Mini Course / Dynamics Learning Seminar

from Friday
June 01, 2018 to Monday
December 31, 2018
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Wednesday
September 12, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Peter Veerman, Portland State University
Diffusion and Consensus on Weakly Connected Directed Graphs

We outline a complete and self-contained treatment of the asymptotics of (discrete and continuous) consensus and diffusion on directed graphs. Let $G$ be a weakly connected directed graph with directed graph Laplacian $\mathcal{L}$. In many or most applications involving digraphs, it is possible to identify a direction of flow of information. We fix that direction as the direction of the edges. With this convention, consensus (and its discrete-time analogue) and diffusion (and its discrete-time analogue) are dual to one another in the sense that $\dot{x}=-\mathcal{L}x$ for consensus, and $\dot{p}=-p\mathcal{L}$ for diffusion. As a result, their asymptotic states can be described as duals.

We give a precise characterization of a basis of row vectors $\lbrace\overline{γ}_i\rbrace_{i=1}^k$ of the left null-space of $\mathcal{L}$ and of a basis of column vectors $\lbraceγ_i\rbrace_{i=1}^k$ of the right null-space of $\mathcal{L}$. This characterization is given in terms of the partition of $G$ into strongly connected components and how these are connected to each other. In turn, this allows us to give a complete characterization of the asymptotic behavior of both diffusion and consensus in terms of these eigenvectors.

As an application of these ideas, we present a treatment of the pagerank algorithm that is dual to the usual one, and give a new result that shows that the teleporting feature usually included in the algorithm actually does not add information.


Wednesday
October 03, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Nguyen-Bac Dang, Stony Brook University
Degree growth of tame automorphisms (part I)

$\hspace {5mm}$In this mini-course, I will present the tame automorphism group preserving an affine quadric threefold. The main focus of this talk is the understanding of the degree sequences induced by the elements of this group, mainly how one can prove that the dynamical degrees have a spectral gap. Precisely, I will explain the construction of the square complex due to Bisi-Furter-Lamy on which the tame group acts by isometry.
$\hspace {5mm}$Using this, I will show how one can understand the degree growth of tame automorphisms whose action on this complex fix a vertex.


Wednesday
October 10, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Lasse Rempe-Gillen, University of Liverpool
Hubbard tress of transcendental entire functions

It is well-known that all essential dynamical information of a postcritically finite polynomial can be encoded into a finite combinatorial tree (together with dynamics), known as a “Hubbard tree”. This tree is obtained by using the dendrite structure of the corresponding Julia sets.

In this talk, we consider the case of postsingularly finite (PSF) entire functions. For such, a Hubbard tree does not exist in general, but we introduce a notion of “homotopy Hubbard trees”, which are invariant up to homotopy relative the postsingular set. We show that, for every PSF entire function, and more generally for any subhyperbolic entire function, there is a unique such homotopy Hubbard tree. This is joint work with Pfrang, and extends work of Pfrang, Rothgang and Schleicher for exponential maps. Our proof uses joint work with Benini on “dreadlocks” of entire functions.

We also describe precisely when the homotopy Hubbard tree can be represented by an invariant tree within the Julia set; in particular, this is the case whenever the function in question has no asymptotic values.


Wednesday
October 17, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Nguyen-Bac Dang, Stony Brook University
Degree growth of tame automorphisms (part II)

In the first part, I explained the construction of the square complex due to Bisi-Furter-Lamy and explained the growth of the degree for the elements whose action on it fixed a vertex. In the second part of this mini course, I will focus on the automorphisms whose action on the complex is hyperbolic. First I will present the valuative estimates derived from the work of Shestakov-Umirbaev which will allow me to relate the degree with the distance in the square complex introduced by Bisi-Furter-Lamy.

Then I will present the main steps of my proof through an explicit example.


Wednesday
November 14, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dimitrios Ntalampekos, Stony Brook University
Non-removability of the Sierpinski gasket

I will sketch a proof that the Sierpinski gasket is non-removable for quasiconformal maps. The first step of the argument is the construction of a non-Euclidean sphere, called a "flap-plane", that arises by attaching infinitely many rectangles to the Euclidean plane. The second step of the construction is to embed quasisymmetrically this flap-plane back to the Euclidean plane, by using the Bonk-Kleiner uniformization theorem. I will provide some details of these arguments, and discuss the properties of the "flap-planes", as well as, possible generalizations.


Wednesday
November 28, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Dimitrios Ntalampekos, Stony Brook University
Non-removability of the Sierpinski gasket (part II)

TBA


Wednesday
December 05, 2018

2:30 PM - 3:30 PM
Math Tower P-131
Kenneth Bromberg, University of Utah
TBA

TBA


Show events for:
Instructions for subscribing to Stony Brook Math Department Calendars