Click here for my front page and contact information. Last updated: September 2024.

About me:

I am a Professor in the Mathematics Department of the Ohio State University, where I am a member of the Ergodic theory group. I joined Ohio State in 2012 (Assistant Professor 2012-2017, Associate Professor 2017-2020, Professor 2020-). From 2009-2012, I was Chowla Research Assistant Professor at Penn State, where I was a member of the Penn State Center for Dynamics and Geometry.  I completed my PhD in Mathematics at the University of Warwick in 2009, supervised by Professors Mark Pollicott and Peter Walters. My research area is ergodic theory and dynamical systems. I am the organiser of the Ohio State ergodic theory seminar. My research is currently supported by the National Science Foundation.

I currently advise two PhD students - Katelynn Huneycutt and Austin Allen. Previous advisees who currently hold academic positions include Tianyu Wang, who is currently an Assistant Professor at SJTU in Shanghai, Ben Call who is currently an NSF postdoc at University of Illinois Chicago, and Caleb Dilsavor, who is currently a postdoc at Northwestern.

In my professional activities besides research, I have worked, and continue to work on, various initiatives to further Diversity and Inclusion in my department. I am a Senator of the Ohio State University Senate from 2021-2024, reelected for 2024-2026.

I am an Executive Editor of the journal Ergodic Theory and Dynamical Systems.

I play double bass and bass guitar. Some videos are at the bottom of the page.


Research Interests:

My main research area is ergodic theory and dynamical systems, particularly the 'entropy theory' of this subject. I am particularly interested in connections between dynamical systems and other areas of mathematics. Recently, my work has particularly focused on connections between dynamical systems and geometry. I have also worked on the implementation of ideas related to my pure research in the applied setting of Bioinformatics.


Current Grants:

PI, NSF Standard Grant, DMS-2349915, 'Advances in Ergodic Geometry', 2024-2027.

co-PI, NSF conference grant DMS - 2230827 'Midwest Dynamical Systems Conferences 2022/2023' with Roland Roeder (PI) and Osama Khalil. The website for the Midwest Dynamics conference series is here


Preprints: (Available at my arXiv page here)

Gibbs Measures for geodesic flows on CAT(-1) spaces. Joint with Caleb Dilsavor
arXiv version here, 46 pages.

Some other new work has been announced in talk abstracts, click here.


Exposition:

Three short articles written for the Early Career section of the Notices of the American Mathematical Society can be found here. My most recent one is on a productivity theme and is titled Do Mathematics Every Day.

I gave a series of four lectures for graduate students entering the field in Trieste, Italy, July 2024. The first video is here. The rest of the videos, notes, problems, etc, can be found here.

A series of three lectures on the Katok Entropy Rigidity Theorem for negative curvature surfaces (taught as part of my course Math 7222 Ergodic Theory II, OSU, Spring 2021): Lecture 1, Lecture 2, Lecture 3, Notes here.

Notes from mini-course at Beyond Uniform Hyperbolicity 2019: Beyond Bowen's Specification Property (II). Videos of the lectures are published by CIRM here.


Published Papers and Papers Accepted for Publication:

1. 'Equilibrium states for self-products of flows and the mixing properties of rank 1 geodesic flows'
       Joint work with Benjamin Call
       arXiv version here, 26 pages. Published in Journal of the London Mathematical Society, 2022, Vol. 105(2), p.797- 824.

2. 'Fluctuations of time averages around closed geodesics in non-positive curvature'
       Joint work with Tianyu Wang
       arXiv version here, 29 pages. Published in 'Communications in Mathematical Physics', 2021, Vol. 385(2), p.1212-1243.

3. 'Measures of maximal entropy on subsystems of topological suspension semi-flows'
       Joint work with Tamara Kucherenko
       arXiv version here, 10 pages. Published in 'Studia Mathematica', 2021, Vol. 260, p.229-240.

4. 'Beyond Bowen's Specification Property'
       Survey paper joint with Vaughn Climenhaga
       arXiv version here, 72 pages, To appear in Springer Lecture Notes in Mathematics Series, CIRM Jean-Morlet Chair Subseries (Fall 2019 Semester)

5. 'Strong symbolic dynamics for geodesic flow on CAT(-1) spaces and other metric Anosov flows'
       Joint work with Dave Constantine and Jean-Francois Lafont
       arXiv version here. Published in 'Journal de l'École polytechnique - Mathématiques (JEP)', 2020, Vol. 7, p.201-231.

