Curriculum vitae Personal data Horst Thaler, born on 3rd November 1969 in Klagenfurt, Austria. e-mail: horst.thaler@unicam.it http://web.unicam.it/matinf/dispense/thaler/homepage/main.html Education 1998: Diploma in Physics, received from the University of Graz. Diploma thesis on Complex Langevin for semisimple compact connected Lie groups and U(1). 2001: Ph.D. degree in Physics with specialization in mathematical physics, received from the University of Graz. Ph.D. thesis on Probabilistic representation of solutions of Schrödinger equations on Lie groups and symmetric spaces. Fellowships and Employment DAAD fellowship in Bonn (German academic exchange service), autumn-winter 2000/2001. Fellowship in the framework of the EC Programme Training and Mobility of Researchers, May, 2001. Scientific cooperator at the University of Bonn, summer-autumn 2001. Scientific cooperator at the University of Trento, spring-summer 2002. Erwin Schrödinger fellowship with invitation to Bonn by Prof. Sergio Albeverio, Oct. 2002 - Sept. 2004. Scientific cooperator at the University of Bonn, 2005. Tutor of seminars on Sturm s differential equations and Feller semigroups (winter 2004) and Markov chains (winter 2005), Bonn. Professore a contratto (lecturer) giving lectures on Probability Theory for Mathematics and Statistics for Biotechnology, University of Camerino, 2006/07 and 2007/08. Assegnista di ricerca(research assignment) from Dec. 2007, University of Camerino.
Research position within the program Rientro dei Cervelli from June 2008, University of Camerino. Associate Professor for Probability and Statistics, from Sept. 2013, University of Camerino. Teaching 2006/07, Probability Theory (for postgraduates in mathematics) and Statistics for Biotechnology (for undergraduates), University of Camerino. 2007/08, Probability Theory (for postgraduates in mathematics) and Statistics for Biotechnology (for undergraduates), University of Camerino. 2008/09, Probability Theory (for postgraduates in mathematics). 2009/10, Probability Theory (for undergraduates in mathematics) and Stochastic Processes (for postgraduates in mathematics), University of Camerino. 2010/11, Probability Theory (for undergraduates in mathematics) and Stochastic Processes (for postgraduates in mathematics), University of Camerino. 2011/12, Probability Theory (for undergraduates in mathematics) and Stochastic Processes (for postgraduates in mathematics), University of Camerino. 2014/15, Mathematics and Geometric Laws of Forms (for undergraduates in design) and Geometry and Linear Algebra (for undergraduates in architecture), University of Camerino. Master students 2010, Laura Mazzoni, master thesis: Dalla percolazione ai modelli random-cluster. 2011, Damiano Gentiletti, master thesis: The mathematics between synapses and how numbers can help us to fight epilepsy. Seminar talks at the institute of mathematics and informatics in Camerino 2006 on AdS/CFT correspondence in the Euclidean context. 2005 on The Feynman graph representation of convolution semigroups. at the institute of applied mathematics in Bonn, 2
2000 on Gradient formulas on loop-spaces. and on Probabilistic representation of solutions to Schrödinger equations on Lie groups. 2001 on Clifford bundles.* 2003 on The conformal group in 2 dimensions.* 2004 on AdS/CFT correspondence for 2-dimensional Euclidean field theory. at the institute of theoretical physics in Graz, 1998 on Fiber bundles.* 1996 on Covariant derivatives.* (* these were cycles of lectures, each comprising 3 talks) Other activities Coordinator of the double degree program in applied mathematics between the Universities of Camerino and Clausthal. Conferences, Invitation XVIII International Congress on Mathematical Physics, Santiago de Chile, July 27 - August 1, 2015. Talk On the role of the central limit theorem in 3d quantum gravity. 32 nd Workshop on Foundations and constructive aspects of QFT, University of Wuppertal, May 31 - June 1, 2013. Workshop 5 th International Conference on Stochastic Analysis and its Applications, Hausdorff Center for Mathematics, Bonn, Sept. 5-9, 2011. Workshop Eurocomb 11, Budapest, Rényi Institute, Aug. 29 - Sept. 2, 2011. Presentation of a poster. Workshop Seminal Interactions between Mathematics and Physics organized by the Center of Mathematics and Theoretical Physics, Rome, Sept. 22-25, 2010. XVI International Congress on Mathematical Physics, Prague, Aug. 1-8, 2009. Summer School Combinatorics and Statistical Mechanics, Erwin Schrödinger International Institute for Mathematical Physics, Vienna, July 7-18, 2008. Conference The manifold geometries of quantum field theory, Hausdorff Center 3
for Mathematics, Max Planck Institute for Mathematics, Bonn, June 30 - July 4, 2008. Talk on AdS/CFT by means of functional integrals. Workshop Combinatorics and Statistical Physics, Erwin Schrödinger Institute, Vienna, May 19-30, 2008. Second School and Workshop on Mathematical Methods in Quantum Mechanics, Bressanone, Febr. 26 - March 3, 2007. Talk on AdS/CFT correspondence in the Euclidean context. Workshop Advanced Geometric Methods in Physics, Florence, April 14-18, 2005. Conference Mathematical problems in dynamics and statistical physics, University of Camerino, Sept. 27 - Oct. 2, 2004. Talk on AdS/CFT correspondence for 2-dimensional Euclidean field theory. 