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Tushar Das

Professor
Mathematics & Statistics University of Wisconsin-La Crosse

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Tushar Das

Professor

Mathematics & Statistics

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.

~~~

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:

Research Articles Published and Forthcoming:

Invited Research Lectures (selected):

Book Reviews (selected):

 

Kudos

presented

Tushar Das, Mathematics & Statistics, presented "Thermodynamic expansions for the Hausdorff dimension of continued fraction Cantor sets via transfer operator perturbation" at the Online Seminar in Diophantine Approximation and Related Topics on Tuesday, March 12, online.

Submitted on: Mar. 12

published

Tushar Das, Mathematics & Statistics, authored the article "A variational principle in the parametric geometry of numbers" in Advances in Mathematics Volume 437, February 2024, no. 109435 published on Monday, Feb. 19 by Elsevier. In joint work with Fishman (UNT), Simmons (York) and Urbanski (UNT) we extend the parametric geometry of numbers to Diophantine approximation for systems of m linear forms in n variables, and establish a new connection to the metric theory via a variational principle that computes fractal dimensions of a variety of sets of number-theoretic interest, e.g. of the set of singular systems of linear forms, thus resolving a conjectures of Kadyrov, Kleinbock, Lindenstrauss and Margulis. As a corollary, the divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs have equal Hausdorff and packing dimensions. Other applications include exact dimension formulas with respect to the uniform exponent of irrationality for simultaneous and dual approximation in two dimensions, completing a cornucopia of partial results from 1977-2016. TD's research was supported in part by the American Institute of Mathematics SQuaREs program (2016-2019), and a 2017-2018 UWL Faculty Research Grant.

Submitted on: Feb. 19

presented

Tushar Das, Mathematics & Statistics, presented "On the mensuration of conformal fractals" at Conference on Functional Analysis and Fractals (CFAF 2024) IIIT Allahabad in India on Feb. 16 online. A plenary talk delivered around midnight (a personal best, so far) in La Crosse and around 11 a.m. in India, which presented research highlights and open problems in the dimension theory of conformal dynamical systems. The research exposed was supported in part by grants from the Institute of Advanced Study and the American Institute of Mathematics.

Submitted on: Feb. 19

presented

Tushar Das, Mathematics & Statistics, presented "A few of my favorite fractal things (thanks John Coltrane)" at Spring 2024 Big Ideas in Dynamics Virtual Learning Conference on Jan. 26 online. Big Ideas in Dynamics is a vibrant research community, sponsored by the American Institute of Mathematics and the National Science Foundation and organized by Noelle Sawyer (Southwestern) and Ben Call (UIC), which is focused on collaborative virtual learning among graduate students and postdoctoral researchers in dynamical systems across the globe.

Submitted on: Jan. 28

presented

Tushar Das, Mathematics & Statistics, presented "Exact dimensions of the prime continued fraction Cantor set" at AMS (American Mathematical Society) Special Session on Ergodic Theory, Symbolic Dynamics, and Related Topics at the 2024 Joint Mathematics Meetings on Jan. 4 in San Fransisco, CA. This research was supported in part by the American Institute of Mathematics (AIM) SQuaREs program and an AMS travel grant.

Submitted on: Jan. 9

presented

Tushar Das, Mathematics & Statistics, presented "Old and new in the dimension theory of continued fraction Cantor sets" at the American Institute of Mathematics (AIM) Special Session on Little School Dynamics at the 2024 Joint Mathematics Meetings on Jan. 4 in San Fransisco, CA. This research was supported in part by the AIM SQuaREs program and a travel grant from American Mathematical Society (AMS).

Submitted on: Jan. 9

presented

Tushar Das, Mathematics & Statistics, presented "Playing games on fractals: Dynamical & Diophantine" at UW Madison Mathematics Colloquium on Sept. 8 in Van Vleck Hall.

Submitted on: Sept. 16, 2023

published

Tushar Das, Mathematics & Statistics, authored the article "On the dimension spectra of infinite conformal iterated function systems" in Journal of Fractal Geometry published on Sept. 1 by European Mathematical Society (EMS) Press. This research was supported in part by an American Institute of Mathematics SQuaREs (Structured Quartet Research Ensembles) grant.

Submitted on: Sept. 2, 2022

awarded

Tushar Das, Mathematics & Statistics, spent three weeks of June researching at the Institute for Advanced Study (IAS) in Princeton, NJ, via the Summer Collaborators Program. The team included Das (UW-La Crosse), Dr. Giulio Tiozzo (U Toronto), Dr. Mariusz Urbanski (U North Texas), and Dr. Anna Zdunik (U Warsaw). They made significant progress on the first of their multi-volume monograph in-progress titled "Open Dynamical Systems Avoiding Arbitrary Balls: Statistics, Geometry, and Thermodynamic Formalism."

Submitted on: July 21, 2022

Memberships & affiliations

[{"organization":"Complex Analysis and Operator Theory (Springer Nature Journal)","position":"Editorial Board Member","url":"https://www.springer.com/journal/11785"}]

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