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Series: Research Horizons Seminar

The Horn inequalities give a characterization of eigenvalues of self-adjoint n by n matrices A, B, C with A+B+C=0. The original proof by Klyachko and Knutson-Tao, requires tools from algebraic geometry, among other things. Our recent work provides a proof using only elementary tools that made it possible to generalize the Horn inequalities to finite von Neumann factors. No knowledge of von Neumann algebra is required.

Series: Research Horizons Seminar

Due to Alexander, it is well known that every closed oriented 3-manifold has an open book decomposition. In this talk, we will define open book decompositions of 3-manifolds. We will discuss various examples and sketch the proof of Alexander's theorem. Further, we will discuss the importance of the open books in manifold theory, in particular in contact geometry.

Series: Research Horizons Seminar

In this talk, we give an insight into the mathematical topic of shape optimization. First, we give several examples of problems, some of them are purely academic and some have an industrial origin. Then, we look at the different mathematical questions arising in shape optimization. To prove the existence of a solution, we need some topology on the set of domains, together with good compactness and continuity properties. Studying the regularity and the geometric properties of a minimizer requires tools from classical analysis, like symmetrization. To be able to define the optimality conditions, we introduce the notion of derivative with respect to the domain. At last, we give some ideas of the different numerical methods used to compute a possible solution.

Series: Research Horizons Seminar

The Apery sequence is a sequence of natural numbers 1,5,73,1445,...which is used to prove the irrationality of zeta(3). Can you compute its asymptotic expansion to all orders of 1/n? The talk will not assume a lot, but promises to compute, and also justify.

Series: Research Horizons Seminar

We examine some problems in the coupled motions of fluids and flexible solid bodies. We first present some basic equations in fluid dynamics and solid mechanics, and then show some recent asymptotic results and numerical simulations. No prior experience with fluid dynamics is necessary.

Series: Research Horizons Seminar

Orthogonal polynomials play a role in myriads of problems ranging from approximation theory to random matrices and signal processing. Generalizations of orthogonal polynomials - such as biorthogonal polynomials, cardinal series, Muntz polynomials, are used for example, in number theory and numerical analysis. We discuss some of these, and some potential research projects involving them.

Series: Research Horizons Seminar

This talk is not an appetizer to pizza, but rather an appetizer to the main course: Hua Xu's and Trevis Litherland's thesis defenses which will respectively take place on Thursday the 30th of October and November the 6th, in Skiles 269, at 3pm. I will present the history and origins of the problems they have been tackling ("Ulam's problems"). Various interactions with other fields such as Analysis, Algebra (Young Tableaux) or Bioinformatics (Sequence Comparison) will be touched upon. Then, some elementary but rather useful probabilistic techniques will also be introduced and shown how to be applied.

Series: Research Horizons Seminar

We consider the pseudodifferential operators H_{m,\Omega} associated by the prescriptions of quantum mechanics to the Klein-Gordon Hamiltonian when restricted to a compact domain \Omega in {\mathbb R}^d. When the mass m is 0 the operator H_{0,\Omega} coincides with the generator of the Cauchy stochastic process with a killing condition on \partial \Omega. (The operator H_{0,\Omega} is sometimes called the fractional Laplacian with power 1/2.) We prove several universal inequalities for the eigenvalues (joint work with Evans Harrell).

Series: Research Horizons Seminar

In the study of one dimensional dynamical systems it is often assumed that the functions involved have a negative Schwarzian derivative. However, as not all one dimensional systems of interest have this property it is natural to consider a generalization of this condition. Specifically, we consider the interval functions of a real variable having some iterate with a negative Schwarzian derivative and show that many known results generalize to this larger class, that is to functions with an eventual negative Schwarzian derivative. The property of having an eventual negative Schwarzian derivative is nonasymptotic therefore verification of whether a function has such an iterate can often be done by direct computation. The introduction of this class was motivated by some maps arising in neuroscience.

Series: Research Horizons Seminar

In this introduction to knot theory we will focus on a class of knots called rational knots. Here the word rational refers to a beautiful theorem by J. Conway that sets up a one to one correspondence between these knots and the rational numbers using continued fractions. We aim to give an elementary proof of Conway's theorem and discuss its application to the study of DNA recombination. No knowledge of topology is assumed.