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Series: Research Horizons Seminar

I will give an overview of how lattices in R^n are providing a powerful new mathematical foundation for cryptography. Lattices yield simple, fast, and highly parallel schemes that, unlike many of today's popular cryptosystems (like RSA and elliptic curves), even appear to remain secure against quantum computers. What's more, lattices were recently used to solve a cryptographic "holy grail" problem known as fully homomorphic encryption.
No background in lattices, cryptography, or quantum computers will be necessary for this talk -- but you will need to know how to add and multiply matrices.

Series: Research Horizons Seminar

A 1986 article with this title, written by M. Zuker and published by the AMS, outlined several major challenges in the area. Stating the folding problem is simple; given an RNA sequence, predict the set of (canonical, nested) base pairs found in the native structure. Yet, despite significant advances over the past 25 years, it remains largely unsolved. A fundamental problem identified by Zuker was, and still is, the "ill-conditioning" of discrete optimization solution approaches. We revisit some of the questions this raises, and present recent advances in considering multiple (sub)optimal structures, in incorporating auxiliary experimental data into the optimization, and in understanding alternative models of RNA folding.

Series: Research Horizons Seminar

Series: Research Horizons Seminar

In the last few years many problems of mathematical and physical interest, which may not be Hamiltonian or even dynamical, were solved using techniques from integrable systems. I will review some of these techniques and their connections to some open research problems.

Series: Research Horizons Seminar

I will discuss algebraic (sums of squares based) certificates for
nonnegativity of polynomials and their use in optimization. Then I will
discuss some recent results on degree bounds and state some open
questions.

Series: Research Horizons Seminar

I'll introduce the Hilbert transform in a natural way justifying it as a canonical operation. In fact, it is such a basic operation, that it arises naturally in a range of settings, with the important complication that the measure spaces need not be Lebesge, but rather a pair of potentially exotic measures. Does the Hilbert transform map L^2 of one measure into L^2 of the other? The full characterization has only just been found. I'll illustrate the difficulties with a charming example using uniform measure on the standard 1/3 Cantor set.

Series: Research Horizons Seminar

The derivation of the properties of macroscopic systems (e.g. the air in a
room) from the motions and interactions of their microscopic constituents is
the principal goal of Statistical Mechanics. I will introduce a simplified
model of a gas (the Kac model). After discussing its relation with more
realistic models, I'll present some known results and possible extension.

Series: Research Horizons Seminar

After some brief comments about the nature of mathematical modeling in
biology and medicine, we will formulate and analyze the SIR infectious
disease transmission model. The model is a system of three non-linear
differential equations that does not admit a closed form solution.
However, we can apply methods of dynamical systems to understand a great
deal about the nature of solutions. Along the way we will use this
model to develop a theoretical foundation for public health
interventions, and we will observe how the model yields several
fundamental insights (e.g., threshold for infection, herd immunity,
etc.) that could not be obtained any other way. At the end of the talk
we will compare the model predictions with data from actual outbreaks.

Series: Research Horizons Seminar

I will review a little bit of the theory of algebric curves, which essentialy
amounts to studying the zero set of a two-variable polynomial. There are
several amazing facts about the number of points on a curve when the ground
field is finite. (This particular case has many applications to cryptography
and coding theory.) An open problem in this area is whether there exist
"supersingular" curves of every genus. (I'll explain the terminology, which has
something to do with having many points or few points.) A new project I have
just started should go some way toward resolving this question.

Series: Research Horizons Seminar

In this talk, I will use the shortest path problem as an example to illustrate how one can use optimization, stochastic differential equations and partial differential equations together to solve some challenging real world problems. On the other end, I will show what new and challenging mathematical problems can be raised from those
applications. The talk is based on a joint work with Shui-Nee Chow and Jun Lu. And it is intended for graduate students.