# Mathematical lectures by 15-th and 17-th of october

As part of the visit of Italian mathematicians Valter Moretti and Sonia Mazzucchi to the laboratory of topological methods in dynamics this week, three talks will be devoted to applications of functional analysis in mathematical physics and quantum theory.

Everyone is welcome. The first talk will be in Russian (with translation into English for Italians), the second and third talks will be in English (a translation into Russian can be arranged at the request of the audience).

**First talk**

**Seminar**"Functional analysis and its applications", laboratory of topological methods in dynamics, leaders are O.E. Galkin and I.D. Remizov

Tuesday October 15, 1:30 pm, HSE (N. Novgorod, 25/12 B. Pecherskaya St.), gathering at the audience 105

Speaker: Ph.D. Oleg Evgenievich Galkin, Associate Professor, HSE and Lobachevski State University

Title: Parabolic Equations for Measures and Gaussian Semigroups

Annotation:

The report is devoted to the Euclidean analog of the Schrödinger equation for the anharmonic oscillator. We consider generalizations of such an equation to the class of Borel measures in an infinite-dimensional Hilbert space. The Cauchy problem for this equation is solved. In particular cases, explicit formulas are obtained for fundamental solutions (which are a generalization of the Möhler formula) and the uniqueness of a solution to the Cauchy problem is proved. An analog of the Ornstein - Uhlenbeck measure is constructed. The definition of Gaussian semigroups is given and their relationship with the considered parabolic equations is described.

**Second talk**

Thursday October 17, 7:00 p.m., HSE building (Sormovskoye Shosse, 30), gathering at the auditorium 205, reports will be a floor below directly under 205 auditorium.

Speaker: Valter Moretti, Full Professor of Mathematical Physics, Department of Mathematics,University of Trento (Italy).

Abstract:

This talk concerns a research program developed with some collaborators from 2012 to 2018 about a long standing issue with the mathematical formulation of Quantum Theories in the Hilbert space. Up to now, no quantum physical system exist which are described in a real or quaternionic Hilbert space, in spite of a result by M.P. Solér who established in 1995 that Quantum Theories, from the viewpoint of the general lattice theory, may be formulated in real, complex, and quanternionic Hilbert spaces.

References:

[1] V. Moretti and M. Oppio, Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one// Rev. Math. Phys. 31, (2019) 1950013 DOI: 10.1142/S0129055X19500132 arXiv:1709.09246

[2] V. Moretti and M. Oppio, The correct formulation of Gleason's theorem in quaternionic Hilbert spaces// Ann. Henri Poincaré 19 (2018), 3321-3355 DOI: 10.1007/s00023-018-0729-8 arXiv:1803.06882

[3] V. Moretti and M. Oppio: Quantum theory in real Hilbert space: How the complex Hilbert space structure emerges from Poincaré symmetry// Rev. Math. Phys. 29 (2017) 1750021 DOI: 10.1142/S0129055X17500210 arXiv:1611.09029

[4] R. Ghiloni, V. Moretti and A. Perotti: Spectral representations of normal operators via Intertwining Quaternionic Projection Valued Measures// Rev. Math. Phys. 29 (2017) 1750034 arXiv:1602.02661

[5]R. Ghiloni, V. Moretti and A. Perotti: Continuous slice functional calculus in quaternionic Hilbert spaces// Rev. Math. Phys. 25 (2013) 1350006 arXiv:1207.0666**Third talk**

Speaker: Sonia Mazzucchi, Associate Professor of Probability, Department of Mathematics, University of Trento (Italy)

Title: An introduction to generalized Feynman-Kac formulae and the mathematical theory of Feynman path integrals.

Oleg Galkin

Senior Research Fellow

Ivan Remizov

Senior Research Fellow