Holonomy is a prime example of mathematical intuition and creativity - it generalises our school knowledge about the sum of angles in a triangle and led to  `Berger’s holonomy theorem’ from 1954 which turned out to be a  most successful research programme for differential geometry for over 50 years. We are going to tell the story of this development, how holonomy relates to curvature and advanced symmetry concepts, including a small detour to theoretical physics and what spinors and Calabi-Yau manifolds have to do with it. We conclude by a small outlook to recent results

One of the simplest classical results of Ramsey Theory proved by I. Schur in 1917, states that for any finite partition of the positive integers, one of the cells of the partition contains numbers x, y, z, such that x + y = z. We will discuss various approaches to the proof of this result as well as some applications and generalizations. In particular, we will explain how Schur's theorem applies to the Fermat equation over finite fields. We will also discuss Schur-Brauer strengthening of the classical van der Waerden theorem on arithmetic progressions, and a far-reaching generalization of Schur's theorem due to Hindman which states that for any finite partition of positive integers, one of the cells of the partition contains an infinite increasing sequence (xi) together with all finite sums formed by elements with distinct indices. Finally, we will discuss a non-commutative version of Schur's theorem and some open problems that it leads to.

We show how the sheaf-theoretic approach can be effectively applied to obtain a quantum deformation of relevant spaces as quotients of algebraic groups. In particular, we study quantum principal bundles corresponding classically to the projection of an algebraic group onto its quotient by the maximal parabolic subgroup. Finally  we discuss differential calculi on quantum principal bundles in a sheaf theoretic framework.

Probabilistic method is a powerful tool in combinatorics which allows us to prove existence of an object or structure without explicitly contracting it. Instead, we come up with a random model and then show that it has the desired properties with positive probability. Initially the method was used to find certain graphs or colorings with specific properties. We are now at the stage where similar scheme can be successfully applied to prove state-of-the-art theorems in algebra, Riemannian geometry and algebraic topology. I will review some of these recent applications.  

The problem of sampling a probability distribution known only up to a normalization constant is fundamental in numerous applications within computational science and engineering. This issue is extensively explored in the fields of applied mathematics, machine learning, and in particular in Bayesian statistics used for parameter identification. Recent research, which is still in its infancy, demonstrates that algorithms based on gradient flows in the space of probability measures open up new directions in sampling algorithms. Detailed tailoring of the gradient flow specification allows to fit the particular algorithm precisely.

https://www.mimuw.edu.pl/~miekisz/kalamajska.pdf

In the talk we discuss work initiated by Kechris-Pestov-Todorcevic in 2005 that relates dynamics of automorphism groups of countable structures, which are important examples of topological groups, and a Ramsey property for the corresponding class of finite structures. We say that a topological group is extremely amenable if its every continuous action on a compact Hausdorff space has a fixed point. In particular, extremely amenable groups are amenable. Extreme amenability of the automorphism group of rationals with the usual order is equivalent to the Ramsey property for the class of finite linear orders, i.e. to the classical Ramsey theorem, which was proved by Pestov. We discuss this and many more examples. 

A spectral triple is one of the main concepts in Noncommutative geometry. Given a class of C*-algebras, such as group C*-algebras, a natural question is whether one can define any interesting spectral triples. For example, one is interested in examples where there is, in addition, the structure of a compact quantum metric space. Important examples are found in work of Connes and Christ-Rieffel. Attention has also recently been given to constructions in the context of twisted group C*-algebras, and to permanence properties such as taking an inductive limit of spectral triples. In the talk I will present these concepts and illustrate them with a construction of spectral triples for noncommutative solenoids obtained in collaborative work with Carla Farsi, Therese Basa-Landry and Judith Packer.

