Georgia Institute of Technology College of Sciences

Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a rational map f, the Julia set Jf  is a beautiful fractal defined as the repeller of the dynamics of f. Fractal invariants of Julia sets, such as Hausdorff dimensions, have information about the complexity of the dynamics of rational maps. Ahlfors-regular conformal dimension, abbreviated by ARconfdim, is the infimum of the Hausdorff dimension in a quasi-symmetric class of Ahlfors-regular metric spaces. The ARconfdim is an important quantity especially in geometric group theory because a natural metric, called a visual metric, on the boundary of any Gromov hyperbolic group is determined up to quasi-symmetry. In the spirit of Sullivan's dictionary, we can use ARconfdim to understand the dynamics of rational maps as well. In this talk, we show that the Julia set of a post-critically finite hyperbolic rational map f has ARconfdim 1 if and only if there is an f-invariant graph G containing the post-critical set such that the dynamics restricted to G has topological entropy zero.  

A 3-connected graph is minimally 3-connected if removal of any edge destroys 3-connectivity. We present an algorithm for constructing minimally 3-connected graphs based on the results in (Dawes, JCTB 40, 159-168, 1986) using two operations: adding an edge between non-adjacent vertices and splitting a vertex of degree at least 4. To test sets of vertices and edges for 3-compatibility, which depends on the cycles of the graph, we develop a method for obtaining the cycles of $G'$ from the cycles of $G$, where $G'$ is obtained from $G$ by one of the two operations above.  We eliminate isomorphic duplicates using certificates generated by McKay's isomorphism checker nauty. The algorithm consecutively constructs the non-isomorphic minimally 3-connected graphs with $n$ vertices and $m$ edges from the non-isomorphic minimally 3-connected graphs with $n-1$ vertices and $m-2$ edges, $n-1$ vertices and $m-3$ edges, and $n-2$ vertices and $m-3$ edges. In this talk I will focus primarily on the theorems behind the algorithm. This is joint work with Joao Costalonga and Robert Kingan.

The work deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in the presence of the transport term. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used. We discuss how the introduction of the transport term impacts the regularity of solutions.

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk will be awesome.

BlueJeans link: https://bluejeans.com/851535338

The talk considers the equivalence relations of topological conjugacy and measure isomorphism on diffeomorphisms of compact manifolds of small dimension. It is shown that neither is a Borel equivalence relation.  As a consequence, there is no inherently countable method that,  for general diffeomorphisms $S$ and $T$, determines whether $S\sim T$. It is also shown that the Time Forward/Time Backward problem for diffeomorphisms of the 2-torus  encodes most mathematical questions, such as the Riemann Hypothesis.

This work is joint with B Weiss and A Gorodetski.

I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function in small intervals of the critical line. This problem has interesting connections with the extreme value statistics of IID and log-correlated random variables.

We will discuss a new method for proving the existence of diffusion in some systems with Normally Hyperbolic Invariant Manifolds (NHIM). We apply this approach to the generalized standard map to show the existence of drift orbits for an explicit range of actions.  The method consists of verifying a finite number of conditions on a computer (keywords: NHIM, shadowing, scattering map, Chirikov Standard model, Parameterization Method, Interval Newton Method).  

Live cell imaging and single particle tracking techniques have become increasingly popular amongst the mathematical biology community. We study endocytosis, the cellular internalization and transport of bioparticles. This transport is carried out in membrane-bound vesicles through the use of motor proteins. Lysosomes, known for endocytosis, phagocytic destruction, and autophagy, move about the cell along microtubules. Single particle tracking methods utilize stochastic models to simulate intracellular transport and give rise to rigorous analysis of the resulting properties, specifically related to transitioning between inactive to active states. This confidence in the stochastic modeling of particle tracking is useful not only for particle-containing lysosomes, but also broad questions of cellular transport studied with single particle tracking.

Meeting Link: https://gatech.bluejeans.com/348270750

In 1973, Brown, Erdős and Sós proved that if H is a 3-uniform hypergraph on n vertices which contains no triangulation of the sphere, then H has at most O(n^{5/2}) edges, and this bound is the best possible up to a constant factor. Resolving a conjecture of Linial, also reiterated by Keevash, Long, Narayanan, and Scott, we show that the same result holds for triangulations of the torus. Furthermore, we extend our result to every closed orientable surface S.

Joint work with Alexandr Polyanskii, István Tomon and Dmitriy Zakharov, see https://arxiv.org/abs/2010.07191