Profile for Tushar Das
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Tushar Das
Professor
Mathematics & Statistics
University of Wisconsin-La Crosse
Tushar Das
Professor
Mathematics & Statistics
Specialty area(s)
Dynamical systems, topology, hyperbolic geometry, fractal geometry, and Diophantine approximation.
Brief biography
I was born and raised in Calcutta/Kolkata, a somewhat large chaotic attractor supported along the east bank of the river Hooghly that feeds into the mouth of the lower Ganges delta in eastern India. I studied mathematics at the University of St. Andrews in Scotland, and after getting my bachelors degree there went on to graduate work at the University of North Texas in Denton. Post Ph.D. I spent a year as a postdoc at Oregon State before joining UWL in the Fall of 2013.
I play a number of active roles aimed at broadening participation and enhancing access and equity throughout UWL. These include being a Math & Stats Department Equity Liaison, co-faculty advisor for Womxn and Minorities in Mathematics (WaMM), co-faculty advisor for UWL Math & Stats Club, member of the Institute for Social Justice Advisory Board, and co-creator and permanent advocate for STEM Community for Anti-Racist Education (STEM-CARE).
I experience great personal joy in challenging my students and inspiring them to achieve farther than their own expectations. As a result, the vast majority of my time is involved with my teaching and intensive mentoring of our burgeoning constellation of graduate school bound majors, many of whom are from underrepresented groups in STEM such as women and first generation students. I am extremely proud that several of my students and mentees continue to be accepted with full funding to excellent graduate programs that include Ohio State, Tufts, UCLA, UC Irvine, UC Davis, UC Santa Cruz, Boston University, U of Oregon, Oregon State, U Mass Amherst, U of Kansas, Iowa State, U of Iowa, U Conn, U of Vermont, U Colorado, Colorado State and Bryn Mawr. Support and recognition from my students led to my 2020 Eagle Teaching Excellence Award.
I believe in Federico Ardila-Mantilla’s axioms, to which I have made a tiny addition:
Axiom 1: Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
Axiom 2: Everyone can have joyful, meaningful, and empowering mathematical experiences.
Axiom 3: Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
Axiom 4: Every student and every teacher deserves to be treated with dignity and respect.
My tiny addition to the last axiom was inspired by a poem from several centuries ago. This version is from Ursula Le Guin's 1997 translation of Laozi's Tao Te Ching.
Good people teach people who aren’t good yet;
the less good are the makings of the good.
Anyone who doesn’t respect a teacher
or cherish a student
may be clever, but has gone astray.
There’s a deep mystery here.
My mathematical research started out with applying ideas from statistical physics (thermodynamic formalism) to study dynamical systems in one real variable that had chaotic attractors called Cantor sets [who was Georg Cantor?]. I went on to study holomorphic dynamics and the beautiful fractals associated with the names of Julia and Mandelbrot, and the related but very different limit sets of Fuchsian and Kleinian groups that tesselate hyperbolic space. [Look for Maurits Escher's Circle Limits to get an idea of how to visualize these in two dimensions.] For a few low-dimensional examples of such spaces: think of the surface of a coral reef, your lungs, or even some kale. Other good examples are trees (graphs with no loops) in neural networks like the web or in your brain. Following my PhD thesis, I worked on generalizing various aspects of this theory to different scenarios where there is a presence of negative curvature, which lead to a monograph published by the American Mathematical Society.
Much of my current research focuses on areas of number theory that share boundaries with dynamical systems and fractal geometry. In particular, the theory of Diophantine approximation [who was Diophantus?], which involves studying complicated irrational numbers that are algebraic (e.g. the square root of two) or transcendental (e.g. pi) through much simpler numbers, namely fractions (e.g. 99/70 or 355/133). You may be surprised that this branch of esoteric pure/theoretical mathematics plays a surprising role in studying the stability of planetary systems [e.g. look up KAM theory]! If so, read Eugene Wigner's essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences, as well as The Usefulness of Useless Knowledge by Abraham Flexner, founder of the Institute for Advanced Study in Princeton, NJ.
I also have interests in the history of mathematics, both in itself and also as part of the broader history of culture and ideas. For a taste, try Jacqueline Stedall's excellent The History of Mathematics: A Very Short Introduction.
Education
Ph.D. in Mathematics, University of North Texas, USA (2012).
– Advisor: Prof. Mariusz Urbanski
– Thesis: Kleinian groups in Hilbert spaces
M.S. in Mathematics, University of North Texas, USA (2007).
– Advisor: Prof. Mariusz Urbanski
– Thesis: Smooth and Hölder classification of Cantor sets on the line
B.Sc.(Hons) in Mathematics, University of St. Andrews, UK (2005).
