My fields of interest are large games (i.e. games with a continuum of players), dynamic games and optimal control theory, I'm also interested in applications of this kind of tools to economics and ecology. I can superwise PhD theses related to all those aspects of my scientific interests, both applicational and theoretical. Dynamic games are mathematical tools to model behaviour of at least two interrelated decision makers, called players, each of them aiming their own objective in an environment changing dynamically in response to their decisions, while the optimization of each player is influenced by the decisions of the rest: like in e.g. an open access marine fishery extraction problem, a commuting problem or in a pursuit-evasion problem.

1. "The tragedy of the commons" in dynamic context. The term "the tragedy of the commons" describes a situation in which many individuals use common resources, each of them pursuing their own objective, and this rationality leads to inefficiency in use of the resource. Mathematically, it is a class of games, in the ecological context, dynamic games, including differential games. "The tragedy of the commons" encompasses a variety of problems, e.g. overexploitation of fisheries, air and water pollution, spread of epidemics in presence of individual means decreasing spreading the virus, greenhouse effect or space debris.
2. Differential and multistage games: theoretical works on linear quadratic and linear logarithmic problems with constraints, with possible extensions.
3. Optimal control problems, including problems with infinite time horizon and constrained problems.
4. Additionally, a scholarship from NCN grant on "the tragedy of the commons" in dynamic context is available for a successful candidate.

The main goals of this PhD project is the construction and study of stability of nonperiodic structures which minimize certain energy functionals – hamiltonians of interacting particles. Examples are based on nonperiodic-tilings and substitution systems like Thue-Morse or Fibonacci sequences. The research will involve ergodic theory and Gibbs measures of classical lattice models of statistical mechanics.

Perception: a driving factor in multiscale collective dynamics. Collective dynamics models such phenomena as pedestrian flows, vehicular traffic, distribution of goods or emergence of languages. The common factor in these phenomena is that their evolution is driven not necessarily by laws of physics, but by less understood concepts of communication and perception. The two main aims of the project are: *Development of a multiscale framework for models with singular perception in the spirit of Hilbert's 6th problem* and *Derivation of perception through machine-learning assisted optimization*. The engame is to uderstand the concept of perception, as it appears in cognitive psychology, on a mathematical level.

Measures may be used to define the Dirichlet form on low dimensional sets. From this starting point I will pass to more complicated optimization problems involving measures like the scalar Free Material Design. A separate topic is convergence of gradient flows depending upon a parameter. One approach is based on notions from the Calculus of Variations. Another one exploits tools of the Dynamical Systems.

I am going to propose research in which modern Mathematics leads to modern applications: a multiscale approach to the reality.

In the analysis of elliptic and parabolic partial differential equations potential estimates provide elegant and powerful way of transfering regularity from data to the solution. This is remarkable how deep consequences they imply, in particular in the case of strongly nonlinear operators. We shall be interested in developing the potential theory and regularity of solutions to problems with measure data and involving operators of very general growth.

We will talk about smooth, almost smooth, sifted and balanced numbers on one side and the numbers having the constant Dirichlet and elliptic curves signatures on the other. The signature is related to the values of Dirichlet character and Frobenius traces of elliptic curve for prime arguments. Such special numbers play  significant role in solving some hard computational problems - the integer factoring problem (IFP) is the good example. We will focus on the deterministic polynomial time algorithms and the algorithms with oracles for IFP. The behaviour of special numbers if related to the complexity of such algorithms on one side and to the evaluation of number of positive integers that can be factored with the aid of the given algorithm (with oracle). Some open problems will be pointed out.

Application of a derivation of the Shannon entropy and the Fisher information metric from the first principles of convex geometry with use of homological algebra [M.’21], to study complexity of polytopes and problems in linear programming, entropy and the Kolmogoroff complexity or topological complexity of networks and dynamical systems. In the further perpective, for anyone more interested in applied mathematics: application the thus obtained theoretical results as methods to solve the fundamental problems related to replicators in abiogenesis or evolutionary and molecular biology, or Bayesian belief revision in medical diagnostics and epidemiological models.

Studying the moduli spaces of monodromy representations of the braided fundamental groupoid [Bigdeli—M.’21] in quantum groups, as a quantum extension of the moduli space of flat connections, with focus on the Chern-Simons theory and invariants of braids and links.

I will present open problems of the theory of understanding the bounds of stochastic processes. The results obtained in this way have many applications in probability: empirical processes, selector processes, infinitely divisible processes.

We present some open problems at the interface of convex geometry, probability and information theory.

In the theory of Sobolev spaces there are a lot of open problems. The examples of such a one are: Hardy and Sobolev inequalities, interpolation inequalities, trace operator, density results,interpolation between weighted Sobolev spaces. They can be stated in the Euclidean as well as in the metric setting. Having them at hands, one can develop existence and regularity theory. All such issues are of my interest.

