## Alexander Plakhov

## A periscope theorem

A parallel bundle of light rays is transformed into another (and codirectional) parallel bundle by several reflections from curved mirrors. This transformation induces a diffeomorphism of two plane domains representing the cross sections of the original and the final bundles. The question is: which diffeomorphisms can be realized this way, and how many reflections are needed? We prove that (a) a gradient diffeomorphism can be realized by 2 reflections, (b) an orientation reversing diffeomorphism can be realized by 4 reflections, and (c) a general diffeomorphism can be realized by (at most) 6 reflections.

This is a joint work with S. Tabachnikov (Penn State, USA) and D. Treschev (Academy of Science, Russia).

There still are interesting open problems related to this problem. In particular, given a PDE of 2nd order in a plane domain, does there exist a solution whose gradient is a diffeomorphism of two plane domains?

This is a joint work with S. Tabachnikov (Penn State, USA) and D. Treschev (Academy of Science, Russia).

There still are interesting open problems related to this problem. In particular, given a PDE of 2nd order in a plane domain, does there exist a solution whose gradient is a diffeomorphism of two plane domains?

Paula Rama

## Lexicographic polynomials of graphs and their spectra

For a (simple) graph H and non-negative integers c

_{0},c_{1},…,c_{d}(c_{d}≠ 0), the expression p(H)=∑_{k=0}^{d}c_{k}·H^{k}is called the lexicographic polynomial in H of degree d, where the sum of two graphs is their join and c_{k}·H^{k}is the join of c_{k}copies of H^{k}. Here H^{k}is the k-th power of H with respect to the lexicographic product. The spectrum (if H is regular) and the Laplacian spectrum (in the general case) of p(H) are determined in terms of the spectrum of H and c_{k}'s. Based on this, some properties of graphs being lexicographic polynomials are deduced and their applications in constructions of infinite families of cospectral or integral graphs are shown. Joint work with Domingos M. Cardoso, Paula Carvalho, Slobodan K. Simi\'c, and Zoran Stani\'c.

Tatiana Tchemisova

## On study of properties of special nonlinear problems arising in parametric SIP

We study parametric Semi-infinite Programming (SIP) problems with finitely representable compact index sets and investigate dependence of solutions of these problems on the parameters. We show that differential properties of solutions of the parametric SIP problems can be formulated in terms of solutions of some special auxiliary Nonlinear Programming (NLP) problems that depend on a finite number of integers (parameters). We discover different properties of solutions of these NLP problems w.r.t. the values of the integers that permit us to obtain important conclusions about behavior of solutions of the original parametric SIP problems under parameter perturbations.