– Advisor: Prof. Dr. Bernd O. Stratmann
– Thesis: Riemann surfaces and uniformization
Career
Teaching history
UWL (2013-):
- MTH 150 College Algebra
- MTH 151 Precalculus
- MTH 207 Calculus I
- MTH 208 Calculus II
- MTH 309 Linear Algebra
- MTH 310 Calculus III: Multivariable Calculus
- MTH 395 Hyperbolic Geometry and Complex Analysis
- MTH 395 Further Linear Algebra
- MTH 395 Finite-dimensional vector spaces
- MTH 407 Real Analysis I
- MTH 408 Real Analysis II
- MTH 411 Abstract Algebra
- MTH 415 Topology
- MTH 461 Mathematical Physics
- MTH 495 Further Linear Algebra
- MTH 495 Honors Complex Analysis
- MTH 495 Topology
- MTH 495 Differential Geometry
- MTH 495 Algebraic Topology
- MTH 495 Geometric Measure Theory and Fractal Geometry
- MTH 495 Calculus on Manifolds
- MTH 495 Galois Theory
- MTH 495 Representation Theory in Quantum Physics
- MTH 495 p-adic Numbers: Algebra, Analysis, Topology
- MTH 495 Differential Topology
- MTH 498 Lie Theory I Geometry & Representations
- MTH 498 Algebraic Numbers and Diophantine Approximation
- MTH 498 Measure Theory & Integration
- MTH 498 Differential Geometry
Oregon State (2012-2013):
- Multivariable Calculus I
- Multivariable Calculus II
North Texas (2006-2012):
- Calculus I
- Calculus II
- Linear Algebra and Vector Geometry
- Multivariable Calculus
Professional history
Professor, UW-La Crosse, USA (2022-present).
Associate Professor, UW-La Crosse, USA (2017-2022).
Assistant Professor, UW-La Crosse, USA (2013-2017).
Postdoctoral Scholar, Oregon State University, USA (2012-2013).
Research and publishing
I serve as an editor for Complex Analysis and Operator Theory (CAOT), which is a Springer Nature journal devoted to the publication of current research developments in the closely related fields of complex analysis and operator theory as well as in applications to system theory, harmonic analysis, probability, statistics, learning theory, mathematical physics and other related fields. Papers should be submitted via the Online Manuscript Submission, Review and Tracking System of the journal.
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Preprint versions of my research outcomes are available on arXiv via http://arxiv.org/a/das_t_4 and also via Google Scholar or Researchgate.
My ORCID iD is 0000-0002-3158-4972.
Research Monographs Published:
- Geometry and dynamics in Gromov hyperbolic metric spaces (with D. Simmons and M. Urbanski), AMS Mathematical Surveys and Monographs, Volume 218.
Research Articles Published and Forthcoming:
- A variational principle in the parametric geometry of numbers (with L. Fishman, D. Simmons and M. Urbanski). Advances in Mathematics, Volume 437, February 2024, 109435.
- Hausdorff dimensions of perturbations of a conformal iterated function system via thermodynamic formalism (with L. Fishman, D. Simmons and M. Urbanski). Selecta Mathematica, Vol. 29 (2023), No. 2, Paper 19.
- On the dimension spectra of infinite conformal iterated function systems, (with D. Simmons). Journal of Fractal Geometry, Vol. 9 (2022), No. 1, 73-87.
- Thermodynamic Formalism for Coarse Expanding Dynamical Systems (with F. Przytycki, G. Tiozzo , M. Urbanski, A. Zdunik). Communications in Mathematical Physics, Vol. 384, No. 1 (2021), pp. 165–199.
- Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems (with L. Fishman, D. Simmons and M. Urbanski). Ergodic Theory and Dynamical Systems, Vol. 41, No. 8 (2021), pp. 2311–2348.
- Intersecting limit sets of Kleinian subgroups and Susskind's question (with D. Simmons), Proceedings of the American Mathematical Society, Volume 148, Issue 8 August 2020, pp. 3203–3207.
- A proof of the matrix version of Baker's conjecture in Diophantine approximation (with D. Simmons). Mathematical Proceedings of the Cambridge Philosophical Society, Volume 167, Issue 1 July 2019, pp. 159–169.
- Badly approximable points on self-affine sponges and the lower Assouad dimension (with L. Fishman, D. Simmons and M. Urbanski). Ergodic Theory and Dynamical Systems, Volume 39, Issue 3 March 2019 , pp. 638–657.
- Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures (with L. Fishman, D. Simmons and M. Urbanski), Selecta Mathematica, , Volume 24, Issue 3, pp. 2165–2206.
- Badly approximable vectors and fractals defined by conformal dynamical systems (with L. Fishman, D. Simmons and M. Urbanski), Mathematical Research Letters, Volume 25, July 2018, Number 2, pp. 437-467.
- The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result (with D. Simmons), Inventiones Mathematicae, , Volume 210, Issue 1, pp. 85–134.
- A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation (with L. Fishman, D. Simmons and M. Urbanski), C. R. Acad. Sci. Paris, Ser. I, Volume 355, Issue 8, August 2017, pp. 835–846.
- Dimension rigidity in conformal structures (with D. Simmons and M. Urbanski), Advances in Mathematics, Volume 308, 21 February 2017, pp. 1127–1186.
- Tukia's isomorphism theorem in spaces CAT(-1) (with D. Simmons and M. Urbanski), Ann. Acad. Sci. Fenn. Math., Vol. 41, 2016, pp. 659–680.
- Geometry of limit sets of discrete groups acting on real infinite dimensional hyperbolic space (with B. O. Stratmann and M. Urbanski), Stochastics and Dynamics, Vol. 16, No. 5, 2016, 1650018.
- The Geometry of Baire Spaces (with M. Urbanski), Dynamical Systems, Vol. 26, No. 4, 2011, pp. 537–567.
Invited Research Lectures (selected):
- Spring '24 Big Ideas in Dynamics, AIM (American Institute of Mathematics) Research Community for Graduate Students in Dynamics, January 2024.
- Exact dimensions of the prime continued fraction Cantor set, AMS Special Session on Ergodic Theory, Symbolic Dynamics, and Related Topics @ 2024 JMM San Fransisco, January 2024.
- Playing games on fractals: Dynamical & Diophantine, UW Madison Mathematics Colloquium, September 2023.
- Dimension theory of conformal iterated function systems: old and new results and questions, Dynamics of (Semi-)Group Actions, University of Lodz, June 2023.
- Thermodynamic expansions via perturbation theory, and the dimension theory of continued fraction Cantor sets, Advances in Operator Theory with Applications to Mathematical Physics, Chapman University, November 2022.
- Dimensions, Dynamics, Diophantine approximations. Little School Dynamics Colloquium, American Institute of Mathematics, July 2022.
- Revisiting a gallery of classic number-theoretic fractals with fresh eyes, then looking ahead. Seminario Tartaglia de la Grupo Dinámica Potosina, March 2022.
- Dimension theory for infinite-alphabet conformal iterated function system limit sets. Bremen Online Dynamics Seminar (BODS), January 2022.
- Equilibrium states and stochastic laws for metric expanding branched covering maps, Universidade do Porto (CMUP) Zoominar in Dynamical Systems, May 2021.
- Hausdorff Hensley Good & Gauss, One World Numeration Seminar, May 2021.
- Using templates to study problems in dynamics and number theory, New England Dynamics and Number Theory Seminar, April 2021.
- Playing games with lattice successive minima, UMass Lowell Mathematical Sciences Colloquium, April 2021.
- On the inheritance of loss, or the Diophantine extremality of measures, Yale Group Actions and Dynamics Seminar, April 2021.
- From the division algorithm to fractal geometry via dynamical systems, IIT Tirupati Mathematics & Statistics Webinar , March 2021.
- Dimension spectra and thermodynamic expansions of conformal fractal limit sets , University of Maryland Dynamics Seminar, February 2021.
- New results in the dimension theory of continued fraction Cantor sets, Bristol Ergodic Theory and Dynamical Systems Seminars, November 2020.
- A Tale of Two Variational Principles , St. Andrews Analysis Seminar, November 2020.
- Asymptotic expansions for fractal dimensions via thermodynamic formalism, Penn State 2020 Workshop in Dynamical Systems and Related Topics, October 2020.
- Is your favorite dynamically defined measure strongly extremal? Resistência Dinâmica (PUC-Rio, Brazil), October 2020.
- Successive minima of lattice trajectories and topological games to compute fractal dimensions, Ohio State Ergodic Theory Seminar, October 2020.
- The Varieties of Discrete Experience, and Other Tales of Isometric Actions on Gromov Hyperbolic Metric Spaces, NYC noncommutative geometry seminar, August 2020.
- Thermodynamic formalism and the Hausdorff dimension of continued fraction Cantor sets, One World Fractals and Related Fields Seminar, June 2020.
- A variational principle in the parametric geometry of numbers, UC San Diego Group Actions Seminar, January 2020.
