Abstract: Renormalization provides a powerful tool to approach universality and
rigidity phenomena in dynamical systems. In this talk, I will discuss
recent results on renormalization and rigidity theory of circle
diffeomorphisms (maps) with a break (a single point where the derivative
has a jump discontinuity) and their relation with generalized interval
exchange transformations introduced by Marmi, Moussa and Yoccoz. In a
joint work with K.Khanin, we proved that renormalizations of any two
sufficiently smooth circle maps with a break, with the same irrational
rotation number and the same size of the break, approach each other
exponentially fast. For almost all (but not all) irrational rotation
numbers, this statement implies rigidity of these maps: any two
sufficiently smooth such maps, with the same irrational rotation number
(in a set of full Lebesgue measure) and the same size of the break, are
$C^1$-smoothly conjugate to each other. These results can be viewed as
an extension of Herman's theory on the linearization of circle
diffeomorphisms.
 

The study of discrete dynamical systems boomed in the age of computing. The Mandelbrot set, created by iterating 0 in the function z^2+c  and allowing c to vary, gives us a wealth of questions to explore. We can ask about the number of rational preperiodic points (points whose iterates end in a cycle) for z^2+c. Can this number be uniform as we allow c to vary? It turns out this is a hard question to answer. Instead we will explore places where this question can be answered; twists of rational functions. 

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.