Tushar Das
Specialty area(s)
Dynamical systems, topology, 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.
My 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.] My current research involves generalizing various aspects of this theory to different scenarios where there is a presence of negative curvature. For a few lowdimensional 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.
I also work in areas of number theory that share boundaries with dynamical systems, in particular the theory of Diophantine approximation [who was Diophantus?] that involve 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.
Finally, I 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
Teaching history
UWL (2013):
 MTH 150 College Algebra
 MTH 151 Precalculus
 MTH 207 Calculus I
 MTH 309 Linear Algebra and Differential Equations
 MTH 395 Hyperbolic Geometry and Complex Analysis
 MTH 395 Further Linear Algebra
 MTH 395 Finitedimensional vector spaces
 MTH 407 Real Analysis I
 MTH 408 Real Analysis II
 MTH 411 Abstract Algebra
 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 Representation Theory in Quantum Physics
 MTH 498 Lie Theory I Geometry & Representations
 MTH 498 Algebraic Numbers and Diophantine Approximation
Oregon State (20122013):
 Multivariable Calculus I
 Multivariable Calculus II
North Texas (20062012):
 Calculus I
 Calculus II
 Linear Algebra and Vector Geometry
 Multivariable Calculus
Professional history
Associate Professor, UWLa Crosse, USA (2017present).
Assistant Professor, UWLa Crosse, USA (20132017).
Postdoctoral Scholar, Oregon State University, USA (20122013).
Research and publishing
Resarch Monographs Published:
 Geometry and dynamics in Gromov hyperbolic metric spaces (with D. Simmons and M. Urbanski), AMS Mathematical Surveys and Monographs, Volume 218.
Resarch Articles Published and Forthcoming:
 Intersecting limit sets of Kleinian subgroups and Susskind's question (with D. Simmons). To appear in Proceedings of the American Mathematical Society
 A proof of the matrix version of Baker's conjecture in Diophantine approximation (with D. Simmons). To appear in Mathematical Proceedings of the Cambridge Philosophical Society.
 Badly approximable points on selfaffine sponges and the lower Assouad dimension (with L. Fishman, D. Simmons and M. Urbanski). To appear in Ergodic Theory and Dynamical Systems.
 Extremality and dynamically defined measures, part I: Diophantine properties of quasidecaying 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 437467.
 The Hausdorff and dynamical dimensions of selfaffine 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 835846.
 Dimension rigidity in conformal structures (with D. Simmons and M. Urbanski), Advances in Mathematics, Volume 308, 21 February 2017, pp 11271186.
 Tukia's isomorphism theorem in spaces CAT(1) (with D. Simmons and M. Urbanski), Ann. Acad. Sci. Fenn. Math., Vol. 41, 2016, pp 659680.
 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 537567.
Invited Research Lectures (selected):
 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 Fransisco 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 PoincareBowen rigidity in conformal dynamics, University of Chicago, Dynamics Seminar, January 2014.
 Dynamics of discrete isometric actions on infinitedimensional 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 HigherDimensional 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 JeanFrançois Quint. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, SpringerVerlag (2016), 323 pages, ISBN: 9783319477190. MAA Book Reviews, November 2017.

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.

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.

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.

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.

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.

Fractal Geometry: Mathematical Foundations and Applications, 3rd ed., by K. Falconer. Wiley (2014), 368 pages, ISBN: 9781119942399. MAA Book Reviews, June 2014.