- Beyond finite-dimensional rectifiability, AMS Special Session on Recent Developments in Harmonic Analysis, University of Wisconsin-Madison, September 2019.
- Templates and dimension games with Diophantine targets, University of Maryland Dynamics Seminar, March 2019.
- Dimension games, homogeneous dynamics, and metric Diophantine approximation, 2019 Ergodic Theory Workshop, University of North Carolina at Chapel Hill, January 2019.
- The parametric geometry of numbers and metric Diophantine approximation, Millican Colloquium Lecture, University of North Texas, November 2018.
- Intersecting limit sets for Kleinian subgroup pairs, AMS Special Session on Statistical and Geometrical Properties of Dynamical Systems, San Francisco State University, October 2018.
- Singular systems of linear forms with a prescribed uniform irrationality exponent, 2018 Ergodic Theory Workshop, University of North Carolina at Chapel Hill, April 2018.
- Singular systems of linear forms and divergent trajectories on homogeneous spaces, AMS Special Session on Dynamics, Geometry and Number Theory, University of North Texas, September 2017.
- Does every expanding repeller have an ergodic invariant measure of full Hausdorff dimension?, AMS Special Session on Fractal Geometry and Ergodic Theory, University of North Texas, September 2017.
- A variational principle in the parametric geometry of numbers, 2017 Ergodic Theory Workshop, University of North Carolina - Chapel Hill, April 2017.
- Badly approximable vectors in conformal fractals, 2016 Ergodic Theory Workshop, University of North Carolina - Chapel Hill, April 2016.
- Extremality and measures from conformal dynamical systems, AMS Special Session on Fractal Geometry and Dynamical Systems, Joint Mathematics Meetings, Seattle, January 2016.
- Diophantine extremality and dynamically defined measures, 49th Spring Topology and Dynamics Conference, Bowling Green State University, May 2015.
- Dimension rigidity in conformal structures, Yale University, Geometry and Topology Seminar, April 2015.
- Extremal measures: a new approach, with new results, 2014 Ergodic Theory Workshop, University of North Carolina - Chapel Hill, March 2014.
- Avatars of Poincare-Bowen rigidity in conformal dynamics, University of Chicago, Dynamics Seminar, January 2014.
- Dynamics of discrete isometric actions on infinite-dimensional Gromov hyperbolic spaces, 36th Conference on Stochastic Processes and Their Applications, University of Colorado at Boulder, August 2013.
Book Reviews (selected):
- An Introduction to the Theory of Higher-Dimensional Quasiconformal Mappings, by Frederick W. Gehring, Gaven J. Martin, and Bruce P. Palka. Mathematical Surveys and Monographs Volume 216, American Mathematical Society (2017), 430 pages, ISBN: 9780821843604. MAA Book Reviews, January 2018.
- Random Walks on Reductive Groups, by Yves Benoist and Jean-François Quint. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Springer-Verlag (2016), 323 pages, ISBN: 9783319477190. MAA Book Reviews, November 2017.
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Fractals in Probability and Analysis, by Christopher J. Bishop and Yuval Peres. Cambridge Studies in Advanced Mathematics, Vol. 162, Cambridge University Press (2017), 402 pages, ISBN: 9781107134119. MAA Book Reviews, November 2017.
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Fourier Analysis and Hausdorff Dimension, by Pertti Mattila. Cambridge Studies in Advanced Mathematics, Vol. 150, Cambridge University Press (2016), 440 pages, ISBN: 9781107107359. MAA Book Reviews, July 2017.
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Hyperbolic Manifolds: An Introduction in 2 and 3 Dimensions, by Albert Marden. Cambridge University Press (2016), 515 pages, ISBN: 9781107116740. MAA Book Reviews, July 2017.
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Foundations of Ergodic Theory, by Marcelo Viana and Krerley Oliveira. Cambridge Studies in Advanced Mathematics, Vol. 151, Cambridge University Press (2016), 530 pages, ISBN: 9781107126961. MAA Book Reviews, June 2016.
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Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory, by U. Bottazzini and J. J. Gray. Sources and Studies in the History of Mathematics and Physical Sciences. Springer (2013), 848 pages, ISBN: 9781461457244. MAA Book Reviews, July 2015.
- The Improbability Principle, by D. J. Hand. Scientific American / Farrar, Straus and Giroux (2014), 288 pages, ISBN: 9780374175344. The Mathematical Intelligencer, , Volume 37, Issue 1, pp 107–111.
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Fractal Geometry: Mathematical Foundations and Applications, 3rd ed., by K. Falconer. Wiley (2014), 368 pages, ISBN: 9781119942399. MAA Book Reviews, June 2014.
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