13 th Workshop on Foundations and Constructive Aspects of Quantum Field Theory, University of Göttingen, Jan. 23-24, 2004. Talk on An indefinite metric model for interacting quantum fields on globally hyperbolic space-times. Journeés Mathématique et Physique in Lyon on Renormalization, Feb. 7, 2003. Workshop Quantum Probability and Applications, University of Trento, Feb. 22-25, 2003. International Conference on Stochastic Analysis and Applications, St. Petersburg, June 4-10, 2001. Invitation to the Institute of Mathematics in Hull by Prof. Z. Brzeźniak, England, Dec. 2000. Research interests Stochastic analysis and its applications, geometry, quantum physics and field theory. Main results The Langevin equation, a stochastic differential equation (SDE), gives under appropriate conditions the equilibrium distribution of a physical system in the large time limit. The complex Langevin equation used in [2] tries to handle the situation where the physical model is described by some complex action. The resulting SDE and its resulting generator become singular in this case and it is important to find conditions under which the Langevin equation will still give the right answers. This has been investigated in [2] where the process lives on Lie groups. 4
A fruitful interplay between stochastic analysis and the structural properties of Lie groups was employed in [3,5] in order to find solutions to Schrödinger equations. The idea for this approach is due to Halim Doss and essentially uses a rotation of the Wiener process to the complexification of the given manifold. Such complexifications exist for Lie groups and certain symmetric spaces which allows to define the process which enters the probabilistic representation of the solutions. The interest in quantum fields on curved space-times stems from the question how a curved geometry combines with quantum effects. In [4] a model of an interacting quantum field theory is constructed, essentially by perturbing a generalized Gaussian process by a Poisson process, the latter of which is responsible for the non-trivial character of the fields. The scattering behavior of these fields shows a combined effect of non-trivial quantum scattering and non-stationary gravitational effects. In [6,7,12] the so called AdS/CFT-correspondence has been studied. It should relate quantum fields on anti de-sitter spaces to conformal quantum fields living on the boundary of these spaces. For this a mathematical rigorous formalism has been set up which includes truly interacting fields. A different line of research deals with convolution semigroups and their representation in terms of Feynman graphs [8] which gives new numerical recipes for studying diffusion-jump processes. Finally, an enumeration problem has been addressed in [9,10] for coloured harddimer configurations which have appeared in a model of quantum gravity that has been conceived by Benedetti, Loll and Zamponi (BeLoZa). The goal was to better understand the asymptotic behaviour of these objects. Among other things a central limit theorem and a kind of law of large numbers were proved. My current research focuses on three issues. First, there is the goal to underpin the result obtained in [6] where triviality for quantum fields with polynomial interaction and ultra-violet cut-off has been established. According to this result it has to be conjectured that AdS/CFT-correspondence will in general give trivial results when quantum fields on a fixed background spacetime are considered [11]. Secondly, the combinatorics of coloured hard-dimer configurations is further investigated in conjunction with the Laplace transform of the one-step propagator that appears in the model of BeLoZa. The goal is to find explicit estimates for this propagator and also to verify the critical point in the phase diagram as conjectured by BeLoZa [13]. A third topic concerns the study of ergodicity of diffusions that are driven by vector fields with unbounded outward drift in certain directions [12]. 5
Publications and thesis [1] Ph.D. thesis on Probabilistic representation of solutions of Schrödinger equations on Lie groups and symmetric spaces (2001), österreichische Nationalbibliothek, http://www.onb.ac.at/ev/catalogues. [2] Gausterer H., Thaler H.: Complex Langevin for semisimple compact connected Lie groups and U(1), J. Phys. A. 31, no. 11, 2541-2549 (1998). [3] Thaler, H.: Solutions of Schrödinger equations on compact Lie groups via probabilistic methods, Potential Analysis 18, 119-140 (2003). [4] Gottschalk, H., Thaler, H.: An indefinite metric model for interacting quantum fields on globally hyperbolic space-times, Annales Henri Poincaré 4, 637-659 (2003). [5] Thaler, H.: The Doss trick on symmetric spaces, Lett. Math. Phys. 72, 115-127 (2005). [6] Gottschalk, H., Thaler, H.: A comment on the infra-red problem in the AdS/CFT correspondence, Proc. Int. Conf. Recent Developments in QFT, Leipzig 2007, arxiv:0709.4486. [7] Gottschalk, H., Thaler, H.: AdS/CFT correspondence in the Euclidean context, Commun. Math. Phys. 277, 83-100 (2008). [8] Gottschalk, H., Smii, B., Thaler, H.: The Feynman graph representation of convolution semigroups and its applications to Lévy statistics, Bernoulli 14, 322-351 (2008). [9] Bernabei, M.S., Thaler, H.: Central limit theorem for coloured hard-dimers, Journal of Probability and Statistics, Volume 2010, Article ID 781681, 13 pages (2010). [10] Bernabei, M.S., Thaler, H.: A noncommutative enumeration problem, International Journal of Combinatorics, Volume 2011, Article ID 403140, 9 pages (2011). [11] Gottschalk, H., Thaler, H.: A triviality result in the AdS/CFT correspondence for Euclidean quantum fields with exponential interaction, Comm. Math. Phys. 324, no.1, 63-75 (2013). [12] Thaler, H.: An ergodic diffusion with unbounded inward and outward drift, preprint arxiv:1205.4906. [13] Bernabei, M.S., Thaler, H.: Next steps in understanding the asymptotics of 3d quantum gravity, preprint arxiv:1412.3250. 6