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Series: CDSNS Colloquium

Using techniques from local bifurcation theory, we prove the existence of various types of temporally periodic solutions for damped wave equations, in higher dimensions. The emphasis is on understanding the role of external bifurcation parameters and symmetry, in generating the periodic motion. The work presented is joint with Brian Pigott

Series: Combinatorics Seminar

A tight k-uniform \ell-cycle, denoted by TC_\ell^k, is a k-uniform hypergraph whose vertex set is v_0, ..., v_{\ell-1}, and the edges are all the k-tuples {v_i, v_{i+1}, \cdots, v_{i+k-1}}, with subscripts modulo \ell. Motivated by a classic result in graph theory that every n-vertex cycle-free graph has at most n-1 edges, Sos and, independently, Verstraete asked whether for every integer k, a k-uniform n-vertex hypergraph without any tight k-uniform cycles has at most \binom{n-1}{k-1} edges. In this talk I will present a construction giving negative answer to this question, and discuss some related problems. Joint work with Jie Ma.

Friday, February 16, 2018 - 15:00 ,
Location: Skiles 271 ,
Yian Yao ,
GT Math ,
Organizer: Jiaqi Yang

I
will report on the parameterization method for computing normally
hyperbolic invariant tori(NHIT) for diffeomorphisms. To this end, a
Newton-like method for solving the invariance equation based on the
graph transform method will be presented with details.
Some notes on numerical implementations will also be included if time
allows.
This is a work by Marta
Canadell and Alex Haro.

Series: Math Physics Seminar

The expectation values of the first and second moments of the quantum mechanical spin operator can be used to define a spin vector and spin fluctuation tensor, respectively. The former is a vector inside the unit ball in three space, while the latter is represented by an ellipsoid in three space. They are both experimentally accessible in many physical systems. By considering transport of the spin vector along loops in the unit ball it is shown that the spin fluctuation tensor picks up geometric phase information. For the physically important case of spin one, the geometric phase is formulated in terms of an SO(3) operator. Loops defined in the unit ball fall into two classes: those which do not pass through the origin and those which pass through the origin. The former class of loops subtend a well defined solid angle at the origin while the latter do not and the corresponding geometric phase is non-Abelian. To deal with both classes, a notion of generalized solid angle is introduced, which helps to clarify the interpretation of the geometric phase information. The experimental systems that can be used to observe this geometric phase are also discussed.Link to arxiv: https://arxiv.org/abs/1702.08564

Friday, February 16, 2018 - 10:10 ,
Location: Skiles 006 ,
Libby Taylor ,
Georgia Tech ,
libbyrtaylor@gmail.com ,
Organizer: Kisun Lee

Algebraic geometry has a plethora of cohomology theories, including the derived functor, de Rham, Cech, Galois, and étale cohomologies. We will give a brief overview of some of these theories and explain how they are unified by the theory of motives. A motive is constructed to be a “universal object” through which all cohomology theories factor. We will motivate the theory using the more familiar examples of Jacobians of curves and Eilenberg-Maclane spaces, and describe how motives generalize these constructions to give categories which encode all the cohomology of various algebro-geometric objects. The emphasis of this talk will be on the motivation and intuition behind these objects, rather than on formal constructions.

Series: Stochastics Seminar

Place Poi(m) particles at each site of a d-ary tree of height n. The particle at the root does a simple random walk. When it visits a site, it wakes up all the particles there, which start their own random walks, waking up more particles in turn. What is the cover time for this process, i.e., the time to visit every site? We show that when m is large, the cover time is O(n log(n)) with high probability, and when m is small, the cover time is at least exp(c sqrt(n)) with high probability. Both bounds are sharp by previous results of Jonathan Hermon's. This is the first result proving that the cover time is polynomial or proving that it's nonpolymial, for any value of m. Joint work with Christopher Hoffman and Matthew Junge.

Series: School of Mathematics Colloquium

Evolution of random systems as well as dynamical systems with chaotic (stochastic) behavior traditionally (and seemingly naturally) is described by studying only asymptotic in time (when time tends to infinity) their properties. The corresponding results are formulated in the form of various limit theorems (CLT, large deviations, etc). Likewise basically all the main notions (entropy, Lyapunov exponents, etc) involve either taking limit when time goes to infinity or averaging over an infinite time interval. Recently a series of results was obtained demonstrating that finite time predictions for such systems are possible. So far the results are on the intersection of dynamical systems, probability and combinatorics. However, this area suggests some new analytical, statistical and geometric problems to name a few, as well as opens up possibility to obtain new types of results in various applications. I will describe the results on (extremely) simple examples which will make this talk quite accessible.

Wednesday, February 14, 2018 - 14:00 ,
Location: Skiles 006 ,
Anubhav Mukherjee ,
GaTech ,
Organizer: Anubhav Mukherjee

We will discuss the relationship between diffeomorphis groups of spheres and balls. And try to give an idea of existense of exotic structures on spheres.

Series: Analysis Seminar

We study Balian-Low type theorems for finite signals in $\mathbb{R}^d$, $d\geq 2$.Our results are generalizations of S. Nitzan and J.-F. Olsen's recent work and show that a quantity closelyrelated to the Balian-Low Theorem has the same asymptotic growth rate, $O(\log{N})$ for each dimension $d$. Joint work with Michael Northington.

Series: Research Horizons Seminar

Some basic problems, notions and results of the Ergodic theory will be introduced. Several examples will be discussed. It is also a preparatory talk for the next day colloquium where finite time properties of dynamical and stochastic systems will be discussed rather than traditional questions all dealing with asymptotic in time properties.