Teoria iteracji przekształceń holomorficznych przeżywa od lat 1980-tych okres żywiołowego rozwoju. Jednym z jej najważniejszych działów jest obecnie badanie przestępnych funkcji całkowitych i meromorficznych na płaszczyźnie zespolonej. Oprócz metod układów dynamicznych, dziedzina ta wykorzystuje techniki i narzędzia pochodzące m.in. z analizy zespolonej, topologii i geometrycznej teorii miary.
W czasie wykładu omówię niektóre zagadnienia dotyczące topologicznej i geometrycznej struktury zbiorów Julii i innych zbiorów niezmienniczych pojawiających się przy badaniu dynamiki takich przekształceń. W szczególności, przedstawię nowe wyniki dotyczące metody Newtona poszukiwania zer funkcji zespolonych.
Do zrozumienia większości wykładu wystarczy znajomość matematyki w zakresie pierwszych 2-3 lat studiów (z uwzględnieniem Funkcji Analitycznych).
Trees are ubiquitous in mathematics. In particular, they appear as basic objects in combinatorics and probability, and as dendrites in topology and dynamics. In this talk, we will see that finite trees have a canonical embedding in the plane (via Shabat polynomials as Grothendieck dessin d'enfants), that conformal maps can be used to compute these embeddings, and that self-similar Julia sets from complex dynamics can be viewed as limits of such finite trees. We will also discuss motivation from probability theory and statistical physics, particularly stochastically self-similar trees (such as the Aldous' Continuum Random Tree), the Brownian map, and conjectured relations to Liouville Quantum Gravity.
A spherical quadrilateral (membrane) is a bordered surface homeomorphic to a closed disc, with four distinguished boundary points called corners, equipped with a Riemannian metric of constant curvature 1, except at the corners, and such that the boundary arcs between the corners are geodesic. We discuss the problem of classification of these quadrilaterals and perform the classification up to isometry in the case that at most two angles at the corners are not multiples of Pi. This is a very old problem, related to the properties of solutions of the Heun's equation (an ordinary differential equation with four regular singular points). The corresponding problem for the spherical triangles, related to the properties of solutions of the hypergeometric equation, has been solved by Klein, with some gaps in Klein's classification filled in by Eremenko in 2004. The quadrilateral case for small corners was treated in the Thesis of Smirnov (1918), but for arbitrary corners remains open. This is joint work with A. Eremenko (Purdue) and V. Tarasov (IUPUI).
A sequence of functions $f = (f_i)$ ($-\infty < i < \infty$) on a surface $S$ is said to be equi-areal (or sometimes, equi-Poisson) if it satisfies the relations
One says that $f$ is $n$-periodic if $f_i = f_{i+n}$ for all $i$. The $n$-periodic equi-areal sequences for low values of $n$ turn out to have close connections with interesting problems in both dynamical systems and in the theory of cluster algebras.
In this talk, I will explain what is known about the classification (up to a natural notion of equivalence) of such periodic sequences and their surprising relationships with differential geometry, cluster algebras, and the theory of over determined differential equations.
We start with some results on the spectral synthesis for systems of exponentials on the interval or, equivalently, for systems of reproducing kernels in the Paley-Wiener space. Then we pass to similar results for general de Branges spaces. We discuss related results on the completeness for the restricted shift operator and on the Riesz bases of reproducing kernels in the Fock spaces.
A normal crossings singularity means a transverse self-intersection. Given a singular variety $X$ (defined over the complex numbers, for example), can we find a proper mapping $F$ from a variety $Y$ to $X$ such that $Y$ has only normal crossings singularities, and $F$ preserves all normal crossings singularities of $X$? The answer depends on whether normal crossings is understood in an algebraic or more general local-analytic sense.
An illuminating example is the pinch point or Whitney umbrella $X$: $z^2 + xy^2 = 0$, which has general normal crossings singularities along the nonzero $x$-axis. There is no proper birational mapping that eliminates the pinch point singularity at the origin without modifying normal crossings points.