6. 'Measures of maximal entropy for suspension flows over the full shift'
       Joint work with Tamara Kucherenko
       arXiv version here. Published in 'Mathematische Zeitschrift', 2020, Vol. 294, p.769-781.

7. 'Unique equilibrium states for geodesic flows in nonpositive curvature'
       Joint work with Keith Burns, Vaughn Climenhaga and Todd Fisher
       Arxiv version here. Published in 'GAFA', 2018, Vol. 28 issue 5, 1209-1259. Correction to a proof detail in section 8 is here

8. 'The weak specification property for geodesic flows on CAT(-1) spaces'
       Joint work with Dave Constantine and Jean-Francois Lafont
       arXiv version here. Published in 'Groups, Geometry and Dynamics', 2020, Vol. 14 issue 1, p.297-336.

9. 'Fat flats in rank one manifolds'
       Joint work with Dave Constantine, Jean-Francois Lafont and Ben McReynolds
       Arxiv version here. Published in 'Michigan Mathematical Journal', 2019, Vol. 68 issue 2, p.251-275.

10. 'Unique equilibrium states for Bonatti-Viana diffeomorphisms'
       Joint work with Vaughn Climenhaga and Todd Fisher
       Arxiv version here. Published in 'Nonlinearity', 2018, Vol. 31 issue 6, p,2532-2577.

11. 'Generalized beta-transformations and the entropy of unimodal maps'
       Arxiv version here. Published in 'Commentarii Mathematici Helvetici', 2017, Vol. 92 issue 4, p.777-800.

12. 'A suspension flow over the full shift with two distinct measures of maximal entropy'
       Joint work with Tamara Kucherenko
       Published in 'Topology Proceedings', 2018, Vol. 52, p.321-328.

13. 'Unique equilibrium states for Mane diffeomorphisms'
       Joint work with Vaughn Climenhaga and Todd Fisher
       Arxiv version here. Published in 'Ergodic Theory and Dynamical Systems', 2019, Vol. 39 issue 9, p.2433-2455.

14. 'Unique equilibrium states for flows and homeomorphisms with non-uniform structure'
       Joint work with Vaughn Climenhaga, Arxiv version here
       Published in 'Advances in Mathematics', 2016, Vol. 303, p.745-799.

15. 'Large deviations for systems with non-uniform structure'
       Joint work with Vaughn Climenhaga and Kenichiro Yamamoto, Arxiv version here
       Published in 'Transactions of the American Mathematical Society', 2017, Vol 369, p.4167-4192. Small correction needed.

16. 'Coding sequence density estimation via topological pressure'
       Joint work with David Koslicki, Arxiv version here
       Published in the 'Jounal of Mathematical Biology', 2015, Vol.70 issue 1-2, p.45-69

17. 'Intrinsic ergodicity via obstruction entropies'
       Joint work with Vaughn Climenhaga, Arxiv version here
       Published in 'Ergodic Theory and Dynamical Systems', 2014 Vol 34 issue 6, p.1816-1831

18. 'Equilibrium states beyond specification and the Bowen property'
       Joint work with Vaughn Climenhaga, Arxiv version here
       Published in 'Journal of the London Mathematical Society', 2013, Vol. 87 part 2, p.401-427.

19. 'Intrinsic ergodicity beyond specification : beta-shifts, S-gap shifts, and their factors'
       Joint work with Vaughn Climenhaga, Arxiv version here
       Published in the 'Israel Journal of Mathematics'  2012, Vol.192 part 2, 785-817. Small correction needed.

20. 'Irregular sets, the beta-transformation and the almost specification property'
       Published in 'Transactions of the American Mathematical Society', 2012, Vol. 364 no. 10, p. 5395-5414.
       Arxiv version here

21. 'A criterion for topological entropy to decrease under normalised Ricci flow'
       Published in 'Discrete and continuous Dynamical Systems, series A', 30(4), 2011.
       Arxiv version here

22. 'The irregular set for maps with the specification property has full topological pressure' 
        Published in 'Dynamical Systems : An International Journal', 25(1), 2010, p.25-51.
        Arxiv version here

23. 'A variational principle for topological pressure for certain non-compact sets' 
        Published in 'Journal of the London Mathematical Society', December 2009, Volume 80, part 3, p585-602.
        Arxiv version here

