Profile for Tushar Das

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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, Faculty Advisor for the Womxn and Minorities in Mathematics (WaMM), Faculty Advisor for the UWL Math & Stats Club, and a member of the Institute for Social Justice Advisory Board.

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 16 of my students and mentees have been accepted with full funding to excellent graduate programs that include Madison, Ohio State, Indiana, 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, Colorado State and Bryn Mawr. The love and support I receive from my students led to a 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 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.  

Please get in touch with me about any of what you may have read in this not-so-brief bio. I would love to hear from you!!

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 208 Calculus II
  • MTH 309 Linear Algebra and Differential Equations
  • 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 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

Associate Professor, UW-La Crosse, USA (2017-present).

Assistant Professor, UW-La Crosse, USA (2013-2017).

Postdoctoral Scholar, Oregon State University, USA (2012-2013).

Research and publishing

Preprint versions of my research outcomes are available via http://arxiv.org/a/das_t_4

My ORCID iD is 0000-0002-3158-4972.

Resarch Monographs Published:

Resarch Articles Published and Forthcoming:

Invited Research Lectures (selected):

  • 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 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 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):

 

Kudos

presented

Tushar Das, Mathematics & Statistics, presented "Thermodynamic formalism and the Hausdorff dimension of continued fraction Cantor sets" at One World Fractals and Related Fields Seminar on June 17 in Cyberspace. We described a program that leverages the thermodynamic formalism (originally used to study models of ferromagnetism in statistical mechanics) to unify and extend several results in classical Diophantine approximation, a subfield of number theory in theoretical mathematics. This was a plenary talk originally supposed to be delivered at the Fractals and Related Fields IV (FARF4) conference in France, which has been postponed indefinitely. The talks were broadcast, with between 75-175 online participants at a time, using the open-source web conferencing software BigBlueButton maintained by the CNRS, the French analogue of the NSF.

Submitted on: July 6

named

Tushar Das, Mathematics & Statistics, was named lead collaborator of the 2020 Summer Collaborators program at the Institute for Advanced Study (IAS) in Princeton, NJ on Jan. 15. The faculty of the School of Mathematics at IAS voted to fund the research project "Thermodynamic formalism for rational maps and beyond." The team comprises Tushar Das, UWL; Giulio Tiozzo, University of Toronto; Mariusz Urbanski, University of North Texas and Anna Zdunik, University of Warsaw. Research will being at the IAS in June 2020.

Submitted on: Jan. 31

presented

Tushar Das, Mathematics & Statistics, presented "A variational principle in the parametric geometry of numbers" at the UC San Diego Group Actions Seminar on Jan. 14 in San Diego, CA. The lecture inaugurated a new UCSD Math Dept. seminar series. We described an ongoing research program that resolves several challenges at the interface of Diophantine approximation and homogenous dynamics. Das's research was supported in part by a 2017-18 UWL Faculty Research Grant as well an American Institute of Mathematics SQuaRE grant.

Submitted on: Jan. 15

presented

Tushar Das, Mathematics & Statistics, presented "Beyond finite-dimensional rectifiability" at Special Session on Recent Developments in Harmonic Analysis on Sept. 14, 2019 in University of Wisconsin-Madison, American Mathematical Society Central Sectional Meeting.

Submitted on: Sept. 16, 2019

published

Tushar Das, Mathematics & Statistics, authored the article "A proof of the matrix version of Baker's conjecture in Diophantine approximation " in " Mathematical Proceedings of the Cambridge Philosophical Society" published on July 7, 2019 by Cambridge University Press. The paper resolves a fundamental question regarding Diophantine approximation on a classical algebro-geometric object, viz. the Veronese curve, which resides in the space of all matrices of a given shape. We prove that random points on the Veronese curve are not very well multiplicatively approximable by rational matrices. This research was supported in part by a 2016-17 UWL Faculty Research Grant.

Submitted on: July 8, 2019

presented

Tushar Das, Mathematics & Statistics, presented "Templates and dimension games with Diophantine targets" at University of Maryland Dynamics Seminar on Mar. 7, 2019 in College Park, Maryland, USA. We were invited to deliver two hour-long lectures that described aspects of our recent program to resolve certain conjectures and questions in the fields of homogeneous dynamics, Diophantine approximation and fractal geometry. The reduction of various problems to questions about certain combinatorial objects that we call <> along with a variant of Schmidt's game allows us to answer some of these challenges, while leaving plenty that remain open.

Submitted on: Mar. 31, 2019

published

Tushar Das, Mathematics & Statistics, authored the article "Badly approximable points on self-affine sponges and the lower Assouad dimension" in "Ergodic Theory and Dynamical Systems" published on Mar. 1, 2019 by Cambridge University Press. We leverage a novel connection between Diophantine approximation and the lower Assouad dimension to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. Our results, which are the first to advance beyond the conformal setting, encompass both the case of self-affine Sierpinski sponge and Baranski carpet fractals. This work was partially supported by a 2016-2017 UWL Faculty Research Grant.

Submitted on: Mar. 4, 2019

presented

Tushar Das, Mathematics & Statistics, presented "Dimension games, homogeneous dynamics and metric Diophantine approximation" at the 2019 Ergodic Theory Workshop on Jan. 30, 2019 in University of North Carolina - Chapel Hill.

Submitted on: Jan. 30, 2019

presented

Tushar Das, Mathematics & Statistics, presented "The parametric geometry of numbers and metric Diophantine approximation" at Millican Colloquium Lecture on Nov. 26, 2018 in University of North Texas, Denton, TX. The colloquium lecture was an introduction to a research program we have been developing over the last three years with collaborators at UNT and York, which aims to resolve several conjectures and questions at the interface of number theory, dynamical systems and fractal geometry. The research described was supported in part by a UWL Faculty Research Grant and the American Institute of Mathematics SQuaREs program.

Submitted on: Nov. 26, 2018