Differential Geometry Seminar
The differential geometry semniar is held on Tuesday afternoons, 14h15h. The organisers are Simone Gutt, Mélanie Bertelson and myself. Confirmed speakers, together with their titles and abstracts when available, appear below.
Below the schedule are instructions for speakers.
Februrary 2014

18th February 14h0015h00. Salles des Profs, 9th floor of building NO.Alberto AbbondandoloA local nonsqueezing theorem on infinite dimensional phase spacesI will discuss the implications of a nonsqueezing theorem à la Gromov for symplectomorphisms of symplectic Hilbert spaces and show a proof of such a result for symplectomorphisms mapping the ball into a convex set.

11th February 14h0015h00. Salles des Profs, 9th floor of building NO.Markus UpmeierGeneralised differential cohomologyIn the beginning of my talk I shall give a presentation of the basic properties and terminology in the theory of differential extensions of generalized cohomology theories. Following a number of examples, we will then turn to the study of the multiplicative properties and some of my own results in this direction.

4th February 14h0015h00. Salles des Profs, 9th floor of building NO.Misha VerbitskyErgodic complex structures and Kobayashi metricLet M be a compact manifold. Consider the action of the diffeomorphism group Diff(M) on the (infinitedimensional) space Comp(M) of complex structures. A complex structure is called ergodic if its Diff(M)orbit is dense in the connected component of Comp(M). I will show that on a hyperkaehler manifold or a compact torus, a complex structure is ergodic unless its Picard rank is maximal. This result has many geometric consequences; for instance, it follows that the Kobayashi pseudometric on any K3 surface or on the deformations of its Hilbert scheme vanishes, solving a longstanding conjecture by Kobayashi.
January 2014

21st January 14h0015h00. Salles des Profs, 9th floor of building NO.Mehdi LejmiExistence of HKT metrics on hypercomplex manifolds of real dimension 8.In this talk, I will introduce the notion of a HKTmetric (hyperkähler with torsion) on a hypercomplex manifold. I will give some examples and properties of such manifolds. The goal of the talk is to prove that a compact hypercomplex manifold of real dimension 8, with holonomy of the Obata connection lying in SL(2,H) admits a HKTmetric if and only if H^{1,0} with respect to a fixed complex structure is even. This is sort of an analogue to the fact that every compact complex surface with even b_1 is Kähler. This is a joint work with Misha Verbitsky and Gueo Grantcharov.

14th January 15h0016h00. Salles des Profs, 9th floor of building NO.Markus UpmeierProduct structures in Differential CohomologyDifferential cohomology is a synthesis of ideas from classical gauge theory and (stable) homotopy theory. So far, the main mathematical motivation has been the creation of a uniform framework for dealing with secondary invariants like the ChernSimons invariant. In physical terms, differential cohomology is thought of as an abelian gauge theory and can for example be used in the description of Dirac's magnetic monopole. In my talk, I shall motivate and introduce the basic objects of this theory and identify special cases as familiar gaugetheoretic objects. Finally, I will report also on my own research concerning product structures.

7th January 14h0015h00. Salles des Profs, 9th floor of building NO.Maria SalazarContact isotropic realizations of Jacobi manifoldsNoncommutative Hamiltonian integrable systems (in the sense of Nekhoroshev or MishenkoFomenko) on symplectic manifolds are intimately related to isotropic realisations of regular Poisson structures, which can be seen as desingularisations of the underlying Poisson manifold. It is natural to ask whether an analogous connection holds for integrable Hamiltonian systems on manifolds endowed with related structures e.g. (twisted) symplectic, Poisson or Dirac. This talk is concerned with describing the global geometric objects behind noncommutative integrable Hamiltonian systems on contact manifolds (as defined by Khesin and Tabachnikov, Banyaga and Molizno, and Jovanovic). This is joint work with Daniele Sepe.
December 2013

17th December 12h0013h00. Salles des Profs, 9th floor of building NO.Steve Zelditch (Northwestern, Chicago)Geodesics in the space of Kahler metrics.MabuchiSemmesDonaldson defined a Riemannian metric on the infinite dimensional space of Kahler metrics in a fixed class. The geodesics are solutions of a homogeneous Monge Ampere equation. My talk is about the initial value problem for geodesics and on the approximation of these geodesics by those of the finite dimensional symmetric space of socalled Bergman metrics. I will sketch the proof of a conjecture of Yanir Rubinstein and myself on the quantization approximation for the initial value problem.