So it makes sense to ask: Can we find the smallest class of singularities $S$ with the following properties: (1) $S$ includes all normal crossings singularities; (2) given $X$, there is a proper mapping $F$ from $Y$ to $X$ such that $Y$ has only singularities in $S$, and $F$ preserves all normal crossings singularities of $X$? For surfaces $X$, it turns out that $S$ comprises precisely normal crossings singularities and the pinch point. We can describe $S$ completely also in dimension three, but the problem is open in higher dimension.
(Joint work with Sergio Da Silva, Pierre Lairez, Pierre Milman and Franklin Vera Pacheco.)
As we seek greater knowledge about the energy-minimal deformations in Geometric Function Theory and Nonlinear Hyperelasticity, the questions about Sobolev homeomorphisms and their limits become ever more quintessential.
We shall discuss the following topics:
Theoretical prediction of failure of bodies caused by interpenetration of matter (collapse of domains) is a good motivation that should appeal to Mathematical Analysts and researchers in the Engineering Fields.
In 1959, R. V. Kadison and I. M. Singer asked whether each pure state of the algebra of bounded diagonal operators on ℓ², admits a unique state extension to B(ℓ²). The positive answer was given in May 2013 by A. Marcus, D. Spielman and N. Srivastava, who took advantage of a series of translations of the original question, due to C. Akemann, J. Anderson, P. Casazza, N. Weaver, … Ultimately, the problem boils down to an estimate of the largest zero of the expected characteristic polynomial of the sum of independent random variables taking values in rank 1 positive matrices in the algebra of n-by-n matrices. In turn, this is proved by studying a special class of polynomials in d variables, the so-called real stable polynomials. The talk will highlight the main steps in the proof.
A wave front is the image of a Legendre mapping. A caustic is the set of critical values of a Lagrange mapping. We study local and global properties of Lagrange and Legendre mappings. As a corollary, we get new coexistence conditions of singularities of wave fronts and caustics.
Since the pioneering works of Scheffer and Shnirelman we know that Euler equations – derived more than 250 years ago to describe the motion of an inviscid incompressible fluid – have nontrivial solutions which are compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving without any action of external forces.
Nash and Kuiper proved the existence of $C^1$ isometric embeddings of a fixed flat rectangle in arbitrarily small balls of $\mathbb R^3$. Thus, you should be able to put a fairly large piece of paper in a pocket of your jacket without folding or crumpling it. With Laszlo Szekelyhidi, we pointed out that these two counterintuitive facts share many similarities. This is even more apparent in our recent results, which prove the existence of Hoelder continuous solutions that dissipate the kinetic energy. Our theorem might be regarded as a first step towards a conjecture of L. Onsager, who in a 1949 paper about the theory of turbulence asserted the existence of such solutions for any Hoelder exponent up to $1/3$. The best result in this direction, $1/5$, has been reached by Phil Isett.
Geometry and topology are full of structures that are frequently too big to easily manipulate on a pad of paper, yet small enough that computers can readily handle them. Hyperbolic 3-manifolds provide a stunning example of this. My talk will concern algorithms in topology, and how they are increasingly capable of attacking more interesting and compelling problems, some from outside of topology itself. Persistent homology is one of these new tools. It is a fairly young idea, with the intent of being useful in statistics, particularly the study of large data sets. I will describe some experiments in this field, as well as some basic open problems. I'll also describe how some of these algorithmics can be turned inward, to construct, for example, a `knot table' for knots in 4-dimensional space, analogous to the Tait-Little knot table.
