Vrijeme: 8.7.2020, 17:30

Prostorija: vir

Predavač: Steffen Winter, Karlsruhe Institute of Technology (KIT), Institute of Stochastics

The talk is scheduled in virtual setting, using Zoom platform. The talk will also be live-streamed via YouTube. During the meeting, the questions can be posed via chat or audio for participants in the meeting. Everybody interested is invited to participate in the Zoom meeting, with the limit of 100 participants, or to follow the live-broadcast. The link to Zoom meeting will be available at least one day before the talk, and the link to live broadcast 10-15 minutes before the talk, on this page.

The talk will be in English.

Abstract:

Fractal percolation is a family of random self-similar sets suggested by Mandelbrot in the seventies to model certain aspects of turbulence. It exhibits a dramatic topological phase transition, changing abruptly from a dust-like structure to the appearance of a system spanning cluster. The transition points are unknown and difficult to estimate, and the geometry of these sets is not completely understood. It is a natural question whether geometric functionals such as intrinsic volumes can provide further insights.

We study some geometric functionals of the fractal percolation process F, which arise as suitably rescaled limits of intrinsic volumes of finite approximations of F. We establish the almost sure existence of these limit functionals, clarify their structure and obtain explicit formulas for their expectations and variances as well as for their finite approximations. The approach is similar to fractal curvatures but in contrast the new functionals can be determined explicitly and approximated well from simulations.

Joint work with Michael Klatt (Princeton).

Vrijeme: 24.6.2020, 17:30

Prostorija: vir

Predavač: Vanja Wagner, PMF-MO

The talk is scheduled in virtual setting, using Zoom platform. During the meeting, the questions can be posed via chat or audio for participants in the meeting. Everybody interested is invited to participate in the Zoom meeting, with the limit of 100 participants. The talk will also be live-streamed via YouTube.

The link to Zoom meeting:

https://us02web.zoom.us/j/87879372208

Meeting ID: 878 7937 2208

The link to YouTube live-broadcast:

was available during the talk

The talk will be in English.

Abstract:

We will consider nonlocal quadratic forms on a subset D of R^d which connect two points by a single jump only if the points satisfy a certain geometric condition depending on the geometry of the set D. The main example will be quadratic forms with visibility constraint, which allow jumps between points x, y in D only if the segment [x,y] is contained in D. We will analyse the comparability of such forms to nonlocal forms determined by the same jumping measure, but without the geometric reduction of jumps. For a specific class of jump measures and domains D, we will discuss the regularity of quadratic forms with the mentioned domain reduction and analyse the boundary behaviour of the corresponding pure-jump Markov process on D.

]]>Vrijeme: 17.6.2020, 17:30

Prostorija: vir

Predavač: Daniel Panazzolo, Université de Haute-Alsace

The talk is scheduled in virtual setting, using Zoom platform. During the meeting, the questions can be posed via chat or audio for participants in the meeting. Everybody interested is invited to participate in the Zoom meeting, with the limit of 100 participants. The talk will also be live-streamed.

Link to join Zoom meeting:

https://us02web.zoom.us/j/87441656696

Meeting ID: 874 4165 6696

Link to broadcast live: was available 10 min before the talk and through the talk.

Abstract:

We consider analytic differential operators of order n defined on

two-dimensional manifolds. Namely, linear operators locally of the

form:

Sum_{0<=i+j<=n} f_{ij} (d/dx)^i (d/dy)^j,

with f_{ij} analytic functions. After introducing the notion of elementary

singular point for such operators, we discuss a theorem of resolution of

singularities, generalizing the classical result of Bendixson-Seidenberg

for vector fields in dimension two.

Vrijeme: 11.3.2020, 17:00

Prostorija: 005

Predavač: Jan P. Boroński, AGH University of Science and Technology, Kraków

I shall discuss a positive solution to the following problem, obtained in a joint work with J. Činč and P. Oprocha.

Question (M. Barge, 1989 [Le89]) Does there exist, for every r ∈ [0, ∞], a pseudo-arc homeomorphism whose topological entropy is r?

Until now all known pseudo-arc homeomorphisms have had entropy 0 or ∞. Recall that the pseudo-arc is a compact and connected space (continuum) first constructed by Knaster in 1922 [Kn]. It can be seen as a pathological fractal. According to the most recent characterization [HO18] it is topologically the only, other than the arc, continuum in the plane homeomorphic to each of its proper subcontinua. The pseudo-arc is homogeneous [Bi] and played a crucial role in the classification of homogeneous planar compacta [HO16]. Lewis showed that for any n the pseudo-arc admits a period n homeomorphism that extends to a rotation of the plane, and that any P-adic Cantor group action acts effectively on the pseudo-arc [Le83] (see also [To]). We adapt Lewis' inverse limit constructions, by combining them with a Denjoy-Rees scheme [BCL] (see also [Re], [BKLO]). The positive entropy homeomorphisms that we obtain are periodic point free, except for a unique fixed point.

I shall start my talk by reviewing the role that the pseudo-arc have played in various areas of mathematics, including topology, surface dynamics, complex analysis, isometric theory of Banach spaces and logic, and then will move on to to the history of the problem, to conclude with a discussion of its solution.

References

[BCL] Béguin, F.; Crovisier, S.; Le Roux, F. Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique. Ann. Sci. École Norm. Sup., 40 (2007) 251-308.

[Bi] Bing, R. H. A homogeneous indecomposable plane continuum. Duke Math. J. 15 (1948) 729–742.

[BKLO] Boroński J.P.; Kennedy J., X. Liu, Oprocha P., Minimal noninvertible maps on the pseudocircle, arXiv:1810.07688

[HO16] Hoehn, L. C.; Oversteegen, L. G. A complete classification of homogeneous plane continua. Acta Math. 216 (2016) 177-216.

