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Series: Research Horizons Seminar

We will discuss a few introductory results in real algebraic geometry concerning semi-algebraic sets. A semi-algebraic subset of R^k is the set of solutions of a boolean combination of finitely many real polynomial equalities and inequalities. These sets arise naturally in many areas of mathematics as well as other scientific disciplines, such as discrete and computational geometry or the configuration spaces in robotic motion planning. After providing some basic definitions and examples, we will outline the proof of a fundamental result, the Oleinik-Petrovsky-Thom-Milnor bound of d(2d-1)^{k-1} on the sum of the Betti numbers of a real algebraic variety, as well as indicate the direction of recent and ongoing research generalizing this result.

Series: Research Horizons Seminar

The classic Pick Interpolation Problem asks: Given points z_1, z_n and w_1, w_n in the unit disk, is there a function f(z) that (1) is holomorphic on the unit disk, (2) satisfies f(z_i)=w_i, and (3) satisfies |f(z)|=1 In 1917, Pick showed that such a function f(z) exists precisely when an associated matrix is positive semidefinite.
In this talk, I will translate the Pick problem to the language of Hilbert function spaces and present a more modern proof of the Pick problem. The benefit of this approach is that, as shown by J. Agler in 1989, it generalizes easily to the two-variable setting. At the heart of the proof is a method of representing bounded analytic one and two-variable functions using Hilbert space operators. Time-permitting, I will discuss recent results concerning the structure of such representations for bounded two-variable analytic functions, which is joint work with G. Knese.

Series: Research Horizons Seminar

Hypergeometric functions have played an important role in mathematics and physics in the last centuries. Multivariate extensions of the classical hypergeometric functions have appeared recently in different applications. I will discuss research problems which relate these functions to the representation theory of Lie algebras and quantum superintegrable systems.

Series: Research Horizons Seminar

Abstract: In this talk, I will use two examples, the influence prediction in social media, and the short path in engineering, to illustrate how we use differential equations to establish models for problems in social science and engineering, and how to use mathematics to design efficient algorithms to compute the solutions. The talk is mainly for first or second year graduate students, and it is based on collaborative work with several faculty members and graduate students in SoM, ECE, CoC.

Series: Research Horizons Seminar

This talk is an introduction to mathematical approaches to image processing: using
variational approaches and PDE based method. Various problems and a few different approaches
will be introduced.

Series: Research Horizons Seminar

We present several main models in this area including random polymers. We then explain some open problems big and small as well as a few of our related results.

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!