## Assistant Professor of Mathematics, University of Illinois

### Contact information

Office 371 Altgeld Hall
Phone 217-244-0225
Email jathreya AT illinois DOT edu
Mailing Address Department of Mathematics,
University of Illinois
1409 W. Green Street
Urbana, IL 61801
Curriculum Vitae Current CV

### Teaching

I am teaching Math 522, Lie Groups and Lie Algebras in Fall 2013.

### Illinois Geometry Lab

The above is the logo for the Illinois Geometry Lab, or IGL, a facility which I co-founded and currently direct. It is an initiative to involve undergraduate students in research and outreach projects involving geometry and visualization. Altgeld Hall, the home of the mathematics department, is one of the defining symbols of the University of Illinois, yet many people in our community remain unaware of the cutting-edge research that happens inside. Indeed, many are surprised by the mere notion of mathematics research.

The IGL seeks to spark interest in mathematics through showing the creative side of mathematics, in particular, by visualizing and fabricating mathematical concepts using a hands-on experimental approach. These types of ideas have been pioneered and developed by Illinois mathematics faculty since the late 19th century, and the IGL builds on this rich tradition. Since Fall 2011, the IGL has been working to bring students and faculty together to work on open problems in mathematics and mathematics visualization, spark the interest of elementary, middle, and high-school students through targeted outreach activities, and providing mathematical activities to the community, in particular via our website. Our efforts have been recognized and funded by a University of Illinois Office of Public Engagement Grant, and been featured on the Scientific American website. More recently, we have been highlighted in Postmarks, a University of Illinois newsletter.

In addition to directing the lab, I am currently directing two projects, one on Apollonian circle packings, and another on the structural design of Lithium-Ion batteries.

### Research interests

My research interests are in geometry and dynamical systems. My primary area of research is the study of dynamics of Lie group actions on various parameter spaces. In particular, I study the dynamics of the $$SL(2, \mathbb R)$$ action on moduli spaces of abelian and quadratic differentials, as well as dynamics on the space of lattices, both over the reals and over fields of positive characteristic. These subjects lie at the intersection of low-dimensional topology, ergodic theory, dynamics of group actions, and Diophantine approximation.

### Awards and Support

My research is currently supported by NSF PI grant (DMS 1069153). I am also supported by the University of Illinois Campus Research Board, and I am currently (Spring 2013) a Fellow of the University of Illinois Center for Advanced Study. I am also supported by an In3 (Interdisciplinary Innovation Initiative) grant, together with Shen Dillon from Materials Science; Ioannis Chasiotis and John Lambros from Aerospace Engineering. We are working on a project on structural design of Lithium-ion batteries. I am a member of the GEometric structures And Representation varieties (GEAR) NSF Research Network. I was previously supported by an NSF Postdoctoral Fellowship.

### Recent Talk slides

I currently supervise two PhD students, Yiannis Konstantoulas and Grace Work. I was the Master's thesis advisor for Aaron Wittrig, who graduated in 2012.

### Outreach

In addition to outreach activities through the Illinois Geometry Lab, I enjoy publicizing and sharing mathematical ideas through various forums. I've written piece on Apollonian circle packings for the Geometry Issue of Indian children's science magazine Brainwave. In Fall 2011, I worked with Anne Sautmann, the director of education at the Krannert Art Museum on a workshop on the interactions between art and mathematics for high school teachers. In Summer 2011, I worked with students and teachers at the Chirag School, in the Indian state of Uttarakhand, to develop curriculum focusing on geometry and pattern recognition.

### Expository Talks

I taught a REGS Minicourse on Translation Surfaces, and a more advanced course on the same topic at the GEAR Junior Retreat. Lecture notes for the REGS course can be found here, and the presentation for the GEAR course is below:

There is also a video archive of the GEAR course.