[HO18] Hoehn, L. C.; Oversteegen, L. G. A complete classification of hereditarily equivalent plane continua arXiv:1812.08846

[Kn] Knaster, B. Un continu dont tout sous-continu est indécomposable. Fund. Math. 3 (1922) 247-286.

[Le83] Lewis, W. Periodic homeomorphisms of chainable continua. Fund. Math. 117 (1983) 81-84.

[Le89] Lewis, W. Continuum theory and dynamics problems. Continuum theory and dynamical systems (Arcata, CA, 1989), 99–101, Contemp. Math., 117, Amer. Math. Soc., Providence, RI, 1991.

[Re] Rees, M. A minimal positive entropy homeomorphism of the 2-torus. J. London Math. Soc., 2 (1981) 537-550.

[To] Toledo, J. Inducible periodic homeomorphisms of tree-like continua. Trans. Amer. Math. Soc. 282 (1984) 77–108.

]]>Vrijeme: 19.2.2020, 17:00

Prostorija: 005

Predavač: Sebastian Schwarzacher, Charles University, Prague

Interactions between fluids and solids are happening everywhere in our daily live. Whether it is a football floating through the air, an airplane flying through the wind or blood floating through our body--interaction between fluids and solids are happening. The talk will introduce how mathematically the difference between fluid matter and solid matter can be captured. Later it aims to give an intuition on some mathematical challenges related to fluid-solid interactions. An emphasis is given on elastic deformable solids.

]]>Vrijeme: 22.1.2020, 17:00

Prostorija: 005

Predavač: Rudi Mrazović, PMF-MO

Hall i Paige su 1955. postavili hipotezu prema kojoj konačna grupa ima potpuno preslikavanje (permutaciju F za koju je xF(x) također permutacija) ako i samo ako su njene Sylowljeve 2-podgrupe trivijalne ili necikličke. Tu su hipotezu 2009. konačno dokazali Wilcox, Evans i Bray nastavljajući se na nekoliko ranijih radova i koristeći klasifikaciju konačnih grupa. Na predavanju ćemo prezentirati kako koristeći potpuno drugačiji pristup motiviran metodom kružnice iz analitičke teorije brojeva možemo doći do precizne asimptotike za broj potpunih preslikavanja u bilo kojoj grupi koja zadovoljava Hall-Paigeov uvjet. Predavanje je temeljeno na zajedničkom radu s Seanom Eberhardom i Freddiejem Mannersom.

]]>Vrijeme: 11.12.2019, 17:00

Prostorija: 005

Predavač: Mirko Primc, PMF-MO

J. Lepowsky i R. Wilson interpretirali su i dokazali Rogers-Ramanujanove identitete u okviru teorije reprezentacija afinih Kac-Moodyjevih Liejevih algebri i teorije reprezentacija algebri verteks-operatora. Prvi novi kombinatorni identiteti otkriveni u okviru te teorije bila su dva Capparellijeva identiteta. Na predavanju ću objasniti kako se nekoliko serija kombinatornih identiteta javlja u okviru teorije reprezentacija simplektičkih afinih Liejevih algebri, od kojih neke serije započinju s Capparellijevim identitetima. Formulirat ću i slutnju kako izgledaju svi kombinatorni identiteti koji se javljaju u opisanom pristupu za simplektičke afine Liejeve algebre.

]]>Vrijeme: 23.10.2019, 17:00

Prostorija: 005

Predavač: Jakub Konieczny, The Hebrew University of Jerusalem

Automatic sequences give rise to one of the basic models of computation and have remarkable links to many areas of mathematics, including dynamics, algebra and logic. Distribution of these sequences has long been studied. During the talk we will explore this topic from the point of view of higher order Fourier analysis.

As it turns out, many of the classical automatic sequences are highly Gowers uniform, while others can be expressed as the sum of a structured component and a uniform component much more efficiently than guaranteed by the arithmetic regularity lemma. We investigate the extent to which this phenomenon extends to general automatic sequences and consider some closely related problems that make sense for sparse sequences.

The talk is partially based on joint work with J. Byszewski and with C. Müllner.

Vrijeme: 11.9.2019, 17:00

Prostorija: 005

Predavač: Marek Ptak, Krakow

Generalized multipliers for left-invertible analytic operators will be

introduced. It will be shown that they form a Banach algebra and

characterize the commutant of such operators in its terms. In the special case, we describe the commutant of balanced weighted shift on directed tree only in terms of its weights. The reflexivity results for weighted shifts on directed tree will presented.

Joint work with Piotr Dymek, Artur Planeta.

Vrijeme: 12.6.2019, 17:00

Prostorija: 005

Predavač: Justin Webster, University of Maryland

When a thin elastic structure is immersed in a fluid flow, certain conditions may bring about excitations in the structure. That is, dynamic fluid loading feeds back with the natural modes of the structure. In this case oscillatory behavior may persist until the flow parameters change or energy is dissipated from the system. This interactive phenomenon is referred to as flutter.

Beyond the obvious applications in aeroscience (paneling, flaps, flags, and airfoils), the flutter phenomenon arises in: (i) the biomedical realm (snoring and sleep apnea), and (ii) sustainable energies (piezoelastic energy harvesters). Modeling, predicting, and controlling flutter have been a foremost problems in engineering for nearly 70 years.

In this talk we describe the basics of modeling flutter in the simplest configuration (an aircraft panel) using differential equations and dynamical systems. After discussing the partial differential equation model, we will discuss theorems that can be proved about solutions to these equations using modern analysis (e.g., monotone operator theory, the theory of global attractors). We will relate these results back to experimental results in engineering and recent numerical work. We will also describe very recent problems in the analysis of axial flow configurations, where a portion of the structure is unsupported (e.g., a flag).

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