Przypomnimy postać i dorobek Samuela Eilenberga, wywodzącego się z warszawskiej szkoły matematycznej jednego z architektów matematyki XX wieku. Po krótkim przeglądzie całości jego dorobku skoncentrujemy się na okresie warszawskim: studiach i doktoracie na UW oraz, mimo bardzo młodego wieku, bogatej działalności badawczej i interakcji z innymi matematykami. Zastanowimy się nad wpływem warszawskich korzeni na późniejszą twórczość Eilenberga oraz jego relacjami z dawnymi kolegami i nauczycielami. Wykład będzie ilustrowany skanami mało znanych zdjęć i dokumentów, które wzbogacają naszą pamięć o warszawskiej szkole matematycznej w okresie międzywojennym.
The fascination with "dimensionality" predates even Aristotle. Since the nineteenth century advances in Science and Mathematics unshackled our imagination with higher-dimensional geometries and multi-dimensional (multivariate) problems. These can now be visualized with a system of Parallel Coordinates. The perceptual barrier imposed by our 3-dimensional habitation has been breached.
We describe how this visualization works and demonstrate some of its applications: in air traffic control (collision avoidance; 3 patents), data exploration (patent - example: discovering banks' manipulation of gold market), modeling complex relations (example: interactive visual model of a country's economy), and new representation of surfaces preferable even for some 3-dimensional applications. Results are first discovered visually and then proven mathematically; in the true spirit of Geometry. Our 3-dimensional experience is now the laboratory for insights into complex high-dimensional situations.
The set of eigenvalues for operators on an infinite dimensional Hilbert space can be uncountable. Examples are provided by multiplication operators on spaces of holomorphic functions such as the Hardy and Bergman spaces. One can use concepts and techniques from complex geometry to study such operators. In particular, one can associate a Hermitian holomorphic bundle with such operators which provides a complete unitary invariant.
In joint work with M. Cowen around 1980, we showed how the Chern curvature for this bundle could be calculated concretely in operator-theoretic terms. We will illustrate these results emphasizing classical examples of reproducing kernel Hilbert spaces of holomorphic functions on domains such as the unit ball in Cn. Also, we will show how this framework can be used to tackle and, in many cases, solve problems in operator theory.
The twentieth century saw a remarkable development and consolidation of the theory of groups and their representations. Now the theory is being extended in a number of different directions. The talk will touch briefly on two of these, namely algebraic combinatorics and Hopf algebras, before delving more deeply into a third direction, the theory of quasigroups.
Kryptografia bierze swoje początki z czasów antycznych. Techniki szyfrowania, stworzone w jej ramach, wykorzystywane były na przestrzeni wieków do utajniania informacji w wojskowości, dyplomacji i kontaktach handlowych. W ciągu ostatnich kilku dekad, w związku z dramatycznym rozwojem technik komunikacji cyfrowej, dziedzina ta znacznie rozszerzyła swój pierwotny zakres zastosowań. Obecnie protokoły skonstruowane przez kryptografów służą nie tylko do zapewnienia poufności przesyłanych informacji, ale także mają szereg innych zastosowań takich jak: podpis elektroniczny, aukcje internetowe, głosowanie elektroniczne oraz obliczenia „w chmurze” (ang. cloud computing). Na wykładzie dokonamy krótkiego wprowadzenia do kryptografii oraz przedstawimy przegląd niektórych z tych zagadnień.
W czasie odczytu zostaną przedstawione wyniki, za które uzyskał to wyróżnienie: dowód zbieżności do stanu stacjonarnego rozwiązań równania Boltzmanna, w tym dowód hipotezy Cercignaniego, wyjaśnienie relaksacji do stanu równowagi rozwiązań równania Vlasova (tłumienie Landaua). W odczycie będzie także mowa o innych ważnych wynikach Villaniego otrzymanych w ostatnich latach, m.in. o związkach krzywizny Ricciego z entropią.
Prócz bycia wybitnym uczonym – choć nie bez związku z tym faktem – Cédric Villani stał się czołowym celebrytą francuskich mediów. Wykorzystując ten swój status Villani przyczynia się w niezwykłym stopniu do propagowania znaczenia matematyki we współczesnym świecie. O tym aspekcie osobowości Villaniego będzie też mowa w czasie odczytu.