### Publications

##### (In reverse chronological order)
1. A Poincaré section for horocycle flow on the space of lattices, (joint with Yitwah Cheung), International Math Research Notices, 2013.
2. Buildings, Extensions, and Volume Growth Entropy, (joint with Anish Ghosh and Amritanshu Prasad), New York Journal of Mathematics, Volume 19 (2013) 1-11.
3. Ergodic Properties of Compositions of Interval Exchange Maps and Rotations, (joint with Michael Boshernitzan), Nonlinearity 26 (2013) 417-423.
4. Gap Distributions and Homogeneous Dynamics, to appear in Proceedings of ICM Satellite Conference on Geometry, Topology, and Dynamics in Negative Curvature.
5. Cusp excursions on parameter spaces, Journal of London Math Society, 2013.
6. The distribution of gaps for saddle connection directions, (joint with Jon Chaika), Geometric and Functional Analysis, Volume 22, Issue 6 (2012), 1491-1516.
7. Lattice point asymptotics and volume growth on Teichmuller space, (joint with Sasha Bufetov, Alex Eskin, and Maryam Mirzakhani), Duke Mathematical Journal, Volume 161, Number 6 (2012), 1055-1111.
8. Ultrametric Logarithm Laws, II (joint with Anish Ghosh and Amritanshu Prasad), Monatshefte fur Mathematik, Volume 167, Issue 3-4 (2012), 333-356.
9. Logarithm laws and shrinking target properties, Proceedings of the Indian Academy of Sciences, volume 119, number 4, pages 541-559, November 2009
10. Logarithm laws for unipotent flows, I (joint with Gregory Margulis), Journal of Modern Dynamics, volume 3, number 3, pages 359-378, July 2009
11. Ultrametric Logarithm Laws, I (joint with Anish Ghosh and Amritanshu Prasad), Discrete and Continuous Dynamical Systems - Series S, volume 2, number 2, pages 337-348, June 2009.
12. Deviation of ergodic averages for rational polygonal billiards (joint with Giovanni Forni), Duke Mathematical Journal, volume 144, number 2, pages 285-319, August 2008.
13. Quantitative recurrence and large deviations for Teichmuller geodsic flow, Geometriae Dedicata, volume 119, number 1, pages 121-140, April 2006
14. On the asymptotics of discrete order statistics, with S. Sethuraman, Statistics and Probability Letters, volume 54, number 3, pages 243-249, October 2001
15. Number theory, balls in boxes, and the asymptotic uniqueness of maximal discrete order statistics, with L. Fidkowski, INTEGERS: Electronic Journal of Combinatorial Number Theory, volume 0, 2000.

### Preprints

1. Right-angled billiards and volumes of moduli spaces of quadratic differentials on $$\mathbb{C}P^1$$, joint with Alex Eskin and Anton Zorich.
2. Counting generalized Jenkins-Strebel differentials, joint with Alex Eskin and Anton Zorich.
3. Logarithm laws for strong unstable foliations in negative curvature and non-Archimedian Diophantine approximation, joint with Frederic Paulin.

### Conferences

Jon Chaika, Joe Rosenblatt, and I are organizing a special session on Multidimensional Dynamical Systems, at the American Mathematical Society Central Section Meeting, April 27-28, 2012, in Ames, Iowa. Details can be found here.

Chris Leininger, Steve Bradlow, and I were local organizers for the first GEometric structures And Representation varieties (GEAR) Network Retreat, August 6-10, 2012, in Champaign-Urbana. Here is a Champaign-Urbana restaurant guide I made for the conference.

Yitwah Cheung, Anton Zorich, and I organized a special session on Lie group actions, Teichmüller Flows and Number Theory, at the American Mathematical Society Western Section Meeting, April 25-26, 2009, in San Francisco. Details can be found here.

Anish Ghosh, David Ellwood, Dmitry Kleinbock, and I organized a workshop on Shrinking Target Properties, from Jan 31-Feb 3, 2008 at Brandeis University and the Clay Mathematics Institute. More details can be found here.

### Lion Hunting

More methods of lion-hunting that every graduate student should know. This is joint with Apoorva Khare (Stanford University).

### The Springer GTM Test

 If I were a Springer-Verlag Graduate Text in Mathematics, I would be J.L. Doob's Measure Theory.I am different from other books on measure theory in that I accept probability theory as an essential part of measure theory. This means that many examples are taken from probability; that probabilistic concepts such as independence, Markov processes, and conditional expectations are integrated into me rather than being relegated to an appendix; that more attention is paid to the role of algebras than is customary; and that the metric defining the distance between sets as the measure of their symmetric difference is exploited more than is customary. Which Springer GTM would you be? The Springer GTM Test

Thanks to Radhika Govindrajan for taking the picture above.