3rd December 12h0013h00. Salles des Profs, 9th floor of building NO.Markus Upmeier (ULB)Refinements of the ChernDold Character in Generalized Cohomology.We shall begin by reviewing the most basic features of the theory of generalized cohomology theory and stable homotopy theory. Fundamental examples such as Ktheory, stable homotopy groups, and cobordism theories will be discussed. The relation to spectra using Brown's Representability Theorem will also be indicated. Next, the ChernDold character, a map that compares an arbitrary generalized cohomology theory with singular cohomology, will be introduced. This finally leads to some of my own research concerning refinements of the ChernDold character to a much more sophisticated object, a map of highly structured ring spectra. These are crucial for stateoftheart treatments of generalized differential cohomology in terms of differential function spectra. The present talk will fill the background for my later talks on differential cohomology.
November 2013

19th November, 14h0015h00. Salles des Profs, 9th floor of building NO.Mehdi Lejmi (ULB)The Jflow and stability.The Jflow is a parabolic flow introduced by Donaldson. In this talk, we present a new algebrogeometric stability condition which is equivalent conjecturally to the existence of solutions of the critical equation of the Jflow. We present also exmaples due to FangLai and explain how this is related to the stability condition. This is a joint work with Gabor Székelyhidi.

12th November, 14h0015h00. Salles des Profs, 9th floor of building NO.Mehdi Lejmi (ULB)Stability under deformations of HermiteEinstein almostKähler metrics.In this talk, we introduce the notion of extremal almostKähler metrics. These metrics appear as a natural generalization of Calabi extremal Kähler metrics to the symplectic case. We focus on the particular case of HermiteEinstein almostKähler metrics, give examples of them, study their deformations and look to obstructions for their existence.
October 2013

23rd October 14h0015h00. Salles des Profs, 9th floor of building NO.Alexander Cardona Guio (Universidad de Los Andes, Bogotà, Colombia)Algèbres de Poisson associées aux structures de Dirac tordues et symétries prolongéesDans cet exposé, je présenterai une définition d'algèbre de Poisson des fonctions admissibles associée aux structures de Dirac tordues par une 3forme fermée qui généralise celle donnée par Courant et Weinstein dans le cas nontordue, et je vais présenter les cas standards associés aux structures de Dirac définies par les graphes de 2formes nondégénérées ainsi que d'autres exemples naturels. Je présenterai également la relation entre les symétries prolongées d'algébroides de Courant exactes sur une variété, les algèbres de Poisson des fonctions admissibles associées aux structures de Dirac tordues sous l'action d'un groupe de Lie et la réduction des structures de Dirac. Enfin, je vais montrer que les homomorphismes habituels d'algèbres de Lie entre les symétries infinitésimales de l'action, celles des champs de vecteurs sur la variété et l'algèbre de Poisson des observables (apparaissant en géométrie symplectique), se généralisent en morphismes naturels des algèbres de Leibniz induites par l'action prolongée et les applications moment associées dans le cadre de structures de Dirac tordues.
April 2013

30th April, 14h3016h00. Salles des Profs, 9th floor of building NO.Daniel Sternheimer (Rikkyo University, Tokyo, and Université de Bourgogne, Dijon).Altneuland in mathematical particle physics: back to the drawing board?We describe work in progress and outline a “framework for conjectural frameworks” based on Flato's deformation philosophy, on joint works with or by Flato and coworkers (especially Fronsdal) since the 60's, and on discussions with many mathematicians and physicists in the past years. Namely we return to the old problem of connection between external (Poincaré group) and internal (unitary) symmetries of elementary particles but with a (Drinfeld) twist, suggesting that the internal symmetries might emerge from deforming to Anti de Sitter SO(2,3) and quantizing that (possibly in a new generalised manner) at root of unity. That raises challenging problems, both on the mathematical part and for particle physics.

16th April, 12h3014h00. Salles des Profs, 9th floor of building NO.Jonathan Fine (Open University UK).Finding linear homologyThe middle perversity intersection homology (mpih) Betti numbers of the toric variety associated with a convex polytope are linear functions of the flag vector of the convex polytope. In this talk I will define similar linear functions, which I hope are the Betti numbers for a not yet defined homology theory. This linear homology theory should exist wherever mpih does. Such homology would prove that the candidate Betti numbers are actual Betti numbers, and so nonnegative on all convex polytopes (being the dimension of a vector space).
March 2013