Próby zrozumienia teorii algebr klastrowych Fomina i Zelevinskiego doprowadziły do nowego otwarcia w teorii algebr skończonego wymiaru, związanego z pojęciem tzw. kategorii klastrowych. Sztywne obiekty w tej kategorii mają mutacje, które pokazują ich ukrytą symetrię.
Wychodząc od reprezentacji kołczanów i twierdzenia Gabriela, zdefiniuję asocjahedron; pokażę, jak się on pojawia i jak w naturalny sposób prowadzi do pojęcia kategorii klastrowych. Omówię także najprostsze przykłady mutacji.
The attempts to understand theory of cluster algebras of Fomin and Zelevinsky led to new developments in the theory of Artin algebras based on the notion of cluster category. The rigid objects in that category come equipped with mutations that reveal hidden symmetry.
Departing from the definition of quiver representations and Gabriel theorem I will define the associahedron, show how it appears in that context and how it leads naturally to the notion of cluster categories. I will also describe the simplest examples of mutations.
I will describe a few mathematical questions (many of them open) that arise in the classical (and very simple) setting of the point mass moving by inertia in a bounded planar domain and elastically bouncing off its boundary.
If we choose a finitely presented group at random, what properties does it have?
This question can be made more precise in different ways. For example, twenty years ago, Gromov introduced his density model for random groups and showed that, generically, random groups are infinite and negatively curved on large scales (to be precise, they are Gromov hyperbolic).
Meanwhile, there have been many developments in the study of the large scale geometry of Gromov hyperbolic groups. In this talk we will follow these two strands of research from the basic definitions through to recent results. We will see how these ideas interact to give a richer picture of the large scale geometry of random groups.
Badanie nieskończonych grup metodami geometrycznymi, rozpoczęte przez Maxa Dehna w początkach dwudziestego wieku, zostało odnowione w końcu wieku w pracach Mostowa, Margulisa i przede wszystkim Gromowa. Nowością ich podejścia jest skupienie się na asymptotycznych, zgrubnych własnościach grup i przestrzeni, które jednak niosą wiele interesujących informacji o pewnej klasie grup.
Opowiem o twierdzeniu Mostowa o sztywności, twierdzeniu Gromowa o wzroście wielomianowym, grupach hiperbolicznych Gromowa oraz (nowszych) zastosowaniach podobnych idei dla ważnych grup niskowymiarowej topologii.
Arnold diffusion is a phenomenon of instability for nearly integrable Hamiltonian systems. I plan to discuss various versions of the problem, the main difficulties, the basic conjectures and the present state of art in the domain.
Opowiem o dwóch najpopularniejszych kryptosystemach asymetrycznych: RSA i kryptosystemie El Gamala. Pierwszy z nich wykorzystuje elementarne własności liczb naturalnych, drugi oparty jest na krzywych eliptycznych nad ciałem skończonym.
In the talk, I am going to explain the so-called local regularity theory for weak solutions to the Navier-Stokes equations. Starting with well understood things such as "epsilon"-regularity theory for suitable weak solutions, I shall continue with the reduction of a local regularity problem to the Liouville-type theorems. A conjecture about mild bounded ancient solutions will be discussed. The positive answer to it would rule out blow-ups of Type I and more generally blow-ups for which a certain "reasonable" scale-invariant quantity is bounded. For example, among such type of quantities are the scaled kinetic energy, the scaled dissipation, and etc. The second part of the talk will address the problem of how scale-invariant norms of the velocity field grow when time approaches a potential blow-up.
Teoria połączeń układów dynamicznych jest obecnie jednym z najbardziej rozwijających się kierunków badań w teorii ergodycznej. Przedstawię jedno z jej zastosowań - dowód twierdzenia ergodycznego T. Tao z 2008 r.