Dr Sergey Sergeev

• PhD in Mathematics (Moscow State University, 18 April 2008)
• MSc in Physics (Moscow State University, 31 January 2004)

Sergey Sergeev was born in Moscow in 1981, living there until 2008, when he defended a PhD in Mathematics at the Faculty of Mechanics and Mathematics of M.V. Lomonosov Moscow State University.  Sergey also has MSc in Physics, obtained from the Physics Department of the same University in 2004.

The tropical mathematics has been his research area from the very beginning. In the Physics Department of MSU, Sergey studied under the guidance of A.N. Sobolevskii, G.L. Litvinov and academician V.P. Maslov, his first official supervisor. His main area of interest was tropical convexity (from tropical segments to tropical cones and the Hahn-Banach separation of several tropically convex sets).

In 2008 Sergey became a Research Fellow in Birmingham and worked on an EPSRC funded project “Feasibility and Reachability in Max-linear Systems”  with P. Butkovic, the Principal Investigator for the project, and Visiting Researcher, H. Schneider. Together they wrote several works on tropical linear algebra (on the tropical matrix powers, visualisation scaling and, later, on Z-matrix equations). Sergey also did some research in tropical convexity and the connection between tropical algebra and mean-payoff games, with S. Gaubert.  In 2010 Sergey went to France (Ecole Polytechnique), where he worked in the Max-plus team of S. Gaubert, on the tropical analogue of linear-fractional programming. In 2012 he returned to Birmingham to work as a Research Fellow on another EPSRC funded project.

Since March 2014 Sergey has been a Lecturer in Mathematical Optimisation. Currently he is working on an EPSRC funded grant EP/P019676/1 "Tropical Optimisation", being focused on various forms of tropical optimisation such as tropical pseudoquadratic and bilevel optimisation. He has been active as organiser of one-day minisymposia in Tropical Mathematics such as a minisymposium on Tropical Mathematics and its Applications in the framework of ILAS-2016 conference in Leuven, Belgium and LMS Workshop on Tropical Mathematics held in Birmingham, November 2017. He is teaching Non-Linear Programming II, a module for 4th year students and Ph.D students which is mostly on Dynamic Programming and Optimal Control. 

In June 2018, Sergey co-organized a one-day conference "Tropical Mathematics and Optimization for Railways" (joint with Prof. Clive Roberts).

• Lectures in Non-Linear Programming II
• Supervising MSci projects in tropical mathematics and optimisation
• Supervising Research Skills and Summer Projects (MSc) in Financial Engineering
• Year 1 and Year 2 tutorials
• Supervising PhD students in Tropical Mathematics

Sergey’s main research activity has been in the area of tropical linear algebra and tropical convexity. These are the linear algebra and convex geometry developed over the max-plus semiring: the set of real numbers with adjoined negative infinity element, equipped with the addition “a+ b”:= max(a, b) and the multiplication “ab” = a + b. These areas can be viewed in a more general framework of idempotent/tropical mathematics, which has applications in such diverse areas as algebraic geometry, discrete event systems, scheduling, optimal control and Hamilton-Jacobi PDE’s, string comparison algorithms, static analysis of computer programs.

His main achievements in tropical linear algebra and tropical convexity can be briefly described as follows. Citations refer to the list of selected papers given below.

1. In collaboration with P. Butkovic and H. Schneider, Sergey generalized  the cyclicity theorem on tropical matrix powers to the reducible case in the form of CSR expansions  and described attraction cones (sets of vectors converging to an eigenvector) as solution sets of tropical linear systems of equations [6,7,10]. In the same collaboration they developed the tropical analogue of the theory of Z-matrix equations (of the form x=Ax+b)[11], commuting matrices (with R.D. Katz [8] ) and the core of nonnegative matrix [14]. With G.Merlet and T. Nowak, Sergey applied the CSR expansion idea to obtain new more efficient bounds on the periodicity of tropical matrix powers [18,19].
2. In collaboration with S. Gaubert [3] Sergey introduced the max-plus analogue of cyclic projections method, obtaining separation theorem for several max-plus convex sets, and deducing the max-plus analogue of Helly’s theorem. Using equivalence between tropical polyhedra and mean-payoff games, the group proposed the first known method for computing the whole spectrum of tropical two-matrix eigenproblem[13], and worked (with R.D. Katz) on new algorithms of tropical linear programming and their possible application to static analysis of computer programs [9]. A number of results on tropical convexity were also obtained in an individual paper [5] and in collaboration with V. Nitica [16,17]. Notably, they also developed an analogue of tropical convexity over max-min semiring.

Apart from the tropical linearity, his interests also include nonlinear optimisation, combinatorial games, nonnegative matrices, theory of partially ordered sets, linear algebra over fuzzy semirings, abstract convexity, Perron-Frobenius theory and diagonal similarity matrix scaling.