 ## Plenary speakers

Talk title: The Tikhonov method in gene expression models

Date & time: Monday, September 2, 2019

The Tikhonov method reduces variables of the complex systems under theassumption that some parameters are small. In the presented gene expressionmodel we will consider two processes: dimer formation and binding to thepromoter. Using the Tikhonov theorem we simplify the entire system to theclassical one in different ways depending on the time scales of the involvedprocesses. More precisely, we prove that the solutions of the full system canbe approximated by the solutions of the reduced systems.

Talk title: On mathematical gems of Krystyna Kuperberg, and how they got to me

Date & time: Sunday, September 1, 2019

In the first part of my talk, I would like to offer a discussion of several mathematical discoveries made by Krystyna Kuperberg, and their impact on the field of topology and dynamical systems. A common denominator for them is that they bring together ingenious geometric intuitions and aesthetic execution in construction. Then I would like to share the impact they had on my own research.

Talk title: Integrable versus non-integrable systems in Celestial Mechanics

Date & time: Saturday, August 31, 2019

All bodies of the Solar system, either natural or artificial, are endowed by two motions: a revolution around a central object and a rotation about its own spin-axis. Both kinds of motion can be modeled starting with an integrable dynamical system. The motion describing the revolution is modeled in a first approximation by Kepler’s two-body problem, which is well known to be integrable with solutions represented by conic trajectories. The motion describing the rotation of a rigid body is known to be integrable at least in three celebrated cases discovered by Euler, Lagrange and Kowalewska.

The integrable approximation represents a first step for the understanding of the motion, but as soon as a more elaborated model is considered, a complex dynamics arise both in orbit and rotation. The stability of such non-integrable models can be studied through mathematical methods such as perturbation theories and KAM theorem. We briefly describe such theories and some of their applications to Celestial  Mechanics.

Talk title: Sofia Kowalewska-- the first woman professor holding a university chair

Date & time: Sunday, September 1, 2019

The talk will present the turbulent life of the famous mathematician Sofia Kowalewskaya -- a strong woman endowed with an original mathematical talent and great inner power for bringing her aspirations to completion. After describing her growing up in Palibino, Moscow and Kaluga we will focus on her "freedom periods" in Saint Petersburg, Heidelberg, Berlin, Goettingen and Stockholm, related to her mathematical career. The talk will be given in Polish and English, with slides in English.

Talk title: On Ball-Evans Conjecture

Date & time: Saturday, August 31, 2019

The primary focus of this talk is the question of J.M. Ball (Oxford 1999) and L.C. Evans  regarding whether a Sobolev homeomorphism  between Euclidean domains can be approximated (uniformly and in the norm topology of the Sobolev class) by piece-wise affine invertible mappings. It suffices to construct such an approximation with diffeomorphisms (a result of independent interest). In case of planar domains we provide an affirmative answer to Ball-Evans question. Indeed we show that Sobolev homeomorphisms can be approximated (in the above manner) with diffeomorphisms. Refer to it as "Diffeomorphic Approximation of Sobolev Homeomorphisms". Our  proof relies on a generalization of the  Rado-Kneser-Choquet  Theorem about harmonic mappings to the case of  p-harmonic mappings.
The solution of Ball-Evans conjecture is due to joint work of T. Iwaniec,  L. Kovalev and J. Onninen; Arch. Rat. Mech. Anal. 201 (2011).

Talk title: Asymptotic properties of solutions to nonlinear second–order diﬀerence equations

Date & time: Monday, September 2, 2019

This speech is devoted to the study of some nonlinear second order diﬀerence equations from asymptotic point of view. In the ﬁrst problem, we consider some generalisation of the discrete version of the classical Sturm-Liouville boundary value problem on half line. Assuming diﬀerent types of growth conditions on a nonlinear part of the equation, we get the existence of a solution to our problem by Schauder’s ﬁxed point theorem. In the second problem, the existence of a solution to a nonlinear second-order equation with prescribed asymptotic behaviour is considered. o(ns), where s ≤ 0, is used as measure of approximation of solutions. The ﬁxed point approach is used to get main theorem. In the last problem, we present the suﬀicient conditions for the existence of a sequence of positive homoclinic solutions to a nonlinear boundary value problem on integers with pk-Laplacian. To achieve our goal the variational technique is used.

Talk title: Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models

Date & time: Sunday, September 1, 2019

We will discuss results that relate the following two contexts:

(i) Given a Riemann metric G on a thin plate, we study the question of what is the infimum

of the averaged pointwise deficit of an immersion from being an orientation-preserving isometric immersion of G over all weakly regular immersions. This deficit is measured by the non-Euclidean energies E^h, which can be seen as modifications of the classical nonlinear three-dimensional elasticity.

(ii) We perform the full scaling analysis of E^h, in the context of dimension reduction as the plate's thickness h goes to 0, and the derivation of Gamma-limits of h^{-2n}E^h for all n. We show the energy quantization, in the sense that the even powers 2n of h are indeed the only possible ones (all of them are also attained).

For each n, we identify conditions for the validity of the scaling h^{2n}, in terms of the vanishing of Riemann curvatures of G up to appropriate orders, and in terms of the matched isometry expansions.  We also establish the singular asymptotic behaviour of the minimizing immersions.

The problems that we discuss arise from the description of elastic materials displaying heterogeneous incompatibilities of strains that may be associated with growth, swelling, shrinkage, plasticity, etc. Our results and methods display the interaction of nonlinear pdes, geometry and mechanics of materials in the prediction of patterns and shape formation.

Talk title: Women in PDE's - regularity issues for nonlinear equations and systems

Date & time: Sunday, September 1, 2019

We will discuss various problems arising in the analysis of nonlinear PDE's, especially concerning the regularity of solutions. The beginning of the story is of course  the famous theorem of Sophie Kowalevski about the existence of analytic solutions to the Cauchy problem for a first order system with analytic coefficients.
We will present the impact of results obtained by Olga Ladyzhenskaya, Nina Uraltseva and Karen Uhlenbeck for the regularity theory of nonlinear elliptic partial differential equations and systems.

Talk title: Mathematical thinking and learning

Date & time: Sunday, September 1, 2019

The topic of the lecture is devoted to the process of teaching-learning mathematics. This field, as we all know, was especially close to Ms Zofia Kowalewska. A multi-faceted analysis of the potential causes of the difficulties associated with this process will be conducted. Special attention will be paid to epistemological obstacles whose source is in the specific nature of mathematical concepts and reasoning. We will consider the issue of the development of certain mathematical concepts in synchronous and diachronic perspectives as well as the conclusions resulting from the theory and practice of teaching mathematics. The concept of anthropomathematics will be outlined as a specific approach to the scientific field of mathematics in an educational context.

Talk title:  Karen Uhlenbeck and her mathematics Abel Prize 2019

Date & time: Monday, September 2, 2019

The Abel Prize (a „mathematical Nobel”) for 2019 was awarded to Karen Uhlenbeck.

We will describe her exceptional mathematical career which lead her to the centre of most important developments in mathematics in the last century. Her research started with general problems in the calculus of variations and went through minimal surfaces and harmonic maps to Yang-Mills fields and gauge theory. Her contributions to geometric analysis were crucial in the proofs of revolutionary results of Simon Donaldson in topology with the use of Yang-Mills equations. Gauge theory, invented by physicists and being essential in mathematical physics today, turned out to be an efficient tool for solving problems in topology of 4-manifolds. Some of Karen Uhlenbeck's results used in this field were presented at the International Congress of Mathematicians in Warsaw in 1983.