# Dr. Alexander Alexandrov

Mathematical Physics

Junior Fellow

January 2014 - October 2014

### CV

Born in 1979; 2002 M.S. in Applied Mathematics and Physics (summa cum laude), Moscow Institute of Physics and Technology (MIPT), Department of General and Applied Physics (DGAP). 2005 PhD in Theoretical Physics, Institute for Theoretical and Experimental Physics (ITEP). 1998-2006 research assistant and researcher at ITEP Moscow, Russia; 2006-2008 postdoctoral fellow at IHES, Bures-sur-Yvette, France; 2008-2009 postdoctoral position at Blackett Laboratory, Imperial College, London, UK; 2009-2011 postdoctoral fellow at IPhT CEA, Saclay and LPT ENS, Paris, France; 01-04/2012 visitor at Hausdorff Research Institute for Mathematics, Bonn; since 2011 postdoctoral fellow at the University of Freiburg, Mathematics Institute.

Research interests: mathematical physics, integrability, string theory, matrix models.

### Selected Publications

- Alexandrov, A., Morozov, A., & Mironov, A. (2004). Partition functions of matrix models: first special functions of string theory. International Journal of Modern Physics A, 19(24), 4127-4163.
- Alexandrov, A. S., Mironov, A. D., & Morozov, A. Y. (2007). M-theory of matrix models. Theoretical and Mathematical Physics, 150(2), 153-164.
- Alexandrov, A., Mironov, A., & Morozov, A. (2009). BGWM as second constituent of complex matrix model. Journal of High Energy Physics, 2009(12), 053.
- Alexandrov, A. (2011). Matrix models for random partitions. Nuclear Physics B,851(3), 620-650.
- Alexandrov, A., Kazakov, V., Leurent, S., Tsuboi, Z., & Zabrodin, A. (2011). Classical tau-function for quantum spin chains. arXiv preprint arXiv:1112.3310.

### FRIAS Research Project

**Matrix models, tau-functions and W-operators**

Matrix models constitute an outstanding class of models in modern mathematical physics. This class appears to be extremely universal so that many problems of different, seemingly unrelated, branches of physics and mathematics can be reformulated (and consequently solved) in terms of matrix models. One of possible explanations of this universality is a deep interconnection between matrix models and integrable systems. In its simplest and most investigated form this interconnection can be formulated as follows: the matrix integrals are the tau-functions of the classical solitonic integrable systems.

In recent years it became clear that in addition to the existing power tools of matrix model theory, there exist a new, extremely promising approach to description of the matrix model tau-functions. This approach is based on the explicit construction of the operators describing infinite-dimensional group elements. For KP/Toda case all corresponding algebra operators belong to a small part of the huge W-algebra, so that we call them W-operators.

The main goal of this project is to complete the initial phase of development of the theory of W-operators in matrix models. For this purpose, we will establish the connection “W-operators – matrix models” for the key examples. Simultaneously we will clarify the relation between description in terms of the W-operators and other important elements of the matrix model theory, such as Virasoro/W-constraints, topological recursion and decomposition formulas. Obtained results will be applied to the problems of modern mathematical physics.