24.  'A thermodynamic definition of topological pressure for non-compact sets'
        Published in 'Ergodic Theory and Dynamical Systems', 31(2) 2011, pp. 527-547
        Arxiv version here

0.    PhD Thesis, Warwick University, 2009, Pdf.


Upcoming Talks and travels: 

Talks, etc:

Here are talks and travels from the last couple of years. For my all-time list, and some selected abstracts of the talks, click here


Conferences, Workshops, Seminars Organized:

Conference in honor of Todd Fisher, Brigham Young University, June 11-13 2024, with Keith Burns, Amie Wilkinson, Davi Obata, Ben Webb and Lennard Bakker

Workshop at BIRS-Okanagan (with Keith Burns, Tamara Kucherenko and Agniezka Zelerowicz) on Thermodynamic Formalism and Geometry, Summer 2023.

Organizer of the Ohio State Ergodic Theory and Probability Seminar, 2012 - 2022

Workshop on best practices in Graduate Advising at Ohio State, June 2019, with Angelica Cueto, David Penneys and Krystal Taylor.

AIM workshop on 'Equilibrium states for dynamical systems arising from geometry', July 2019, with Keith Burns, Vaughn Climenhaga and Todd Fisher.

Workshop on dynamical systems Thermodynamic Formalism: Ergodic Theory and Geometry at Warwick University celebrating the 60th birthday of Mark Pollicott, July 2019, with Oliver Jenkinson, Thomas Jordan and Richard Sharp.

Organiser/mentor (with Jon Chaika, Vaughn Climenhaga, Boris Hasselblatt and Bryna Kra) for an AMS Mathematics Research Community

Workshop on Thermodynamic Formalism, ICMS, Edinburgh (Scotland), June 2018. The conference website is here. With Thomas Jordan, Omri Sarig and Mike Todd.

Midwest Dynamical Systems Meeting at Ohio State, October 2015. The conference website is here. An archive of past MWDS conferences is here.

Ohio State Young Scholars Program 2013 'Building Mathematical Capacity Summer Experience' summer camp for rising 10th graders, Mathematics director (with my colleagues John Johnson and Adriana Dawes)

Organizer of the Ohio State Math Department Colloquium, 2013-2014; Co-organiser 2016-2017

AMS special session on Dynamical Systems:Thermodynamic formalism and connections with geometry, Western sectional meeting in Boulder, Colorado, April 2013. With Keith Burns.

Organizer of the Penn State Center for Dynamics and Geometry seminar, 2009 - 2011

Co-organiser (with Mark Pollicott) of Dynamical Systems seminar, University of Warwick, 2008-2009

Organizer of workshop Chaotic Properties of Dynamical Systems, University of Warwick, August 2007. With Thomas Jordan.



Past Awards and Grants:

PI, NSF grant DMS-1954463, 'New Directions in Thermodynamic Formalism for Geodesic Flows Beyond the Closed Riemannian Case', 2020- 2024.

co-PI, NSF conference grant DMS - 1856176 'Midwest Dynamical Systems Conferences 2019/2020' with Laura DeMarco (PI) and Roland Roeder.

PI, NSF CAREER award DMS-1461163, 'Entropy in dynamics: connections with geometry, algebraic numbers, and bioscience', 2015-2021.

I was a participant in an AIM SQuaRE with Keith Burns, Vaughn Climenhaga and Todd Fisher from 2013-2016. The project was entitled "Equilibrium states for geodesic flows in nonpositive curvature".

2011-2015, NSF "standard" grant

August 2007, ASSEC Conference Competition, Winner (with Thomas Jordan) of University of Warwick ASSEC (American Study and Student Exchange Committee) Conference Competition to organise conference 'Chaotic Properties of Dynamical Systems', Warwick Mathematics Institute.

August 2007, Awarded  London Mathematical Society Scheme 1 Conference Grant to support the conference 'Chaotic Properties of Dynamical Systems'



Youtube videos I play bass on:

Quacktronics, Dick's Den, Columbus, June 2024, Quacktronics Play the Beatles full show.

Molly Winters Duo, Comfest, June 2024, Full show.

The Deeptones, Columbus Jazz and Ribs Festival 2022, Do Your Thing.

The Deeptones, Comfest 2022, The Letter (Facebook video).

Clayton Denwood, from studio album Tryin' to Resist 2009, Nothing at all and Tryin' to Resist.

Adam Philips Trio, Live from the Big Pink 2007, Bullet in the Head.