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Series: Research Horizons Seminar

Beginning with the Cauchy formula, we introduce
the Poisson average, and the Carleson embeding
theorem. From there, recent weighted estimates
for the Hilbert and Cauchy transforms can
be introduced.

Series: Research Horizons Seminar

The theory of non-Archimedean analytic spaces closely parallels that
of complex analytic spaces, with many theorems holding in both
situations. I'll illustrate this principle by giving a survey of the
structure theory of analytic curves over non-Archimedean fields, and
comparing them to classical Riemann surfaces. I'll draw plenty of
pictures and discuss topology, pair-of-pants decompositions, etc.

Series: Research Horizons Seminar

Answering a question of R. Stanley, we show that for each t ≥1, there is a least positive integer f(t) so that a planar poset with t minimal elements has dimension at most f(t). In particular, we show that f(t) ≤ 2t + 1 and that this inequality is tight for t=1 and t=2. For larger values of t, we can only show that f(t) ≥ t+3. This research is joint work with Georgia Tech graduate student Ruidong Wang.

Series: Research Horizons Seminar

Hyperbolic 3-manifolds is a great class of 3-dimensional geometric objects with interesting topology, a rich source of examples (practially one for every knot that you can draw), with arithmetically interesting volumes expressed in terms of dialogarithms of algebraic numbers, and with computer software that allows to manipulate them. Tired of abstract existential mathematics? Interested in concrete 3-dimensional topology and geometry? Or maybe Quantum Topology? Come and listen!

Series: Research Horizons Seminar

In dynamical systems, the long term behavior is organized by invariant manifolds that serve as landmarks that organize the traffic. There are two main theorems (established around 40-60 years ago) that tell you that these manifolds persist under small perturbations: KAM theorem and the theory of normally hyperbolic manifolds. In recent times there have been constructive proofs of these results which also lead to effective algorithms which allow to explore what happens in the border of the applicability of the theorems. We plan to review the basic concepts and present the experimental results.

Series: Research Horizons Seminar

Abstract: Four dimensions is unique in many ways. For example, n-dimensional Euclidean space has a unique smooth structure if and only if n is not equal to four. In other words, there is only one way to understand smooth functions on R^n if and only if n is not 4. There are many other ways that smooth structures on 4-dimensional manifolds behave in surprising ways. In this talk I will discuss this and I will sketch the beautiful interplay of ideas (you got algebra, analysis and topology, a little something for everyone!) that go into proving R^4 has more that one smooth structure (actually it has uncountably many different smooth structures but that that would take longer to explain).

Series: Research Horizons Seminar

I will discuss the variational approach to determining the stability of pendant liquid drops. The outline will include some theoretical aspects and questions which currently can only be answered numerically.

Series: Research Horizons Seminar

This is a will be a panel made of two senior grad students, a post doc and a faculty member. The panelists will answer questions and give advice to younger graduate students on a range of topics including how to be a good citizen of the department and choosing an advisor.
The panelists are Dr. Kang, Dr. Kelly Bickel, Albert Bush, and Chris Pryby.

Series: Research Horizons Seminar

The standard map is a widely studied area-preserving system with application to many natural phenomena. When unperturbed, every orbit of this map lies on an invariant circle. In this talk we will explore what happens to these circles when the system is perturbed, employing both analytical and numerical tools. I will conclude by discussing some active areas of current research.

Series: Research Horizons Seminar

We'll look at some of the basics of potential theory in the complex plane. We'll also discuss how potential theory may be used in studying zeros of polynomials and approximation theory.