Let p be a prime number. Suppose that a p-group acts on a finite set X. Then the cardinality of the fixed point set is congruent modulo p to the cardinality of X. A generalization of this simple fact holds for actions of p-groups on a compact Hausdorff topological space. One has to replace the cardinality by the Euler characteristic. In general the fixed points of a group action contain some information about global invariants. I will discuss mainly circle actions. There is a list of parallel theorems which are proven by different methods in topology, symplectic geometry and algebraic geometry. In particular I'll present topological Borel localization theorem, Atiyah-Bott formula for smooth manifolds, Duistermaat-Heckman formula in symplectic geometry and its relation with Bialynicki-Birula decomposition in algebraic geometry.
I will introduce the concept of integrate-and-fire (IF) models of neuron activity which are examples of the so-called hybrid dynamical systems, coupling continuous-time dynamics with discrete events.
In the first part of the talk I will summarize some results on one-dimensional IF models, with periodic and almost-periodic input function. It turns out that analysis of the dynamics of IF models for periodic inputs can be reduced to investigation of degree-one circle maps but in the almost-periodic case the situation is more challenging.
Next we will focus on bidimensional IF models (with the particular example of the adaptive-exponential model), which are able to capture a wide repertoire of biologically realistic behaviours. Characterization of their dynamical properties is attainable by methods of piecewise smooth dynamical systems with elements of differential equations and bifurcation theory.
In the first part of the talk I would like, briefly, to present classical results of the theory of functions of bounded variation, along with their applications. Furthermore, I would like also to discuss some open problems concerning functions of bounded variation, with particular emphasis on the properties of the Nemytskii operator.
We shall explain the basics of the theory of the complex Monge-Ampère equation. We shall explain several a priori estimates and low regularity theory. Special emphasis will be put on th differences and similarities to the real counterpart.
During the talk I will survey older and also more recent results concerning ergodic and mixing properties of smooth flows on surfaces. I will also present some of the most basic tools which are needed there, such as so-called special flows and interval exchange transformations.
One of the most important notions in probability is the notion of martingale. Roughly speaking, it is a process that is a 'fair' game, i.e. from the point of view of the expectations of an agent the future values of this process will not change. The Doob's inequality is one of the most important inequalities in probability. It connects supremum of expected values of a right-continuous martingale or a nonnegative right-continuous submartingale with expected value of its supremum.
During my speech I will show some well-known examples and basic properties of martingales. Using methods of stochastic analysis I will also sketch the method by which we can find optimal constant in Doob's inequality for Bessel process.
In many biochemical reactions occurring in living cells or on their membrane, number of various molecules might be low which results in significant stochastic fluctuations. In addition, most reactions are not instantaneous and there exist natural time delays in the evolution of cell states. In many cases it is a challenge to develop a systematic and rigorous treatment of stochastic dynamics describing those cellular level processes.
During the seminar I will present a methodology to deal with time delays in biological systems and apply it to simple mathematical models of gene expression (both deterministic and stochastic) with delayed degradation. I will show rigorously that time delay of protein degradation does not cause oscillations as it was recently argued.
I will also present a stochastic model of the biochemical reactions occurring on the cell membrane. The aim is to derive approximations of the steady state moments for various proteins distributions. I will show some novel rigorous approaches that provide satisfactory approximation.
At the end of the seminar I will present some open problems related to the stochastic dynamics on the cellular level.
During this talk I will provide a characterization of the strong maximum principle for uniformly strong elliptic operators of second order.
In the talk I will discuss some methods of solving vacuum Einstein equations in general relativity. After decomposing the 4-dimensional "universe" into "space" and "time" parts Einstein equations become evolutionary ODEs for two covariant symmetric 2-tensor. The initial data is, however, not arbitrary, but has to satisfy certain non-linear constraint equations. This is where an interesting interplay of differential geometry and the theory of PDEs enters the game.
In 1937 Cherry constructed the first example of analytic flows on the two-dimensional torus with a non-trivial quasi-minimal set. We will explain the structure of such a flows and we will discuss the metric, topological and ergodic properties of the quasi-minimal set. Because of the form of the first return map for Cherry flows, we will show that this problem is strictly connected with the study of a class of circle functions with a flat interval.
Bellman function method is a powerful technique which can be used in the study of various estimates arising in harmonic analysis and probability theory. Roughly speaking, it relates the validity of a given inequality to the existence of a certain special function, enjoying appropriate majorization and concavity. As an illustration, we will use this approach to provide some sharp estimates for the dyadic maximal operator on Rn.
The representation theory has been one of central topics in mathematics for the last 200 years. We will start by briefly recalling the classification of finite dimensional representations of GL(n), relating them to Young diagrams and corresponding Schur functors. Then, we will present some of the problems arising from composing two Schur functors - so-called plethysm. The topic is strongly related to the description of homogeneous polynomials on irreducible representations, like the symmetric or wedge power.
We introduce an algebraic structure called a D-module and discuss its basic properties. We concentrate on holonomic D-modules and their application to study isolated hypersurface singularities.
The Zariski multiplicity conjecture says that the multiplicity of a reduced complex hypersurface singularity is an embedded topological invariant. Posed 40 years ago, it has been solved only in a few special cases. In this talk, I will review some of the most significant achievements and discuss future perspectives.
In my talk I will explain the connection between the maximum of a branching Brownian motion and KPP equation. I will also show the asymptotics of the maximum for the branching random walk.
The purpose of the talk is to introduce some basic objects of nonlinear analysis and geometric function and mapping theory and present their rudimentary properties and relations between them. We will discuss scalar and vectorial p-harmonic operators, quasiregular mappings and their generalizations (a counterpart of analytic functions in higher real dimensions). The talk should be accessible to the general audience of mathematicians.
Witold Sadowski will tell us about the problem of uniqueness and regularity of solutions of the 3D Navier-Stokes equation.