- You are here:
- GT Home
- Home
- News & Events

Series: CDSNS Colloquium

The format of this talk is rather non-standard. It is actually a combination of two-three mini-talks. They would include only motivations, formulations, basic ideas of proof if feasible, and open problems. No technicalities. Each topicwould be enough for 2+ lectures. However I know the hard way that in diverse audience, after 1/3 of allocated time 2/3 of people fall asleep or start playing with their tablets. I hope to switch to new topics at approximately right times.The topics will probably be chosen from the list below.“A survival guide for feeble fish”. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water flow. This is related tohomogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov.How can one discretize elliptic PDEs without using finite elements, triangulations and such? On manifolds and even reasonably “nice” mm–spaces. A notion of rho-Laplacian and its stability. Joint with S. Ivanov and Kurylev.One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodicdynamics. What happens outside KAM tori has been remaining a great mystery. The main quantative invariants so far are entropies. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however,on a zero measure set. We were able to show that metric entropy can become infinite too, under arbitrarily small C^{infty} perturbations. Furthermore, a slightly modified construction resolves another long–standing problem of theexistence of entropy non-expansive systems. These modified examples do generate positive positive metric entropy is generated in arbitrarily small tubular neighborhood of one trajectory. The technology is based on a metric theory of“dual lens maps” developed by Ivanov and myself, which grew from the “what is inside” topic.How well can we approximate an (unbounded) space by a metric graph whose parameters (degree of vertices, length of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers (the mostdifficult one is for R^2) are given using dynamics and Fourier series.“What is inside?” Imagine a body with some intrinsic structure, which, as usual, can be thought of as a metric. One knows distances between boundary points (say, by sending waves and measuring how long it takes them to reach specific points on the boundary). One may think of medical imaging or geophysics. This topic is related to the one on minimal fillings, the next one. Joint work with S. Ivanov.Ellipticity of surface area in normed space. An array of problems which go back to Busemann. They include minimality of linear subspaces in normed spaces and constructing surfaces with prescribed weighted image under the Gauss map. I will try to give a report of recentin “what is inside?” mini-talk. Joint with S. Ivanov.More stories are left in my left pocket. Possibly: On making decisions under uncertain information, both because we do not know the result of our decisions and we have only probabilistic data.

Series: Other Talks

Rhythm is a great thing. It therefore follows that several rhythms at
once is even greater. Learn 2:3, 3:4, and 4:5, and a little bit about fractions.
Polyrhythms when sped up, lead to harmony and scales. Slower polyrhythms
happen in celestial mechanics. A little bit of music, a little bit of
mathematics.

Series: Dissertation Defense

A subdivision of a graph G, also known as a topological G and denoted by TG, is a graph obtained from G by replacing certain edges of G with internally vertex-disjoint paths. This dissertation has two parts. The first part studies a structural problem and the second part studies an extremal problem. In the first part of this dissertation, we focus on TK_5, or subdivisions of K_5. A well-known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K_5 or K_{3,3}. Wagner proved in 1937 that if a graph other than K_5 does not contain any subdivision of K_{3,3} then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of K_5 then it is planar or it admits a cut of size at most 4. In this dissertation, we give a proof of the Kelmans-Seymour conjecture. We also discuss several related results and problems. The second part of this dissertation concerns subdivisions of large cliques in C_4-free graphs. Mader conjectured that every C_4-free graph with average degree d contains TK_l with l = \Omega(d). Komlos and Szemeredi reduced the problem to expanders and proved Mader's conjecture for n-vertex expanders with average degree d < exp( (log n)^(1/8) ). In this dissertation, we show that Mader's conjecture is true for n-vertex expanders with average degree d < n^0.3, which improves Komlos and Szemeredi's quasi-polynomial bound to a polynomial bound. As a consequence, we show that every C_4-free graph with average degree d contains a TK_l with l = \Omega(d/(log d)^c) for any c > 3/2. We note that Mader's conjecture has been recently verified by Liu and Montgomery.

Monday, April 10, 2017 - 14:00 ,
Location: Skiles 005 ,
Elisabetta Matsumoto ,
GT Physics ,
Organizer: Sung Ha Kang

The properties of euclidean space seem natural and obvious to us, to thepoint that it took mathematicians over two thousand years to see analternative to Euclid’s parallel postulate. The eventual discovery ofhyperbolic geometry in the 19th century shook our assumptions, revealingjust how strongly our native experience of the world blinded us fromconsistent alternatives, even in a field that many see as purelytheoretical. Non-euclidean spaces are still seen as unintuitive and exotic,but with direct immersive experiences we can get a better intuitive feel forthem. The latest wave of virtual reality hardware, in particular the HTCVive, tracks both the orientation and the position of the headset within aroom-sized volume, allowing for such an experience. We use this nacenttechnology to explore the three-dimensional geometries of theThurston/Perelman geometrization theorem. This talk focuses on oursimulations of H³ and H²×E.

