Differential Geometry Seminar

The differential geometry semniar is held on Tuesday afternoons, 14h-15h. The organisers are Simone Gutt and myself. Confirmed speakers, together with their titles and abstracts when available, appear below.

Below the schedule are instructions for speakers.

April 2013

30th April March, 14h30-16h00. 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 March, 12h30-14h00. Salles des Profs, 9th floor of building NO.

Jonathan Fine (Open University UK).

Finding linear homology

The 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 non-negative on all convex polytopes (being the dimension of a vector space).

March 2013

26th March, 13h30-14h20 and 14h30-15h20 (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 n-fold vector bundles

Double 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, Gracia-Saz and the speaker have proved that the corresponding group is of order 96 and is a non-split extension of S_4 by the Klein 4-group. For n-fold 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, 10h30-11h30, Salles de Solvay, 5th floor of building NO.

Will Kirwin (Cologne).

Adapted complex structures and the geodesic flow.

Let M be a compact, real-analytic 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, 14h00-15h00, Salles des Profs, 9th floor of building NO.

Julien Meyer (ULB).

The Hodge theorem for Kähler manifolds

This 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, 14h00-16h15, Salles des Profs, 9th floor of building NO.

14h00-15h00, 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 3-folds to produce large numbers of compact 2-connected Riemannian 7-manifolds with holonomy G2. By applying the classification theory of smooth 2-connected 7-manifolds, we can determine their diffeomorphism type. It turns out that many different twisted connected sums define G2-metrics on the same smooth manifold, raising the question whether these metrics belong to different components of the G2 moduli space.

15h15-16h15, 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 k-surgery 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 n-1. The situation for k equal to n-1 is more difficult, as we will explain, but we provide formulas for the linearized Legendrian contact homology.

September 2012

18th September, 14h00-15h00, 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, 15h00-16h00, 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 2-Grassmannians G_2(C^n).

April 2012

27th April, 14h00-15h00, 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, 14h00-15h00, Salle de Solvay, 5th floor, Buliding NO.

Robert Berman (Chalmers).

“Kähler-Einstein 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 Yau-Tian-Donaldson conjecture concerning Kähler-Einstein 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 Kähler-Einstein 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 non-standard “beta-deformation,” making it back-ground free. Time permitting I will also point out some connections to quantum gravity and speculate on possible relations to the recent work of Ferrari-Klevtsov-Zelditch on random Kähler metrics.

March 2012

30th March, 14h00-15h00, 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 4-manifolds 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 4-manifolds and also unveils interesting phenomena regarding generalized complex structures.
16th March, 14h00-15h00, 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 Chern-Simons theory.
9th March, 14h00-15h00, 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 Gromov-Witten invariants of certain families of symplectic nilmanifolds, including the Kodaira-Thurston 4-manifold. Bryan-Leung (2000) proved that in the case of the hyperKaehler family of K3 surfaces the Gromov-Witten 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 well-known before explaining the more general setting (twistor families associated to nilpotent Lie algebras).
2nd March, 14h00-15h00, 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 non-holomorphic 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, 14h00-15h00, room A2.220 (Chemistry dept.)

Leo Tzou (Helsinki).
17th February, 14h00-15h00, A2.220 (Chemistry dept.)

Thomas Bruun Madsen (Kings College, London).

“From half-flat to exceptional.”

I will talk about work in progress on the construction of (non-compact) metrics with holonomy G_2 via Hitchin's flow equations. In particular, I will discuss a new description of SO(3)xSO(3)-invariant half-flat 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) Bryant-Salamon metric on the spin bundle over a three-sphere
10th February, 14h00-15h00, Salle de Solvay, 5th floor, Building NO.

Stuart Hall (Buckingham).

“Investigating the linear stability of Kähler-Ricci 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 NO

14h00-15h00, 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 non-compact manifolds (e.g. Riemann surfaces with cusps, arithmetic quotients or, more generally, quasi-projective manifolds). We also examine the equidistribution of sections of big line bundles endowed with singular Hermitian metrics.

15h30-16h30, 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.

From here, your travel plans will depend on where you are staying. There are two hotels we typically use to accommodate speakers, either Hotel Agenda Louise or Hotel Opera.

Travel to and from Hotel Agenda

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, 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 Arts-Loi and then change for line 5, direction Hermann-Debroux. The whole metro journey will take about 20-25 minutes.

Delta station is on the campus of the university. The maths department is in building NO. Here is a link to a map of the campus with both Delta metro and building NO marked. The differential geometry group is on the 7th floor and I can normally be found there in my office, O.7.112.

JOEL FINE