26th March, 13h3014h20 and 14h3015h20 (the first talk will be an introduction at the level of a masters course, the second a usual seminar.) Salles des Profs, 9th floor of building NO.Kirill Mackenzie (Sheffield UK).Duality for nfold vector bundlesDouble vector bundles have been implicit in differential geometry for many years: for a vector bundle E over M the tangent TE with its two structures (over E and over TM) gives the formulation of connections in E which was used by Dieudonné, and T(TM), T(T^*M) and T^*(T^*M) arise in Tulczyjew's formulation of geometric mechanics. In general a double vector bundle is a manifold D with two compatible vector bundle structures over bases A and B which are themselves vector bundles over a common manifold M . Such a D can be dualized in two ways and these duals are themselves dual over a base which emerges from the double structure. The two dualizations generate the symmetric group of order 6. For triple vector bundles, GraciaSaz and the speaker have proved that the corresponding group is of order 96 and is a nonsplit extension of S_4 by the Klein 4group. For nfold vector bundles the corresponding group is an extension of S_n+1 by a direct product of cyclic groups of order 2. It is a subgroup of O(k,Z) where k depends on n, but is not a Coxeter group. In the first talk I will describe the pairing of the two duals of a double vector bundle which is the basis of this work, and the results for n = 3,4. In the second talk I will describe the method by which the groups are calculated and the interpretation  on which further developments rest  of the kernel elements as graphs of even valency. This work arose in the study of bracket structures associated to Poisson manifolds and Lie algebroids, but the talk requires no knowledge of these. An acquaintance with the concept of vector bundle is sufficient.

15th March, 10h3011h30, Salles de Solvay, 5th floor of building NO.Will Kirwin (Cologne).Adapted complex structures and the geodesic flow.Let M be a compact, realanalytic Riemannian manifold. An adapted complex structure is a certain complex structure on a neighborhood of M in its tangent bundle (i.e. the Grauert tube). After giving a brief history and introduction, I will describe how adapted complex structures can be understood in terms of the "imaginary time" geodesic flow, and how the construction is related to a general construction known as Thiemann's complexifier method. I will also describe a generalization involving magnetic flows, and, time permitting, various applications.
November 2012

27th November, 14h0015h00, Salles des Profs, 9th floor of building NO.Julien Meyer (ULB).The Hodge theorem for Kähler manifoldsThis is the first in our series of internal seminars, giving graduate and postgraduate students the chance to exlpain topics they have recently learnt themselves. Please remember, these seminars are more informal than normal and the audience should ask plenty of questions to ensure that we all learn as much as possible.
October 2012

16th October, 14h0016h15, Salles des Profs, 9th floor of building NO.14h0015h00, Johannes Nordstrom (Imperial College London)Diffeomorphism types of compact G2 manifolds.In joint work with Corti, Haskins and Pacini, we apply Kovalev's twisted connected sum construction to certain weak Fano 3folds to produce large numbers of compact 2connected Riemannian 7manifolds with holonomy G2. By applying the classification theory of smooth 2connected 7manifolds, we can determine their diffeomorphism type. It turns out that many different twisted connected sums define G2metrics on the same smooth manifold, raising the question whether these metrics belong to different components of the G2 moduli space.15h1516h15, Georgios Dimitroglou Rizell (ULB)Ambient Legendrian Surgery in the critical case.We describe the operation Ambient Legendrian surgery on a Legendrian submanifold L of dimension n. This produces a Legendrian embedding of the manifold L_S obtained by ksurgery on L together with an exact Lagrangian (k+1)handle attachment. There are formulas which compute the (full) Legendrian contact homology of L_S in terms of data on L in the case k less than n1. The situation for k equal to n1 is more difficult, as we will explain, but we provide formulas for the linearized Legendrian contact homology.
September 2012

18th September, 14h0015h00, Salles des Profs, 9th floor of building NO.Ludwig Faddeev (St. Petersburg)Examples of Poisson structures and their quantum counterparts from the theory of integrable models.