• Frobenius Theorem (1870s) on integrable distributions, which gives rise to foliations of the underlying manifolds.
• Engel, von Weber and Cartan breaking away from the integrability assumption and heading towards the other extreme: higher and higher Lie multiplications of distribution' generators yielding more and more dimensions in the tangent bundle.
• One classical specification of this - Goursat distributions (Goursat 1922, Kumpera 1978, till present).
• Other nowadays trendy specifications: special multi-flags (2001 till present). In particular - special 2-flags.
• Singularities of their nonholonomic behaviours encoded by the growth vectors. Procedures to compute the latter.
• What is known and what is open.

• Dynamics of entire and meromorphic functions in the plane. Geometric and topological properties of invariant sets. Invariant measures.
• Random dynamics in one and several dimensions. This includes random complex dynamics as well as random dynamical systems on (real) manifolds. Questions include: properties of stationary measures, stability, ergodic properties of stationary measures, dimension of stationary measures.
• Questions arising from potential theory in the plane: studying the harmonic measure on invariant domains for dynamics on the plane in deterministic and/or random setting.
• Dynamics of rational maps on the Riemann sphere, in particular dimension properties of natural invariant sets.

Graded manifolds are manifolds with an atlas whose local coordinates have associated weights (e.g. in Z_2 or N) and the transformations of coordinates respect the weight. There are many natural examples (e.g. supermanifolds, double vector bundles or higher tangent bundles) with applications in physics. This is a theory which is reacher than just differential geometry.

Let 𝕂 be an algebraically closed field of characteristic zero. We say that a polynomial automorphism f: 𝕂ⁿ → 𝕂ⁿ is special if the Jacobian of f is equal to 1. We show that every (n-1)-dimensional component H of the set Fix(f) of fixed points of a non-trivial special polynomial automorphism f: 𝕂ⁿ → 𝕂ⁿ is uniruled. Moreover, we show that if f is non-special and H is an (n-1)-dimensional component of the set Fix(f), then H is smooth, irreducible and H=Fix(f). Moreover, for 𝕂 = ℂ if f is non-special and Jac(f) has an infinite order in ℂ^*, then the Euler characteristic of H is equal to 1.

We will talk about phenomena which, despite being elementary in formulation and accessible for computer experiments, show unity of mathematics because they occur in problems from seemingly unrelated disciplines. Starting with the classical facts about values of the Riemann zeta function at integral arguments, we will hint on their conjectural generalizations which involve studying numbers of solutions to polynomial equations over finite fields. We will then speak about how such "number of solutions" may depend on a parameter and give a number theorist's outlook on differential equations."

In my short communication I will speak about two recently investigated problems on uniqueness and regularity of PDEs. Next, I'll introduce Bellman's dynamic programming principle and mention a few real world problems (including National Bank's strategies of dealing with debts) which are reduced to the properties of solutions of particular PDEs.

Within this project we would like to provide the mathematical analysis of nonlinear Partial Differential Equations (PDEs) and their solutions. In particular we concentrate on hydrodynamic models, since many phenomena in nature, technology, sociology are described by models seeing it as a flow.

In the comprehensive description of many phenomena the challenge is to take into account: collective behaviour and swarming of objects, microstructure associate with particles or objects interaction with fluid which are immersed in, non-Newtonian rheology of the fluid, changes of the shape and volume of the domain, heat effects, different scales of certain parameters which matters in the system are dominant or negligible. The above mentioned phenomena are a source of nonlocal effects, nonlinearities in the system, dependence on domain changes, and may change the character of the system respectively. Therefore, there is a need to construct and analyse models that take full account of their character. Here, for example: the Navier-Stokes-Smoluchowsky type systems, generalized Navier-Stokes-Vlasov system, generalized Euler systems can be used.

Then we will try to answer some of the following questions: Are the considered systems possess solutions (strong, weak, measure-valued)? Are they global in time, unique or regular? What is their behaviour for large times? How the whole systems can change when some parameters converge to zero or infinity.

The implementation of the project will require the use of advanced methods of the theory of partial differential equation and functional analysis, which the PhD student will learn during the studies. This project can be accomplished in cooperation with scientists from abroad institutions (Oxford, Prague, London).

I plan to present interesting directions which may be investigated as PhD projects. The project is supported by the grant Sonata Bis (leaded by Aneta Wróblewska-Kamińska).

A generic example of a dynamical system is a particle moving in some space such that its motion is governed by a precise rule, e.g., a ball bouncing around on a billiard table. There are potentially many ways of representing a given system. Many deep questions in the theory of dynamical systems boil down to finding a representation which has certain a priori specified desired properties - oftentimes one looks for the most efficient representation in a given canonical family of representation systems. In my presentation, I aim to provide a gentle introduction to these kinds of explorations.

