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, 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 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 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

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 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

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

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.

Resarch Monographs Published:

Resarch Articles Published and Forthcoming:

Invited Research Lectures (selected):

  • 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 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

designed

Eugenia Turov, Chemistry & Biochemistry; and Tushar Das, Whitney George and Nathan Warnberg, all Mathematics & Statistics; designed an anti-racist education workshop for "the purpose of building a Community of Anti-Racist Educators". In addition, Whitney George and Eugenia Turov co-faciliated the weeklong workshop. The workshop was a resounding success as noted by one participant leaving with 'profound insights about their personal struggles with internalized white-supremacy and how it shows up in their classrooms.'

Submitted on: Aug. 30

 

published

Tushar Das, Mathematics & Statistics, authored the article "Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems" in Ergodic Theory and Dynamical Systems published on June 30 by Cambridge University Press. This article, written in collaboration with L. Fishman (UNT), D. Simmons (York) and M. Urbanski (UNT), follows [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math. 24(3) (2018), 2165–2206]. TD's research on this project was supported in part by a 2014-2015 UWL Faculty Research Grant.

Submitted on: July 10

 

presented

Tushar Das, Mathematics & Statistics, presented "Equilibrium states and stochastic laws for coarse expanding dynamics of metric branched covers" at Centro de Matemática - Universidade do Porto (CMUP) Zoominar in Dynamical Systems on May 21 online.

Submitted on: July 9

 

presented

Tushar Das, Mathematics & Statistics, presented "Hausdorff Hensley Good & Gauss" at One World Numeration Seminar (OWNS) on May 4 online. Das was invited to speak about their recent work (arXiv:1910.10259 and arXiv:2007.10554) on the dimension theory of iterated function systems, open problems and research directions. This work was supported in part by an American Institute of Mathematics SQuaREs grant.

Submitted on: May 8

 

presented

Tushar Das, Mathematics & Statistics, presented "Using templates to study problems in dynamics and number theory" at New England Dynamics and Number Theory Seminar on April 19 online. Das was invited to survey results, techniques and open problems surrounding the variational principle in the parametric geometry of numbers https://arxiv.org/abs/1901.06602 Research supported in part by a 2017-2018 FRG and an American Institute of Mathematics SQuaRE grant.

Submitted on: May 3

 

published

Tushar Das, Mathematics & Statistics, authored the article "Thermodynamic formalism for coarse expanding dynamical systems" in Communications of Mathematical Physics published on April 10 by Springer Nature. This paper, written in collaboration with Feliks Przytycki (Polish Academy of Sciences), Giulio Tiozzo (U Toronto), Mariusz Urbanski (U North Texas) & Anna Zdunik (U Warsaw), introduces and develops the thermodynamic formalism for a novel class of dynamical systems: proving existence and uniqueness of equilibrium states for a wide class of potentials, as well as stochastic laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and large deviations.

Submitted on: April 20

 

presented

Tushar Das, Mathematics & Statistics, presented "Playing games with lattice successive minima" at UMass Lowell Mathematical Sciences Colloquium on Tuesday, April 13 online.

Submitted on: April 13

 

presented

Tushar Das, Mathematics & Statistics, presented "On the inheritance of loss, or the Diophantine extremality of measures" at Yale Group Actions and Dynamics Seminar on Monday, April 5 online. Das was invited to give a virtual research lecture on new results and open questions around the extremality of fractal measures, extending a burgeoning thread in the deep work of G. A. Margulis (2020 Abel Prize winner and Erastus L. DeForest Professor of Mathematics at Yale) and his students and collaborators. Research reported on was in collaboration with L. Fishman, D. Simmons and M. Urbanski, and supported in part by 2014-15 and 2017-18 UWL Faculty Research Grants.

Submitted on: April 5

 

presented

Tushar Das, Mathematics & Statistics, presented "From the division algorithm to fractal geometry via dynamical systems" at Indian Institute of Technology (IIT) Tirupati Mathematics and Statistics Department Webinar on Tuesday, March 30 online. TD was invited to give a virtual research colloquium on new results and open questions around the fertile interface of dynamical systems, fractal geometry, and Diophantine approximation. Research reported on was supported in part by an American Institute of Mathematics SQuaREs grant.

Submitted on: Mar. 30