In this talk I will illustrate my ideas and plans about the development of an undergraduate curriculum in the broader area of data science that includes, among other things, a course in image
processing. I will give an overview of the field, discuss typical problems that are studied within the discipline, and present an array of applications in medicine, astronomy, atmospheric science, security, navigation systems, and others: this will include a brief exposition of my own research in the recovery of images from videos affected by optical turbulence. I will be drawing ideas from my own experience in teaching courses and doing research with undergraduates at different academic institutions.
 

For 3 years I served as the Director of Jump Labs, a new endeavor for cutting-edge research and recruiting launched by Jump Trading, a quantitative high frequency trading firm based in Chicago. 
Jump Labs sponsors research in high performance computing and data science via gifts grants involving:

• Mentors from Jump Trading and Jump Venture Capital portfolio companies who guide the research along with University of Illinois professors
• Jump Trading proprietary data: ~50 PB of historical market microstructure data from 60 exchanges around the world
• Supercomputer grid resources
• Office space at Jump Labs in the University of Illinois Research Park

The crux is to create a long term and powerful pipeline for talent acquisition by challenging the faculty and students with real-world problems. The structure aligns relevant industrial research with the passions and expertise of the faculty member and students. Opportunities for publication are encouraged.  In our first two years we sponsored over 60 undergraduate and graduate students and 20 professors spanning 25 projects. The structure seeks to advance relevant research and creates a powerful recruiting pipeline for talent that is long term and low risk.

We will discuss the successes and challenges encountered at Jump Labs in its first three years.

Proximal algorithms offer state of the art performance for many large scale optimization problems. In recent years, the proximal algorithms landscape has simplified, making the subject quite accessible to undergraduate students. Students are empowered to achieve impressive results in areas such as image and signal processing, medical imaging, and machine learning using just a page or two of Python code. In this talk I'll discuss my experiences teaching proximal algorithms to students in the Physics and Biology in Medicine program at UCLA. I'll also share some of my teaching philosophy and approaches to teaching undergraduate math courses. Finally, I'll discuss my own research in optimization algorithms for radiation treatment planning, which is a fruitful source of undergraduate research projects.
 

Let X be an algebraic variety -- that is, the solution set to a system of polynomial equations.  Then the fundamental group of X has several incarnations, reflecting the geometry, topology, and arithmetic of X.  This talk will discuss some of these incarnations and the subtle relationships between them, and will describe an ongoing program which aims to apply the study of the fundamental group to classical problems in algebraic geometry and number theory.

Hilbert's 19th problem asks if minimizers of "natural" variational integrals are smooth. For the past century, this problem inspired fundamental regularity results for elliptic and parabolic PDEs. It also led to the construction of several beautiful counterexamples to regularity. The dichotomy of regularity vs. singularity is related to that of single PDE (the scalar case) vs. system of PDEs (the vectorial case), and low dimension vs. high dimension. I will discuss some interesting recent counterexamples to regularity in low-dimensional vectorial cases, as well as outstanding open problems. Parts of this are joint works with A. Figalli and O. Savin.