe the generalized structures by introducing anti-commuting coordinates (i.e. in terms of geometry of supermanifolds) and how this description helps to elucidate the above peculiarities.

Lecture supported by LMS under Scheme 2.

• Monday March 9th, 2pm: Ian Marshall (Faculty of Mathematics, Higher School of Economics, Ulitsa Vavilova 7, Moscow)

A new model in the Calogero-Ruijsenaars family

Hamiltonian reduction is used to project a trivially integrable system on the Heisenberg double of $SU(n,n)$, to obtain a system of Ruijsenaars type on a suitable quotient space. This system possesses $BC_n$ symmetry and is shown to be equivalent to the standard three-parameter $BC_n$ hyperbolic Sutherland model in the cotangent bundle limit.

• Monday March 2nd: Kirill will give the first of three or so talks on Poisson Lie groups.

Time: 1pm to 2pm in LT A.

This is the lecture originally scheduled for 23/2.

These are designed for postgraduates but all interested are welcome.

A PLG $G$ is a Lie group equipped with a Poisson structure with respect to which the multiplication is a Poisson map. This is a standard compatibility condition but because Poisson structures are contravariant, there are unexpected consequences: group inversion is antiPoisson, not Poisson, and left- and right-translations are neither Poisson nor antiPoisson. In other words the standard actions of the group on itself are not by Poisson diffeomorphisms.

When $G$ is linearized to its Lie algebra $\frak g$, the Poisson structure induces a Lie algebra structure on $\frak g^*$, the vector space dual. Together $\frak g$ and $\frak g^*$ form a Lie bialgebra. The relationship between them is embodied in the (classical) Drinfeld double: the direct sum $\frak g\oplus g^*$ has a Lie algebra structure for which $\frak g$ and $\frak g^*$ are subalgebras.

These ideas are the basis of several strands of current work. In the opposite direction they are used in work on integrable systems.

I will start with Lie bialgebras and work backwards towards PLGs. I'll assume a slight acquaintance with Lie groups and Lie algebras.

• Monday February 23rd: No seminar due to illness

• Monday February 16th: No seminar

• Monday February 9th, 3pm in J11. Theodore Voronov (Manchester) The seminar will conclude at about 4.30; there will be a break midway.

Microformal geometry: thick morphisms'' of supermanifolds, adjoints of nonlinear operators and homotopy algebras

We introduce a generalization of smooth maps of manifolds (or supermanifolds) called thick morphisms''. Such morphisms are defined via formal canonical relations between cotangent bundles and make a formal category, a thickening'' of the usual category of smooth manifolds with the same class of objects. They induce pullbacks of smooth functions, which are formal nonlinear mappings with remarkable properties.

In particular, we shall explain how this new construction makes it possible to obtain an analog of the adjoint operator for the case when the initial operator is nonlinear. (We consider maps of vector spaces or fiberwise maps of vector bundles $\Phi:\, E_1\to E_2$.) This gives a nonlinear pushforward map'' of the spaces of functions on the dual bundles $\Phi_*:\, \mathbf{C}^{\infty}(E_1^*)\to \mathbf{C}^{\infty}(E_2^*)$. (Since the mapping of functions is itself nonlinear, the functions should be even or bosonic''; there is a parallel fermionic'' construction.)

Time permitting, we shall give an application to homotopy Poisson algebras and homotopy Lie (bi)algebroids.

(See preprints: 1409.6475 and 1411.6720.)

### Autumn 2014

• Monday December 1st: 3pm to 4pm

Madeleine will give the third lecture on "Degree 2 N-manifolds and metric double vector bundles"

• Monday November 24th: 3pm to 4pm

Madeleine will give the second of two or three lectures on "Degree 2 N-manifolds and metric double vector bundles"

In the second lecture the plan is to talk about N-manifolds, how degree 1 NQ-manifolds are Lie algebroids, and how degree 2 N-manifolds are equivalent to metric DVB's.

• Monday November 17th: 3pm to 4pm

Note venue: F 38

Madeleine will give the first of two or three lectures on "Degree 2 N-manifolds and metric double vector bundles"

In the first lecture, the plan is to talk only about vector bundles and their equivalence to sheaves of locally finitely generated C^\infty(M)-modules.

• Monday November 10th: 3pm to 4pm

Kirill will give the fourth of three lectures on principal bundles and connections therein.

This will cover Chevalley-Eilenberg cohomology of Lie algebras, its resemblance to de Rham cohomology, and a unification which for $H^2$ and $H^3$ includes the standard identities for connections and curvature.

These lectures are designed primarily for postgraduates, but all interested are welcome.

(Note that the time is 3pm, not 2pm.

• Monday November 1st: 3pm to 4pm

Kirill will give the third of three lectures on principal bundles and connections therein. This will include a more careful treatment of the smooth structures. These three lectures are designed primarily for postgraduates, but all interested are welcome.

(Note that the time is 3pm, not 2pm.

• Monday October 27th: No seminar.

• Monday October 20th: 3pm to 4pm

Kirill will give the second of two or three lectures on principal bundles and connections therein. This will include a more careful treatment of the smooth structures. These three lectures are designed primarily for postgraduates, but all interested are welcome.

(Note that the time is 3pm, not 2pm.)

• Monday October 13th: 3pm to 4pm

Kirill will give the first of two or three lectures on principal bundles and connections therein. This is an introductory overview, designed primarily for postgraduates, but all interested are welcome.

(This replaces the talk scheduled for Monday 5th; note that the time is now 3pm, not 2pm.)

### 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.