Schedule

Schedule

Welcome Reception

The Welcome Reception will take place Wednesday, June 26, 4:30 p.m. in Mensa and in case of good weather on the Red Square, outside Mensa.

Group Photo

The group photo will take place Wednesday, June 26, directly before the Welcome Reception on the Red Square, next to Mensa. If you want to be on the photo, please agree to it at registration.

Conference Dinner

The conference dinner will take place Friday, 28.06. at 7 pm in the restaurant Ristorante Goldener Hecht da Cesare in Erlangen city centre, Glockenstr. 8, 91052 Erlangen.

Guided Tour

On Saturday, June 29,  there is a guided tour of Erlangen, including some of its mathematical history. The tour will be in English and starts and ends in Erlangen city centre:

• start of the tour: 10:00 a.m. at Hugenottenplatz, at the entrance of the church
• end of the tour: at noon at Hugenottenbrunnen (fountain) in Schlossgarten

Titles and Abstracts

• Maria Stella Adamo:
  Reflection positivity twofold:
  OS axioms as well as representations for Lie groups
  Reflection positivity appears in Osterwalder-Schrader’s (OS) reconstruction theorem, which gives conditions to be satisfied by correlations functions of a Euclidean field theory to produce a relativistic quantum field theory à la Wightman. Specifically, under OS axioms, one can construct the analytic continuation of Euclidean correlation functions encoding positivity to produce distributions which verify Wightman’s axioms. In a work in progress with Yuto Moriwaki and Yoh Tanimoto, we show OS axioms and, in particular, reflection positivity, for correlation functions defined through quasi-primary fields of a unitary VOA satisfying polynomial energy bounds. Reflection positivity can be investigated for unitary representations of Lie groups, where analytic continuation issues also appear. In collaboration with Karl-Hermann Neeb and Jonas Schober, we investigated reflection positivity for the integers, the real and circle groups. We used Hankel operators to produce reflection positive (RP) representations in the regular multiplicity-free case for the integers and real groups. For the circle group, we approached the study of RP representations by using RP functions. These functions can be analytically continued to the beta-strip and are in 1-1 correspondence with continuous positive functions in the real line, which verify the beta-KMS condition.
  
• Daniel Beltita:
  On an inverse problem in representation theory of Lie groups
  While a direct problem in representation theory would require the description of the representations of a group in terms of its structure, we plan to discuss an inverse problem in representation theory of Lie groups. More specifically, we focus on the question of the extent to which a Lie group can be distinguished from the other Lie groups in terms of its corresponding unitary representation theory. That question actually has an easy negative answer for several Lie groups. Yet, we provide affirmative answers for some other Lie groups, including the Heisenberg groups.
  The presentation is based on joint work with Ingrid Beltita.
• Helge Glöckner:
  Regularity properties of infinite-dimensional Lie groups
  Consider a Lie group G modelled on a locally convex space, with neutral element e. A time-dependent left-invariant vector field on G is determined by a path c in the Lie algebra L(G) of G. If an integral curve starting at e exists for each smooth path c in L(G) and depends smoothly on c, then G is called regular. The question whether each Lie group (modelled on a sufficiently complete locally convex space) is regular has remained open since the 1980s, when John Milnor first introduced the concept. It is one of the miracles of the theory that regularity could be established for all concrete examples, despite the absence of a general argument. In fact, most examples even have stronger regularity properties: smooth paths can be replaced with continuous paths in the definition of regularity, or with Bochner integrable functions.
  The talk will provide an introduction to the topic and describe some recent results.
• Joachim Hilgert:
  Pairing formulas for resonant and coresonant states
  Let M be a locally symmetric space of negative curvature. Based on ideas of Patterson and Sullivan one can construct geometric measures on the geodesic boundary of M and from that what’s now called Bowen-Margulis-Sullivan measures on the sphere bundle invariant under the geodesic flow. The Thurston-Sullivan „smearing argument´´ allows to estimate the total volume of these invariant measures by L²-norms of certain eigenfunctions of the positive Laplace-Beltrami operator L on M. In the case of compact M one can associate  distributions on the sphere bundle with eigenfunctions of L by a  quantization procedure (following Anantharam-Zelditch, who introduced  them for hyperbolic surfaces, these distributions are called Patterson-Sullivan distributions rather than Bowen-Margulis-Sullivan distributions). In this case, using group theory, the „smearing estimate´´ can be made much more precise by turning it into an equality equating the evaluation of the Patterson-Sullivan distribution with the L²-norm of the corresponding eigenfunction. Polarizing this equality one obtains the pairing formula from the title. In this talk I will discuss the pairing formula, some applications and its analogon for finite graphs.
  
