Friday, March 2, 2018 - 15:05 , Location: Skiles 271 , Adrian P. Bustamante , Georgia Tech , Organizer:
Given a one-parameter family of maps of an interval to itself, one can observe period doubling bifurcations as the parameter is varied. The aspects of those bifurcations which are independent of the choice of a particular one-parameter family are called universal. In this talk we will introduce, heuristically, the so-called Feigenbaun universality and then we'll expose some rigorous results about it.
Series: Combinatorics Seminar
This is Lecture 3 of a series of 3 lectures. See the abstract on Tuesday's ACO colloquium of this week.(Please note that this lecture will be 80 minutes' long.)
Series: Math Physics Seminar
Recent advances in fluid dynamics reveal that the recurrent flows observed in moderate Reynolds number turbulence result from close passes to unstable invariant solutions of Navier-Stokes equations. By now hundreds of such solutions been computed for a variety of flow geometries, but always confined to small computational domains (minimal cells).Pipe, channel and plane flows, however, are flows on infinite spatial domains. We propose to recast the Navier-Stokes equations as a space-time theory, with the unstable invariant solutions now being the space-time tori (and not the 1-dimensional periodic orbits of the classical periodic orbit theory). The symbolic dynamics is likewise higher-dimensional (rather than a single temporal string of symbols). In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns.We illustrate the strategy by solving a very simple classical field theory on a lattice modelling many-particle quantum chaos, adiscretized screened Poisson equation, or the ``spatiotemporal cat.'' No actual cats, graduate or undergraduate, have showninterest in, or were harmed during this research.
Friday, March 2, 2018 - 14:00 , Location: Skiles 006 , Jen Hom , Georgia Tech , Organizer: Jennifer Hom
In this series of talks, we will study the relationship between the Alexander module and the bordered Floer homology of the Seifert surface complement. In particular, we will show that bordered Floer categorifies Donaldson's TQFT description of the Alexander module. No prior knowledge of the Alexander module or Heegaard Floer homology will be assumed.
Friday, March 2, 2018 - 10:00 , Location: Skiles 254 , Marcel Celaya , Georgia Tech , firstname.lastname@example.org , Organizer: Kisun Lee
In this talk we will discuss the paper of Adiprasito, Huh, and Katz titled "Hodge Theory for Combinatorial Geometries," which establishes the log-concavity of the characteristic polynomial of a matroid.
Series: Stelson Lecture Series
How is it possible to send encrypted information across an insecure channel (like the internet) so that only the intended recipient can decode it, without sharing the secret key in advance? In 1976, well before this question arose, a new mathematical theory of encryption (public-key cryptography) was invented by Diffie and Hellman, which made digital commerce and finance possible. The technology advances of the last twenty years bring new and urgent problems, including the need to compute on encrypted data in the cloud and to have cryptography that can withstand the speed-ups of quantum computers. In this lecture, we will discuss some of the history of cryptography, as well as some of the latest ideas in "lattice" cryptography which appear to be quantum resistant and efficient.
Series: Graph Theory Seminar
This is Lecture 2 of a series of 3 lectures by the speaker. See the abstract on Tuesday's ACO colloquium of this week. (Please note that this lecture will be 80 minutes' long.)
Wednesday, February 28, 2018 - 14:00 , Location: Skiles 006 , Hyun Ki Min , GaTech , Organizer: Anubhav Mukherjee
I will introduce the notion of satellite knots and show that a knot in a 3-sphere is either a torus knot, a satellite knot or a hyperbolic knot.
Series: PDE Seminar
Black holes are predicted by Einstein's theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being. In this talk, with tools from hyperbolic PDE, quasilinear elliptic equations, geometric analysis and dynamical systems, we will prove that, through a nonlinear focusing effect, initially low-amplitude and diffused gravitational waves can give birth to a black hole region in our universe. This result extends the 1965 Penrose’s singularity theorem and it also proves a conjecture of Ashtekar on black-hole thermodynamics. Open problems and new directions will also be discussed.
Series: ACO Colloquium
Many hard problems of combinatorial counting can be encoded as problems of computing an appropriate partition function. Formally speaking, such a partition function is just a multivariate polynomial with great many monomials enumerating combinatorial structures of interest. For example, the permanent of an nxn matrix is a polynomial of degree n in n^2 variables with n! monomials enumerating perfect matchings in the complete bipartite graph on n+n vertices. Typically, we are interested to compute the value of such a polynomial at a real point; it turns out that to do it efficiently, it is very helpful to understand the behavior of complex zeros of the polynomial. This approach goes back to the Lee-Yang theory of the critical temperature and phase transition in statistical physics, but it is not identical to it: thinking of the phase transition from the algorithmic point of view allows us greater flexibility: roughly speaking, for computational purposes we can freely operate with “complex temperatures”. I plan to illustrate this approach on the problems of computing the permanent and its versions for non-bipartite graphs (hafnian) and hypergraphs, as well as for computing the graph homomorphism partition function and its versions (partition functions with multiplicities and tensor networks) that are responsible for a variety of problems on graphs involving colorings, independent sets, Hamiltonian cycles, etc. (This is the first (overview) lecture; two more will follow up on Thursday 1:30pm, Friday 3pm of the week. These two lectures are each 80 minutes' long.)