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Series: School of Mathematics Colloquium

Kickoff of the Tech Topology Conference from December 6-8, 2013. For complete details see

ttc.gatech.edu

We will start by defining the Jones polynomial of a knot and talking about some of its classical applications to knot theory. We will then define a fancier version ("categorification") of the Jones polynomial, called Khovanov homology and mention some of its applications. We will conclude by talking about a further refinement, a Khovanov homotopy type, sketch some of the ideas behind its construction, and mention some applications. (This last part is joint work with Sucharit Sarkar.) At least the first half of the talk should be accessible to non-topologists.

Series: School of Mathematics Colloquium

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse.

Series: School of Mathematics Colloquium

In this talk, I shall sketch the study of the problem of Arnold diffusion from variational point of view. Arnold diffusion has been shown typical phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom:
$$
H(x,y)=h(y)+\epsilon P(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3.
$$
Under typical perturbation $\epsilon P$, the system admits ``connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\min\alpha$.

Series: School of Mathematics Colloquium

This talk deals with problems that are asymptotically related to best-packing and best-covering. In particular, we discuss how to efficiently generate N points on a d-dimensional manifold that have the desirable qualities of well-separation and optimal order covering radius, while asymptotically having a prescribed distribution. Even for certain small numbers of points like N=5, optimal arrangements with regard to energy and polarization can be a challenging problem.

Series: School of Mathematics Colloquium

(Joint with: M. Benedicks and D. Schnellmann) Many interesting dynamical systems possess a unique SRB ("physical")measure, which behaves well with respect to Lebesgue measure. Given a smooth one-parameter family of dynamical systems f_t, is natural to ask whether the SRB measure depends smoothly on the parameter t. If the f_t are smooth hyperbolic diffeomorphisms (which are structurally stable), the SRB measure depends differentiably on the parameter t, and its derivative is given by a "linear response" formula (Ruelle, 1997). When bifurcations are present and structural stability does not hold, linear response may break down. This was first observed for piecewise expanding interval maps, where linear response holds for tangential families, but where a modulus of continuity t log t may be attained for transversal families (Baladi-Smania, 2008). The case of smooth unimodal maps is much more delicate. Ruelle (Misiurewicz case, 2009) and Baladi-Smania (slow recurrence case, 2012) obtained linear response for fully tangential families (confined within a topological class). The talk will be nontechnical and most of it will be devoted to motivation and history. We also aim to present our new results on the transversal smooth unimodal case (including the quadratic family), where we obtain Holder upper and lower bounds (in the sense of Whitney, along suitable classes of parameters).

Series: School of Mathematics Colloquium

In this talk, my goal is to give an introduction to some of the mathematics
behind quasicrystals. Quasicrystals were discovered in 1982, when Dan
Schechtmann observed a material which produced a diffraction pattern made of
sharp peaks, but with a 10-fold rotational symmetry. This indicated that the
material was highly ordered, but the atoms were nevertheless arranged in a
non-periodic way.
These quasicrystals can be defined by certain aperiodic tilings, amongst which
the famous Penrose tiling. What makes aperiodic tilings so interesting--besides
their aesthetic appeal--is that they can be studied using tools from many areas
of mathematics: combinatorics, topology, dynamics, operator algebras...
While the study of tilings borrows from various areas of mathematics, it
doesn't go just one way: tiling techniques were used by Giordano, Matui, Putnam
and Skau to prove a purely dynamical statement: any Z^d free minimal action on
a Cantor set is orbit equivalent to an action of Z.

Series: School of Mathematics Colloquium

Probabilistic methods in dynamical systems is a popular area of research. The talk will present the origin of the interplay between both subjects with Poincar\'e's unpredictability and Kolmogorov's axiomatic treatment of probability, followed by two main breakthroughs in the 60es by Ornstein and Gordin. Present studies are concerned with two main problems: transferring probabilistic laws and laws for 'smooth' functions. Recent results for both type of questions are explained at the end.

Series: School of Mathematics Colloquium

During the 17th Century the French priest and physicist Edme Mariotte observed
that objects floating on a liquid surface can attract or repel each other, and he attempted
(without success!) to develop physical laws describing the phenomenon. Initial steps
toward a consistent theory came later with Laplace, who in 1806 examined the
configuration of two infinite vertical parallel plates of possibly differing materials, partially
immersed in an infinite liquid bath and rigidly constrained. This can be viewed as an
instantaneous snapshot of an idealized special case of the Mariotte observations. Using the
then novel concept of surface tension, Laplace identified particular choices of materials and
of plate separation, for which the plates would either attract or repel each other.
The present work returns to that two‐plate configuration from a more geometrical
point of view, leading to characterization of all modes of behavior that can occur. The
results lead to algorithms for evaluating the forces with arbitrary precision subject to the
physical hypotheses, and embrace also some surprises, notably the remarkable variety of
occurring behavior patterns despite the relatively few available parameters. A striking
limiting discontinuity appears as the plates approach each other.
A message is conveyed, that small configurational changes can have large
observational consequences, and thus easy answers in less restrictive circumstances
should not be expected.

Series: School of Mathematics Colloquium

The problem of finding rational solutions to cubic equations is central in number theory, and goes back to Fermat. I will discuss why these equations are particularly interesting, and the modern theory of elliptic curves that has developed over the past century, including the Mordell-Weil theorem and the conjecture of Birch and Swinnerton-Dyer. I will end with a description of some recent results of Manjul Bhargava on the average rank.

Series: School of Mathematics Colloquium

The study of mechanical linkages is a very classical one, dating back to
the Industrial Revolution. In this talk we will discuss the geometry
of the configuration spaces in some simple idealized examples and, in
particular, their curvature and geometry. This leads to an interesting
quantitative description of their dynamical behaviour.