# Tushar Das

Associate Professor

Mathematics And Statistics

1016 Cowley Hall

608.785.8392

tdas@uwlax.edu

## Specialty area(s)

Dynamical systems, topology, 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 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].

Finally, I have interests in the history of mathematics, both in itself and also as part of the broader history of culture and ideas.

## Teaching history

**UWL**:

- MTH 151, MTH 411, MTH 495 Calculus on Manifolds II, Spring 2018.
- MTH 151, MTH 309, MTH 495 Calculus on Manifolds I, Fall 2017.
- MTH 150, MTH 309, MTH 495 Algebraic Topology, Spring 2017.
- MTH 151, MTH 309, MTH 495 Topology, Fall 2016.
- MTH 151, MTH 207, MTH 495 Algebraic Topology, MTH 495 Differential Geometry, MTH 495 Further Linear Algebra, MTH 495 Geometric Measure Theory and Fractal Geometry, Spring 2016.
- MTH 151, MTH 407, MTH 495 Topology, Fall 2015.
- MTH 151, MTH 408 Real Analysis II, and MTH 495 Honors Complex Analysis, Spring 2015.
- MTH 151, MTH 407 Real Analysis, MTH 395 Hyperbolic Geometry and Complex Analysis, and MTH 395 Further Linear Algebra, Fall 2014.
- MTH 207 Calculus I, Spring 2014.
- MTH 309 Linear Algebra, and MTH 151 Precalculus, Fall 2013.

**Oregon State**:

- Multivariable Calculus I, Winter 2013.
- Multivariable Calculus I and II, Fall 2012.

**North Texas**:

- Multivariable Calculus I, Fall 2010.
- Linear Algebra and Vector Geometry, Spring 2010.
- Calculus II, Spring 2009.
- Calculus I, Spring 2011, Fall 2009, Summer 2009, Fall 2008, Spring 2008, Fall 2007.

## 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 and Under Review:**

*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 Under Review**:

- 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, in press.
- The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result (with D. Simmons), Inventiones Mathematicae, in press.
- A proof of the matrix version of Baker's conjecture in Diophantine approximation (with D. Simmons), submitted and under review.
- Dimension rigidity in conformal structures (with D. Simmons and M. Urbanski), Advances in Mathematics, Vol. 308 (2017), 1127-1186. ArXiV preprint (2016)
*Badly approximable points on self-affine sponges and the lower Assouad dimension*(with L. Fishman, D. Simmons and M. Urbanski), Ergodic Theory and Dynamical Systems, in press.- Badly approximable vectors and fractals defined by conformal dynamical systems (with L. Fishman, D. Simmons and M. Urbanski), Mathematical Research Letters, in press.
- Extremality and dynamically defined measures, part II: Measures from conformal dynamical systems (with L. Fishman, D. Simmons and M. Urbanski), submitted and under review.
- Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures (with L. Fishman, D. Simmons and M. Urbanski), Selecta Mathematica New Series, in press.
- Tukia's isomorphism theorem in spaces CAT(-1) (with D. Simmons and M. Urbanski), Ann. Acad. Sci. Fenn. Math., Vol. 41, 2016, 659-680.
- 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, 537-567.

**Invited research talks (selected)**:

*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.

## Education

Ph.D. in Mathematics, University of North Texas, USA (2012).

M.S. in Mathematics, University of North Texas, USA (2007).

B.Sc.(Hons) in Mathematics, University of St. Andrews, Scotland, UK (2005).