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Tushar Das

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 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.

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 Finite-dimensional 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 (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

Resarch Monographs Published:

Resarch Articles Published and Forthcoming:

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 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):