Geometry and Quantum Theory (GQT)


1 January 2011

Events 2011

GQT colloquium Amsterdam
December 9th, 2011


  • 13:00‒14:00 Klaas Landsman: The mathematics of mass: from the Higgs boson to Schrödinger’s cat
  • 14:30‒15:30 Jesper Grodal: Finite loop spaces
  • 15:45‒16:45 Guido Carlet: WDVV equations and 2+1 integrable systems

Contact: Tilman Bauer

Klaas Landsman (Radboud Universiteit Nijmegen)
The mathematics of mass: from the Higgs boson to Schrödinger’s cat

Since the Higgs boson has not been found (yet), the speaker is among many who take a critical look at the Higgs mechanism that underlies the generation of mass in the Standard Model of elementary particle physics. This leads to fascinating mathematical questions involving C*-algebras, ground states, symmetry (breaking), phase transitions, and, perhaps unexpectedly, the insight that the Higgs mechanism is predicated on a solution of the infamous measurement problem in quantum mechanics (which, in a world premiere, we will present).

Jesper Grodal (University of Copenhagen)
Finite loop spaces

Hilbert’s 5th problem, in its most basic form, asks if every compact topological group, which admits the structure of a smooth manifold, is a Lie group. In this form, it was answered affirmatively by von Neumann in 1929. If one takes a homotopical interpretation of the word “admits”, the question is more subtle, and one is led to the notion of a finite loop space. These turn out not quite to be Lie groups, but nevertheless posses a rich enough structure to admit a classification. My talk will outline this story, which starts with a 1941 paper of Hopf: Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen” and ends close to the present.

Guido Carlet (University of Copenhagen):
WDVV equations and 2+1 integrable systems

Frobenius manifolds, originating as a geometrical formulation of WDVV equations of two-dimensional topological field theory, provide an important tool to study certain dispersionless integrable systems in 1+1 dimensions. Recently there has been some progress in the program of extending the Frobenius manifold techniques to the case of integrable systems with two spatial variables, the so-called 2+1 integrable hierarchies. In this case the associated Frobenius manifold is infinite-dimensional. We will recall some basic definitions and constructions in the theory of Frobenius manifolds and, considering the example of the 2D Toda hierarchy, we will show that they extend to the infinite-dimensional case, provided that we require some analytical properties of the Lax symbols.

November 14th, 2011 – Lecture Jun Tomiyama: A proposal for non-commutative spectral synthesis

Time: 13.00-14.00 hr, room 412 of the Snellius building (Niels Bohrweg 1, Leiden)

Abstract (pdf format)

GQT Colloquium Utrecht
October 14th, 2011

The colloquium starts in the afternoon at 13.15 and will be held in Buys Ballot Laboratorium room 001.

13.15-14.15: Simone Gutt Symplectic Dirac operators and Mp^c structures
14.30-15.30: Ieke Moerdijk Infinity-Categories and Infinity-Operads
15.30-16.00: coffee and tea
16.00-17.00: Carel Faber Modular forms and the cohomology of moduli spaces
17.05-18.00: drinks

Simone Gutt: My talk will be based on some ongoing work with Michel Cahen and John Rawnsley. Symplectic spinors have been introduced by Kostant on symplectic manifolds admitting a metaplectic structure, (the metaplectic group being a double cover of the symplectic group, the analogue of the Spin group in Riemmannian geometry). In that context, K. and L. Habermann have introduced two natural Dirac operators, choosing a compatible almost complex structure and a linear connexion preserving the symplectic 2-form. The commutator of those operators is elliptic. We extend the construction of symplectic Dirac operators to any symplectic manifold through the use of Mp^c structures. The group Mp^c is the analogue of the Riemmannian Spin^c and is a circle extension of the symplectic group. Mp^c-structures exist on any symplectic manifold and equivalence classes are parametrized by elements in H^2(M,Z). For any Mp^c structure, choosing a compatible almost complex structures and a linear connection, there are two natural Dirac operators, acting on the sections of a spinor bundle, whose commutator is elliptic. Using the Fock description of the spinor space allows the definition of a notion of degree and the construction of a dense family of finite dimensional subbundles stabilized by this operator.