• Karl Heinrich Hofmann:
  Recollections on the development Lie theory and Karl-Hermann Neeb in it
• Bas Janssens:
  Localization of positive energy representations for gauge groups and diffeomorphism groups
  Let G be a Lie group, equipped with a 1-parameter group of automorphisms. A positive energy representation is a projective unitary representation of G that extends to the semidirect product R×G in such a way that the generator of the R-action has spectrum bounded from below. We consider two situations:
  1) G is the group of compactly supported diffeomorphisms of a manifold M.
  2) G is the group of compactly supported gauge transformations of a principal fibre bundle over M.
  The 1-parameter group of automorphisms is induced from a fixed-point free flow on M (in case 1), or from a lift of this flow to the principal bundle (in case 2). We show that positive energy representations are either trivial (in case 1), or localize at a 1-dimensional, R-equivariantly embedded submanifold of M (in case 2).
  This is joint work with Karl-Hermann Neeb and Milan Niestijl.
• Timothée Marquis:
  On the centre of Iwahori-Hecke algebras
  Iwahori-Hecke algebras C_q(W) are deformations of the group algebra of a Coxeter group W. They are intimately related with the representation theory of groups with a BN-pair whose associated building is locally finite (such as Kac-Moody groups over finite fields). We recently proved that the centre of C_q(W) is trivial (in the sense that it only consists of the constant functions) whenever W is of (irreducible) indefinite type — when W is of finite or affine type, this is not true anymore. This is a consequence of a purely Coxeter group-theoretic result on conjugacy classes of W. After reviewing the context and motivation for this latter result, I will explain the key ideas behind its (partly geometric, partly combinatorial) proof.
  This is joint work with Sven Raum.
• Vincenzo Morinelli:
  A New Analytical-Geometric Approach to Algebraic Quantum Field Theory
  In this presentation, we will provide an overview of the analysis developed in recent years, in collaboration with K.-H. Neeb and G. Ólafsson, regarding a new analytical-geometric approach to Algebraic Quantum Field Theory. We will explore the deep relationship between the geometry of standard subspaces, the geometry of Euler elements in the Lie algebra of a Lie group and the geometry of an Algebraic Quantum Field Theory. We will also discuss the implications of this analysis for representation theory, standard subspace theory and the algebraic approach to Quantum Field Theory.
  A survey of the project can be found at:https://publications.mfo.de/handle/mfo/4092
• Bent Ørsted: 
  Intertwining differential operators
  The wave equation, Dirac’s equation, and Maxwell’s equations are basic in classical mathematical physics. They are also examples of differential operators with many symmetries, namely they are intertwining operators for representations of Lie groups acting in space-time such as the conformal group of Minkowski space. We shall give more examples from the representation theory of semi-simple Lie groups related to Verma modules and principal series representations.
• Hadi Salmasian:
  Weyl algebras, quantum groups, and interpolation polynomials
  
  The algebra of invariant polynomial differential operators on a multiplicity-free G-space, where G is a reductive group, has a distinguished basis that generalize the classical Capelli operators. Kostant proposed to compute the spectrum of this basis, and subsequent work by Kostant-Sahi and then Sahi resulted in a family of symmetric polynomials that are usually called interpolation Jack polynomials. In this talk we describe q-deformations of the above results, connecting interpolation Macdonald polynomials to spectra of suitably defined q-differential operators associated to quantum symmetric spaces of U_q(gl_n). This talk is based on a joint project with G. Letzter and S. Sahi.
• Cornelia Vizman:
  Universal central extension of the Lie algebra of exact divergence-free vector fields
  Two conjectures on universal central extensions of Lie algebras of vector fields are due to Claude Roger [2]. The one about the universal central extension of the Lie algebra of Hamiltonian vector fields is proven in [1].
  In this talk I will explain how to solve the second one: the universal central extension of the Lie algebra of exact divergence free vector fields. Unlike the Hamiltonian setting, the divergence free setting requires a detour into the realm of Leibniz algebras.
  With Bas Janssens, Delft University of Technology, and Leonid Ryvkin, University Claude Bernard Lyon 1 and University of Göttingen.
  [1] B. Janssens, C. Vizman, Universal central extension of the Lie algebra of Hamiltonian vector fields, IMRN, 2016.16 (2016) 4996-5047.
  [2] C. Roger, Extensions centrales d’algèbres et de groupes de Lie de dimension infinie, algèbre de Virasoro et généralisations, Rep. Math. Phys. 35 (1995) 225-266
• Stefan Wagner:
  Factor systems as a computational framework for noncommutative principal bundles
  Free actions, in the sense of Ellwood, provide a natural framework for C*-algebraic noncommutative principal bundles, which are becoming increasingly prevalent in various applications to noncommutative geometry and mathematical physics. One of the key features of free actions of compact quantum groups are their associated factor systems, which make them accessible to classification, K-theoretic considerations, and computations in general. In this talk we give an introduction to the theory of factor systems for free actions of compact quantum groups, followed by a discussion of various applications.