18th September, 15h0016h00, Salles des Profs, 9th floor of building NO.Daniel Sternheimer (Dijon and Keio)A short presentation of deformation quantization and of bold mathematical ideas around relativistic symmetries towards a conjectural physics framework based on such deformations.
May 2012

11th May, time and room to be announced.Sebastian Klein.Totally geodesic submanifolds in Riemannian symmetric spaces of rank 2.The objective of the talk is to describe a method for the classification of totally geodesic submanifolds in Riemannian symmetric spaces of rank 2 via the root space decomposition of the space. In the first part of the talk I will describe relations between the root system of a symmetric space and the root system of a totally geodesic submanifold, as well as relations between the root spaces of a symmetric space and the root spaces of a totally geodesic submanifold. These relations serve as a fundament for the classification of totally geodesic submanifolds. In the second part of the talk I will describe the application of these results to the classification of totally geodesic submanifolds in one specific series of symmetric spaces of rank 2, namely the complex 2Grassmannians G_2(C^n).
April 2012

27th April, 14h0015h00, room 2.NO.707.Julien Keller (Marseilles).“Chow stability and projectivisations of stable bundles”We will discuss GIT stability of projectivisations of Giseker stable vector bundles living over a projective surface carrying a constant scalar curvature Kähler metric. We will give an example of a smooth manifold which is Chow stable but not asymptotically Chow stable. This is joint work with Julius Ross (Univ de Cambridge).

24th April, 14h0015h00, Salle de Solvay, 5th floor, Buliding NO.Robert Berman (Chalmers).“KählerEinstein metrics emerging from free fermions and statistical mechanics”From a statistical mechanical point of view it is natural to view differential geometry as an emergent phenomena: the smooth shapes that we see are emergent effects of some underlying microscopic model, as the number of particles tends to infinity. On the other hand, from a mathematical point of view it is also natural to view differential geometry as a limit of algebraic geometry, as the “degree” tends to infinity. Naively, this just amounts to the fact that any smooth curve can be approximated by a polynomial curve, but, in fact, this idea goes much deeper and is related to the fundamental YauTianDonaldson conjecture concerning KählerEinstein metrics on projective algebraic varities. In this talk I will explain how these two different points of view on differential geometry can be merged, leading to a new statistical mechanics approach to KählerEinstein metrics. It turns out that  from a physical point of view  the underlying microscopic theory consists of a gas of free fermions subject to a nonstandard “betadeformation,” making it background free. Time permitting I will also point out some connections to quantum gravity and speculate on possible relations to the recent work of FerrariKlevtsovZelditch on random Kähler metrics.
March 2012

30th March, 14h0015h00, Salle de Solvay, 5th floor, Building NO.Rafael Torres (Oxford).“Constructions of generalized complex structures in dimension four”Recent constructions of exotic smooth structures on 4manifolds can be used to expand our understanding of generalized complex structures. The talk will be a description of the produce of merging these two research areas together, which yields existence results for a myriad of 4manifolds and also unveils interesting phenomena regarding generalized complex structures.

16th March, 14h0015h00, Salle de Solvay, 5th floor, Building NO.Konrad Waldorf (Regensburg).“Introduction to gerbes and higher holonomies”In this talk I give an introduction to the theory of bundle gerbes, which have been invented by M. Murray in 1995. Bundle gerbes generalize line bundles in the sense that they provide a geometrical realization of cohomology classes in degrees higher than two. Like line bundles, bundle gerbes can be equipped with connections leading to a notion of higher holonomies, taken around closed manifolds of dimension higher than one. The main motivation for bundle gerbes and their higher holonomies is their application to quantum field theories, in particular WZW models and ChernSimons theory.

9th March, 14h0015h00, Salle de Solvay, 5th floor, Building NO.Jonny Evans (ETH Zurich).“Pseudoholomorphic curves and nilpotent Lie algebras.”In joint work with Jarek Kedra, we explore the genus 1 GromovWitten invariants of certain families of symplectic nilmanifolds, including the KodairaThurston 4manifold. BryanLeung (2000) proved that in the case of the hyperKaehler family of K3 surfaces the GromovWitten invariants are coefficients of a quasimodular form. Our computations yield some other interesting arithmetic answers. I will explain how this works in the simplest cases (tori!) where it is wellknown before explaining the more general setting (twistor families associated to nilpotent Lie algebras).

2nd March, 14h0015h00, Salle de Solvay, 5th floor, Building NO.Dan Popovici (Toulouse).“Deformation Limits of Compact Kaehler Manifolds.”If in a holomorphic family of compact complex manifolds all the fibres, except one, are supposed to be Kähler, the remaining (limit/central) fibre has long been conjectured to be of class C (i.e. bimeromorphically equivalent to a compact Kaehler manifold). We shall explain a strategy for tackling this (still open) conjecture in which only one of three major ingredients has yet to be proved: Demailly's conjecture on transcendental Morse inequalities. A sequence of nonholomorphic but almost holomorphic complex line bundles can naturally be associated with any real closed (1,1)form on the central fibre of the family and what is at stake is to construct sufficiently many almost holomorphic sections of these line bundles when the original form is supposed to satisfy a weak positivity assumption.
February 2012

24th February, 14h0015h00, room A2.220 (Chemistry dept.)Leo Tzou (Helsinki).