Series: Geometry Topology Seminar

A foundational result in the study of contact geometry and Legendrian knots is Eliashberg and Fraser's classification of Legendrian unknots They showed that two homotopy-theoretic invariants - the Thurston-Bennequin number and rotation number - completely determine a Legendrian unknot up to isotopy. Legendrian spatial graphs are a natural generalization of Legendrian knots. We prove an analogous result for planar Legendrian graphs. Using convex surface theory, we prove that the rotation invariant and Legendrian ribbon are a complete set of invariants for planar Legendrian graphs. We apply this result to completely classify planar Legendrian embeddings of the Theta graph. Surprisingly, this classification shows that Legendrian graphs violate some proven and conjectured properties of Legendrian knots. This is joint work with Danielle O'Donnol.

Series: CDSNS Colloquium

Given a dynamical system (in finite or infinite dimension) it is very natural to look for finite dimensional invariant subspaces on which the dynamics is very simple. Of particular interest are the invariant tori on which the dynamics is conjugated to a linear one. The problem of persistence under perturbations of such objects has been widely studied starting form the 50's, and this gives rise to the celebrated KAM theory. The aim of this talk is to give an overview of the main difficulties and strategies, having in mind the application to PDEs.

Series: Combinatorics Seminar

The flow polytope associated to an acyclic graph is the set of all
nonnegative flows on the edges of the graph with a fixed netflow at each
vertex. We will examine flow polytopes arising from permutation matrices,
alternating sign matrices and Tesler matrices. Our inspiration is the
Chan-Robins-Yuen polytope (a face of the polytope of doubly-stochastic
matrices), whose volume is equal to the product of the first n Catalan
numbers (although there is no known combinatorial proof of this fact!). The
volumes of the polytopes we study all have nice product formulas.

Series: Combinatorics Seminar

The theme of this talk is walks in a random environment of "signposts"
altered by the walker. I'll focus on three related examples:
1. Rotor walk on Z^2. Your initial signposts are independent with the
uniform distribution on {North,East,South,West}. At each step you rotate
the signpost at your current location clockwise 90 degrees and then follow
it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps
you will visit order n^{2/3} distinct sites. I'll outline an elementary
proof of a lower bound of this order. The upper bound, which is still open,
is related to a famous question about the path of a light ray in a grid of
randomly oriented mirrors. This part is joint work with Laura Florescu and
Yuval Peres.
2. p-rotor walk on Z. In this walk you flip the signpost at your current
location with probability 1-p and then follow it. I'll explain why your
scaling limit will be a Brownian motion perturbed at its extrema. This part
is joint work with Wilfried Huss and Ecaterina Sava-Huss.
3. p-rotor walk on Z^2. Rotate the signpost at your current location
clockwise with probability p and counterclockwise with probability 1-p, and
then follow it. This walk “organizes” its environment of signposts. The
stationary environment is an orientation of the uniform spanning forest,
plus one additional edge. This part is joint work with Swee Hong Chan, Lila
Greco and Boyao Li.

Friday, April 7, 2017 - 15:05 ,
Location: Skiles 254 ,
Prof. Rafael de la Llave ,
School of Math, Georgia Tech ,
Organizer: Jiaqi Yang

It is well known that periodic orbits give all the information about dynamical systems, at least for expanding maps, for which the periodic orbits are dense. This turns out to be true in dimensions 1 and 2, and false in dimension 4 or higher.We will present a proof that two $C^\infty$ expanding maps of the circle, which are topologically equivalent are $C^\infty$ conjugate if and only if the derivatives or the return map at periodic orbits are the same.

Series: Stochastics Seminar

We discuss scaling methods
which can be used to solve low mode control problems for nonlinear
partial differential equations. These methods lead naturally to a
infinite-dimensional generalization of the notion of saturation,
originally due to Jurdjevic and Kupka in the finite-dimensional setting
of ODEs. The methods will be highlighted by applying them to specific
equations, including reaction-diffusion equations, the 2d/3d
Euler/Navier-Stokes equations and the 2d Boussinesq equations.
Applications to support properties of the laws solving randomly-forced
versions of each of these equations will be noted.