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Series: Stelson Lecture Series

How is it possible to send encrypted information across an insecure channel (like the internet) so that only the intended recipient can decode it, without sharing the secret key in advance? In 1976, well before this question arose, a new mathematical theory of encryption (public-key cryptography) was invented by Diffie and Hellman, which made digital commerce and finance possible. The technology advances of the last twenty years bring new and urgent problems, including the need to compute on encrypted data in the cloud and to have cryptography that can withstand the speed-ups of quantum computers. In this lecture, we will discuss some of the history of cryptography, as well as some of the latest ideas in "lattice" cryptography which appear to be quantum resistant and efficient.

Series: Stelson Lecture Series

We will see how a result in von Neumann algebras (a theory developed by von Neumann to give themathematical framework for quantum physics) gave rise, rather serendipitously, to an elementary but very usefulinvariant in the theory of ordinary knots in threel space. Then we'll look at some subsequent developments of the theory, and talk about a thorny problem which remains open.

Series: Stelson Lecture Series

General Audience Lecture. Reception to follow in Klaus Atrium.

This talk is the story of an encounter of three distinct fields:
non-Euclidean geometry, gas dynamics and economics. Some of the most
fundamental mathematical tools behind these theories appear to have a close
connection, which was revealed around the turn of the 21st century, and has
developed strikingly since then.

Series: Stelson Lecture Series

Mathematics Audience Lecture

This talk explains how the solution to a regularity/geometry problem
arising from a question of optimization has led to unexpected new results
in the well-established field of the analysis of cut loci.

Series: Stelson Lecture Series

Mathematics lecture

This talks introduces a novel framework for phase retrieval, a problem which
arises in X-ray crystallography, diffraction imaging, astronomical imaging
and many other applications. Our approach combines multiple structured
illuminations together with ideas from convex programming to recover the
phase from intensity measurements, typically from the modulus of the
diffracted wave. We demonstrate empirically that any complex-valued object
can be recovered from the knowledge of the magnitude of just a few
diffracted patterns by solving a simple convex optimization problem inspired
by the recent literature on matrix completion. More importantly, we also
demonstrate that our noise-aware algorithms are stable in the sense that the
reconstruction degrades gracefully as the signal-to-noise ratio decreases.
Finally, we present some novel theory showing that our entire approach may
be provably surprisingly effective.

Series: Stelson Lecture Series

General audience lecture

This talk is about a curious phenomenon. Suppose we have a data matrix,
which is the superposition of a low-rank component and a sparse component.
Can we recover each component individually? We prove that under some
suitable assumptions, it is possible to recover both the low-rank and the
sparse components exactly by solving a very convenient convex program. This
suggests the possibility of a principled approach to robust principal
component analysis since our methodology and results assert that one can
recover the principal components of a data matrix even though a positive
fraction of its entries are arbitrarily corrupted. This extends to the
situation where a fraction of the entries are missing as well. In the second
part of the talk, we present applications in computer vision. In video
surveillance, for example, our methodology allows for the detection of
objects in a cluttered background. We show how the methodology can be
adapted to simultaneously align a batch of images and correct serious
defects/corruptions in each image, opening new perspectives.

Series: Stelson Lecture Series

Mathematics lecture

New technologies have been introduced into the front tracking method to improve its performance in extreme applications, those dominated by a high density of interfacial area. New mathematical theories have been developed to understand the meaning of numerical convergence in this regime. In view of the scientific difficulties of such problems, careful verifaction, validation and uncertainty quantification studies have been conducted. A number of interface dominated flows occur within practical problems of high consequence, and in these cases, we are able to contribute to ongoing scientific studies. We include here turbulent mixing and combustion, chemical processing, design of high energy accelerators, nuclear fusion related studies, studies of nuclear power reactors and studies of flow in porous media. In this lecture, we will review some of the above topics.

Series: Stelson Lecture Series

This lecture is more for the general audience.

Reception to follow in Klaus Atrium.

The changing status of knowledge from descriptive to analytic, from empirical to theoretical and from intuitive to mathematical has to be one of the most striking adventures of the human spirit. The changes often occur in small steps and can be lost from view. In this lecture we will review vignettes drawn from the speaker's personal knowledge that illustrate this transformation in thinking. Examples include not only the traditional areas of physics and engineering, but also newer topics, as in biology and medicine, in the social sciences, in commerce, and in the arts. We also review some of the forces driving these changes, which ultimately have a profound effect on the organization of human life.

Series: Stelson Lecture Series

This lecture will be more for the mathematical audience

Whether the 3D incompressible Navier-Stokes equations can develop
a finite time singularity from smooth initial data is one of the
seven Millennium Problems posted by the Clay Mathematical Institute.
We review some recent theoretical and computational studies of the
3D Euler equations which show that there is a subtle dynamic depletion of
nonlinear vortex stretching due to local geometric regularity of
vortex filaments. The local geometric regularity of vortex filaments
can lead to tremendous cancellation of nonlinear vortex stretching.
This is also confirmed by our large scale computations for some of
the most well-known blow-up candidates. We also
investigate the stabilizing effect of convection in 3D incompressible
Euler and Navier-Stokes equations. The convection term is the main source
of nonlinearity for these equations. It is often considered destabilizing
although it conserves energy due to the incompressibility condition. Here
we reveal a surprising nonlinear stabilizing effect that the convection
term plays in regularizing the solution. Finally, we present a new class
of solutions for the 3D Euler and Navier-Stokes equations, which exhibit
very interesting dynamic growth property. By exploiting the special
structure of the solution and the cancellation between the convection
term and the vortex stretching term, we prove nonlinear stability and
the global regularity of this class of solutions.

Multiscale Modeling and Computation - The Interplay Between Mathematics and Engineering Applications

Series: Stelson Lecture Series

This lecture is more for the general audience. Reception following lecture. Organizers: Chongchun Zeng and Hao Min Zhou

Many problems of fundamental and practical importance contain multiple scale solutions. Composite and nano materials, flow and transport in heterogeneous porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the wide range of length scales in the underlying physical problems. Direct numerical simulations using a fine grid are very expensive. Developing effective multiscale methods that can capture accurately the large scale solution on a coarse grid has become essential in many engineering applications. In this talk, I will use two examples to illustrate how multiscale mathematics analysis can impact engineering applications. The first example is to develop multiscale computational methods to upscale multi-phase flows in strongly heterogeneous porous media. Multi-phase flows arise in many applications, ranging from petroleum engineering, contaminant transport, and fluid dynamics applications. Multiscale computational methods guided by multiscale analysis have already been adopted by the industry in their flow simulators. In the second example, we will show how to develop a systematic multiscale analysis for incompressible flows in three space dimensions. Deriving a reliable turbulent model has a significant impact in many engineering applications, including the aircraft design. This is known to be an extremely challenging problem. So far, most of the existing turbulent models are based on heuristic closure assumption and involve unknown parameters which need to be fitted by experimental data. We will show that how multiscale analysis can be used to develop a systematic multiscale method that does not involve any closure assumption and there are no adjustable parameters.