17th February, 14h0015h00, A2.220 (Chemistry dept.)Thomas Bruun Madsen (Kings College, London).“From halfflat to exceptional.”I will talk about work in progress on the construction of (noncompact) metrics with holonomy G_2 via Hitchin's flow equations. In particular, I will discuss a new description of SO(3)xSO(3)invariant halfflat SU(3)structures on S3xS3; such structures play an essential role when we look for solutions to the Hitchin flow. The theoretical framework is supplemented by some concrete examples, including the (complete) BryantSalamon metric on the spin bundle over a threesphere

10th February, 14h0015h00, Salle de Solvay, 5th floor, Building NO.Stuart Hall (Buckingham).“Investigating the linear stability of KählerRicci solitons” (joint with Thomas Murphy)Ricci solitons are generalisations of Einstein metrics that are fixed points of the Ricci flow. In this talk we will discuss the notion of linear stability which roughly determines whether a soliton is attracting or repelling as a fixed point. We will focus on the Kähler case where more subtle questions can be asked about what happens if one changes the complex structure of a soliton.
January 2012

20th January, Salle des Profs, 9th floor, building NO14h0015h00, George Marinescu (Cologne).“Equidistribution of zeros of holomorphic sections of high tensor powers of line bundles.”We present some equidistribution results for sequences of random sections of high tensor powers of positive line bundles over noncompact manifolds (e.g. Riemann surfaces with cusps, arithmetic quotients or, more generally, quasiprojective manifolds). We also examine the equidistribution of sections of big line bundles endowed with singular Hermitian metrics.
15h3016h30, Carl Tippler (Nantes).“Deformations of extremal Kähler toric manifolds.”Using the method of Székelyhidi, we reduce the existence of extremal Kähler metrics on complex deformations of extremal Kähler manifolds to a finite dimensional GIT problem. We compute stable points in the case of toric manifolds, providing new examples of extremal Kähler surfaces.
Instructions for speakers
Expenses
We will normally pay your hotel bill (room and breakfast) directly ourselves. We will need the following inorder to pay your travel expenses:
 Your original tickets.
 Your IBAN, BIC and bank's name and address.
 Your personal address.
 A photocopy of your passport or identity card.
Travel
If you are arriving by plane, at Zaventem airport, then you should take the train to Gare du Midi, the main station in Brussels. The trains leave roughly every thirty minutes and the journey takes half an hour. If you are arriving by train, it is almost certain your train will also stop at Gare du Midi.
We typically use Hotel Agenda Louise for our speakers. To travel to Hotel Agenda from Gare du Midi, the main train station in Brussels, take the Metro to Louise, on line 2 or 6, direction Simonis (Elisabeth), a journey of aproximately 5 minutes. Be careful not to take the metro in the oposite direction, which has almost the same name: Simonis (Leopold). From Louise metro station you can walk to the hotel along Avenue Louise and turn right down Rue de Florence. The Hotel is on your right. The walk should take at most 10 minutes. Follow these links for a map of the Brussels Metro and a map of the walk from Louise metro to Hotel Agenda Louise
To travel from Hotel Agenda to the maths department, you can walk, take the metro or take a tram.
From Hotel Agenda to the department by metro
For the metro, walk back to Louise Metro station and then take the metro to Delta station. To do this take line 2 or 6 direction Simonis (Elisabeth) to ArtsLoi and then change for line 5, direction HermannDebroux. The whole metro journey will take about 30 minutes.
Delta station is on the same campus as the maths department. The department is in building NO, the differential geometry group is on the 7th floor and I can normally be found there in my office, O.7.112. Here is a link to a map of the campus with both Delta metro and building NO marked.
From Hotel Agenda to the department by tram
To get here by tram, walk from the hotel back to Avenue Louise and turn right, (the oposite direction from the metro). Catch the 94 tram, direction Musée du tram, as far as the ULB Solbosch campus. Warning: this is not the campus where the maths department is! To get to the maths department, you now walk along Avenue de l'Université to Campus de la Plaine. For more details, see this map of the tram journey and walk. Avenue de l'Université leaves you at accèss 2 on this map of the campus. The maths department is in building NO, the differential geometry group is on the 7th floor and I can normally be found there in my office, O.7.112.