Carel Faber: This will be an overview over the cohomology classes on moduli spaces of curves or abelian varieties (of low genus respectively dimension) corresponding to modular forms. Elliptic cusp forms give classes for curves of genus 1, and their generalizations, Siegel cusp forms, give classes for abelian varieties. One obtains a formula for moduli of pointed curves of genus 2 in these terms. For curves of higher genus, Teichmüller modular forms enter the picture. The related cohomology classes are still quite mysterious. (Joint work with Jonas Bergström and Gerard van der Geer.)

GQT colloquium Nijmegen
June 17th, 2011

Location: RU Nijmegen, Linneausgebouw, LIN 5
10:45-11:15 – Coffee and Tea
11:15-12:15 – Yong-Geun Oh
12:15-13:45 – Lunch
13:45-14:45 – Alexander Braverman
14:45-15:00 – Announcements of the GQT board
15:00-15:30 – Coffee and tea
15:30-16:30 – Erik van den Ban
16:30-18:00 – Reception (in the Huygens building)

Titles and abstracts:
Yong-Geun Oh (University of Wisconsin-Madison)
Counting embedded curves in Calabi-Yau threefolds and Gopakumar-Vafa invariants
The Gopakumar-Vafa BPS invariant is an integer invariant of the cohomology of D-brane moduli spaces on Calabi-Yau threefolds in string theory. By a conjectured relationship to Gromov-Witten theory, it is expected that the invariant coincides with the “number” of embedded (pseudo-)holomorphic curves. In this talk, we will explain how the latter integer invariants can be defined for a generic choice of compatible almost complex structures. If time permits, we will comment on the corresponding analog to the open string case and some implications on the wall-crossing phenomenon and open questions towards complete solution to the Gopakumar-Vafa conjecture.

Alexander Braverman (Brown University):
Instanton counting and geometric representation theory
This talk will be a survey of known results about (equivariant) cohomology and K-theory of various moduli spaces closely related to the moduli space of principal G-bundles (here G is a simply connected semi-simple algebraic group over complex number) on the projective plane trivialized at infinity (this is also the same as the moduli space on “instantons on R^4 framed at infinity”). The purpose of the talk will be two-fold: 1) Explain more precisely what kind of moduli spaces we should be talking about and explain why physicists what to integrate various cohomology (or K-theory) classes over these moduli spaces and what kind of answers they expect to get (and why). 2) Explain how one can understand the cohomology (or K-theory) of the above moduli spaces by representation-theoretic means and how this helps to prove predictions from 1 (in some cases).

Erik van den Ban (Utrecht):
Cusp forms on reductive symmetric spaces. (joint work with Job Kuit)
For a real reductive Lie group G, there exists a notion of cusp form, introduced by Harish-Chandra, who used this terminology to emphasize the analogy with the general theory of automorphic forms. The cusp forms coincide precisely with the discrete part of the spectral decomposition of the space of square integrable functions on G. This is in contrast with the theory of automorphic forms, where the discrete part of the spectral decomposition is built from cusp forms and residues of Eisenstein series, which in turn are built from cusp forms of lower rank subgroups. The class of reductive symmetric spaces contains that of the real reductive groups. It would be interesting to have a notion of cusp forms in this setting too. However, the definition of cusp forms is somewhat problematic due to the divergence of certain integrals. A number of years ago, M. Flensted-Jensen suggested an idea for such a definition, but the convergence of cuspidal integrals remained an unsolved issue in general. In this talk, based on ongoing joint work with Job Kuit, I will discuss this convergence problem, and present the solution for symmetric spaces of split rank 1. It turns out that in general the discrete spectrum splits into a cuspidal and a residual part.

EWM summer school
June 6th, 2011
The European Women in Mathematics organise a summer school for PhD students.

Special day on Lie groups
May 17th, 2011

This workshop is organized at the occasion of the thesis defense of Job Kuit, the day before.

