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Series: Research Horizons Seminar

Note: This is a special time for Research Horizons.

Special seminar title: The idea of studying the geometry and topology of
finite metric spaces has arisen due to the fact that almost all kinds of
data sets arising in science or the commercial world are equipped with a
metric. This has led to the development of cohomology theories applicable
to finite metric spaces, which allow one to construct "measurements" of the
shape of the data sets. We will define these theories and discuss their
properties. We will also describe their applications, and suggest
directions of future research on them.

Series: Research Horizons Seminar

Numerical relativity has opened the door to unveil phenomena
associated with strong dynamical gravity. I will present results from
three studies of black holes that have been only possible thanks to
state of the art computational tools and powerful computer hardware.

Series: Research Horizons Seminar

An important question in modern complex analysis is to obtain a
characterization of the sequence of points in the disc {z_j} that
interpolates any given target sequence {a_j} with an element of a space of
analytic functions. In this talk we will discuss this question and
reformulate it as a problem in linear algebra and then show how this can be
solved with relatively straightforward tools. Connections to open
questions will also be given.

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 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

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.