A well known notion of CAT(0) spaces is a metric counterpart of non-positive curvature as studied in the theory of Riemannian manifolds. This variant of metric non-positive curvature, as well as the Gromov hyperbolicity - a coarse property corresponding to the negative curvature - have been extremely useful when applied to studies of groups. Equipping a group with an appropriate action on a CAT(0) or a Gromov hyperbolic space allows one to show many strong features of the group. On the other hand, using local nature of non-positive curvature, we obtain powerful tools for constructing new, often exotic, examples of groups.

In my presentation I will focus on combinatorial versions of non-positive curvature. These include: small cancellation, CAT(0) cubical, systolic, and Helly complexes and groups. A combinatorial approach has many advantages compared to the metric variants. For example, various algorithmic problems, including the so-called biautomaticity, can be solved using the combinatorial tools, while there are examples of e.g. CAT(0) groups which are not biautomatic.

Geometric group theory is a topic that lies at the intersection of modern algebra, topology and geometry. It has applications in areas such as non-ncommutative geometry, to index problems including the Baum-Connes and Novikov conjectures, or in combinatorics and computer science, e.g. in the form of explicit constructions of expanders. I will describe some recent results concerning the interplay between geometric methods in group theory and noncommutative geometry, as well as some possible research directions in that area.

For the last two decades, the interest in partial differential equations (PDEs) with non-local (integro-differential) operators has grown rapidly. The fundamental non-local operator is the fractional Laplacian. The specific feature of non-local operator is that its value on a function u at point x depends on values of u on the whole space. The road map for investigating non-local PDEs is modeled on the theory of local PDEs. However, specific properties of non-local operators pose certain problems particularly challenging even on the conceptual level. The recent surge in interest in this topic is caused by the growing number of scientific publications which reveal that in the large part of physical, biological, chemical, and mathematical finance models the substitution of classical differential operators by non-local operators in related PDEs leads to the solutions that better describe the phenomena, both locally and globally.

In the presentation I would like to sketch a probabilistic approach to the above-mentioned PDEs. It is based on one of the most intriguing correspondence in mathematics between a class of integro-differential operators and a class of Markov stochastic processes. The last object is purely probabilistic but may also be considered as a Borel measure on the space of possible trajectories of a particle under the dynamic given by the associated operator. I will conclude with a few interesting directions which may be investigated as PhD projects.

K-theory is in the mainstream of modern mathematics, and graph C*-algebras proved to be tremendously successful in studying the K-theory of operator algebras. They are currently at the research frontier of noncommutative topology enjoying a substantial ongoing research output. The goal of this presentation is threefold: to introduce the founding ideas of noncommutative topology, to explain the concept of graphs and graph algebras, and to exemplify noncommutative topology by the tangible world of graph C*-algebras, known as “operator algebras that one can see”. In topology, pushouts are formal recipes for collapsing and gluing topological spaces. For instance, shrinking the boundary circle of a disc to a point yields a sphere, shrinking the equator of a sphere to a point gives two spheres joined at the point, collapsing the boundary of a solid torus to a circle, or gluing two solid tori over their boundaries, produces a three-sphere. In noncommutative topology, such procedures are expressed in terms of pullbacks of C*-algebras. Indeed, the Gelfand-Naimark theorem establishing the anti-equvalence of the category of compact Hausdorff spaces and commutative unital C*-algebras transforms pushouts into pullbacks. It turns out that one can visualize a pullback of C*-algebras of graphs as a pushout of these graphs thus providing much needed intuition to the abstract setting of operator algebras. We will discuss how to make this visualization rigorous by conceptualizing abundant examples from noncommutative topology.

We are interested in the boundary behaviour for harmonic functions and their generalizations in various settings, including the Euclidean spaces as well as Riemannian manifolds and Carnot groups. The discussed generalizations of harmonic functions include the p-harmonic functions, as well as harmonic and p-harmonic mappings. Our starting point is the celebrated Fatou theorem for harmonic functions which asserts that a nonnegative harmonic function defined on a Euclidean ball B has nontangential limits at almost every, with respect to surface measure, boundary point of B. Recently there has been another interesting development in the studies of the boundary behaviour for solutions of uniformly elliptic PDEs in the divergence form, namely the so-called quantitative Fatou property and its characterization of uniform rectifiability of the boundary of the domain and relation to the Carleson measures. Thus, the project relates geometric properties of PDEs, harmonic analysis and the geometric measure theory.

A conformal structure on a manifold is a class of Riemannian (or pseudo-Riemannian) metrics, where two metrics g₁ and g₂ are considered equivalent if g₁=fg₂ for some positive-valued function f. In the conformal geometry one can measure angles between vectors but not a distance between points.

The conformal structures naturally arise in mathematical physics (general relativity in particular), integrable systems and geometric approach to differential equations.

I will present a number of open problems related to the conformal structures that include variational approach to conformal geodesics (which are certain curves inviariantly defined by a conformal metric) and a characterization of integrability of differential equations through geometric properties of conformal metrics. Generalizations to the so-called causal geometries and sub-Riemannian geometries (with a particular interest in applications to control theory) will be presented as well.