11:00 – 11:50, T. Kobayashi (Univ. of Tokyo): Restrictions of Verma modules to symmetric pairs and some applications to differential geometry
13:00 – 13:50, S. Helgason (MIT): TBA
14:10 – 15:00, H. Schlichtkrull (Univ of Copenhagen): Decay on homogeneous spaces of reductive type

Thesis defense of Job Kuit
May 16th, 2011

On Monday May 16, 14:30, Job Kuit will defend his thesis: “Radon transformation for reductive symmetric spaces: support theorems”.

GQT Colloquium
April 8th, 2011
Place: Minnaert building room 211, de Uithof, Utrecht.

10.45 – 11.15 Coffee & tea
11.15 – 12.15 Gerard van der Geer Modular forms for genus three
12.15 – 14.00 Lunch
14.00 – 15.00 Ulrike Tillmann Cobordism theory: old and new
15.00 – 15.30 Coffee & tea
15.30 – 16.30 Arno Kuijlaars Multiple orthogonal polynomials and random matrix eigenvalues
16.30 – Drinks in Library Math. building

Ulrike Tillmann
With the work of Thom in the early 1950s cobordism theory became an important tool for the classification of smooth manifolds. Cobordisms played a very different role in the axiomatic approach to topological quantum field theory by Atiyah and Segal in the late 1980s. In this lecture I will bring these two themes together and explain how recent results shed new light on both of these.

Arno Kuijlaars
The Gaussian Unitary Ensemble is the simplest unitary invariant random matrix ensemble. Its eigenvalues can be analyzed in full detail with the use of Hermite polynomials. General unitary ensembles are associated with orthogonal polynomials with exponential weights. In the fundamental work of Deift et al. (1999) the steepest descent analysis of the Riemann-Hilbert problem for these polynomials was developed in order to establish universality for eigenvalue spacings. Multiple orthogonal polynomials play a similar role in certain other random matrix ensembles such as the random matrix model with external source, and the two-matrix model. I will discuss the Riemann-Hilbert problem and a vector equilibrium problem that describes the global distribution of eigenvalues and is crucial for the asymptotic analysis.

Inaugural Lecture Nicolai Reshetikhin
February 18th, 2011

Title: The harmony of Mathematics and Physics
Date/Time: Friday, February 18, 2011, 16:00
Location: Aula of the University of Amsterdam, Singel 411

GQT Colloquium
February 18th, 2011
The GQT Colloquium on Friday, February 18, 2011, will be combined with the inaugural lecture of Nicolai Reshetikhin.

Venue: Zaal AB.44, gebouw A, Roetersstraat 15, 1018 WB Amsterdam.

10:50-11:15 Coffee and tea
11:15-12:15 Talk Marc Levine
12:15-13:45 Lunch break
13:45-14:45 Talk Michel van den Bergh

After the last lecture everybody is invited to attend the inaugural lecture of Nicolai Reshetikhin, See “The harmony of Mathematics and Physics”.

Mark Levine
Some applications of algebraic cobordism
Abstract: Algebraic cobordism is a version in algebraic geometry of the topological theory of complex cobordism. Originally, it was constructed to help understand rather subtle divisibility properties of characteristic classes, which in turn leads to some interesting results on homogeneous varieties over non-algebraically closed field. Later on, Pandharipande saw how to apply this theory to solve problems in Donaldson-Thomas theory. I will describe the theory of algebraic cobordism and say a bit about its original series of applications to homogeneous varieties, then turn to the work on Donaldson-Thomas theory and conclude with some more recent work of Pandharipande, Lee and Tzeng using algebraic cobordism to study problems in Gromow-Witten theory.

Michel van den Bergh
Formality and deformations
Abstract: Kontsevich’s celebrated formality theorem shows that every Poisson bracket on a C^infty manifold can be quantized into a start product. For a smooth algebraic variety this is not quite true. We will discuss the algebraic version of the formality theorem and its relation to deformation theory. We will also discuss how algebraic formality may be used to prove a conjecture by Caldararu which gives an intriguing connection between algebraic geometry and Lie theory.