Cancer treatment has entered a new phase in which it is already known that we should not just focus on killing the cancer cells which, over time, develop drug resistance, but try to influence the tumor microenvironment, i.e., the cells of the immune system, as well as the vascularization of the tumor. This leads to novel combination therapies in which drugs or cytotoxic effects are combined with indirect therapies such as immunotherapy or anti-angiogenic treatment. In this situation, cases of so-called synergy have been experimentally confirmed where one therapy enhances the effect of another. However, the problem arises how to administer these drugs. This is particularly complicated in the case of combination therapies as in addition to dose and frequency of each drug also the question of order matters. Full, maximum doses of all drugs are not the answer, because side effects, not only of chemotherapy (killing healthy cells, especially the bone marrow) as well as immunotherapy and angiogenic inhibitors can lead to the death of the patient during the process of fighting the cancer. So the natural questions arises how optimal dosing protocols should look like in order to achieve the greatest effectiveness with the lowest possible toxicity of the drugs. Testing combined therapy protocols in clinical trials is difficult and expensive, and, to begin with, it is impossible to test all possible options. In this talk we will show an optimal control approach to answer some of these questions. We will present various models for combination therapies including antiangiogenic treatment, chemo-, radio- and immunotherapies addressing the phenomenon of synergy. We will address various obstacles like evolving heterogeneity of the tumor and evolving drug resistance as well as the role of drug delivery (pharmacokinetics and pharmacodynamics) in the models.  It will be shown that in many cases “more is not necessarily better,” but with an appropriately selected lower dose better results can be achieved than with traditional maximum allowable dose regimens with rest periods bringing the concept of the metronomic chemotherapy into play. We will show not only the results, but also some limitations of the theory and discuss projects not fully completed. For a specific example, the treatment of chronic myeloid leukemia (CML) through a combination of tyrosine kinase inhibitors and immuno-modulatory therapies, we compare optimal solutions with best-in class solutions that only allow the use of a limited range of dosages and a priori specified timing changes.

After a little introduction to ergodic theory and topological dynamics: basic concepts and examples, we will focus on (bounded) functions u which preserve the multiplicative structure of the set of natural numbers:  u(mn)=u(m)u(n) whenever m and n are coprime. Such functions are called multiplicative and we are interested in their statistical properties or, more precisely, their degree of randomness. We will discuss those functions which are expected to be extremely random: the Moebius and Liouville functions. We will concentrate on relations between the  old Chowla conjecture on correlations of such functions  with the recent Sarnak's conjecture on the orthogonality  of these functions with all deterministic observables.

In the first part of the talk we recall the proof of the famous Lovasz’ theorem on the chromatic number of the Kneser graph. Then we discuss various attempts to generalize this result.

In Geometric Group Theory, we study groups by exploring their actions on spaces — for example on metric spaces (via isometries) or on complexes (via automorphisms). Despite the fancy terminology, "Nonpositive Curvature" (NPC) in the title refers to local conditions, either metric ones (for a metric space), or combinatorial ones (for complexes) implying features that are typical for manifolds with nonpositive scalar curvature. One such important condition is asphericity, that is, contractibility of the universal cover. A group and a corresponding space (on which the group acts) equipped with such NPC structure are amenable to a treatment by various geometric tools, and exhibit many interesting geometric, topological, algebraic, and algorithmic properties. Nonpositive curvature is useful for constructing interesting examples of groups (and spaces) as well as for exploring known ones. I will present several versions of NPC, examples of NPC groups, and a few important open questions in the field.

https://www.mimuw.edu.pl/~miekisz/pietruska.pdf

I will sketch how ideas from ordinary geometry, when combined with simple (though un-intuitive) ideas from quantum physics, lead to the topic of non-commutative geometry. I will then indicate a common path for generalizing specific concepts from ordinary geometry to the non-commutative setting. I will illustrate how this path is used by applying it to lead to the concept of a non-commutative (or "quantum") compact metric space. I will then indicate some specific examples of quantum compact metric spaces.

I will discuss the derivation and reasons behind introduction of a dissipative effect into, otherwise well established,  Aw-Rascle model of vehicular traffic. The dissipative offset in the closure relation between the velocities leads to degenerate system of hyperbolic-parabolic partial differential equations. I will present various results concerning the existence of strong, weak, and measure-valued solutions, their uniqueness and asymptotic limits. The talk will be concluded with discussion of open problems and possible research directions.