Analysis in metric spaces

Geometric Measure Theory

 My research is in analysis and geometry and the interplay between the two -- including in general metric spaces. If I give an example: one can integrate the speed of a curve (analysis) to recover the length of a rectifiable curve (geometry). A theme in my research is to push the boundaries of how non-Euclidean a space can be while still allowing a great deal of analysis and geometry. Apart from countless applications in mathematics and sciences, humans study these objects for their intrinsic beauty. See below for more on my research.

In more detail:

Analysis on Metric Spaces. Already by 70's and 80's a great deal of potential theory was developed in Non-Euclidean settings (think maximal function on doubling spaces). However, in 90's multiple successful approaches lead to the introduction of first order calculus (where one talks about "derivatives") in the abstract setting of metric-measure spaces. Questions I am interested in include Poincare inequalities, analysis on graphs, etc. There is deep connections between geometric properties of a space and the degree of analysis that it supports. Discovering such connections has been an active theme of recent research.

Geometric Measure Theory (GMT), born in 60's, allows to do differential geometry on spaces and maps that for the most part are smooth manifolds -- the singular parts have negligible size (e.g. Hausdorff dimension). Among its early applications was the solution of the Plateau problem: given a boundary, what is the surface with the least area that spans it? GMT has applications to calculus of variations, harmonic analysis, and even computer science. My work in GMT, mostly with Piotr Hajlasz, revolved around the the area and co-area formulas for Lipschitz maps into metric spaces. The notion of metric derivative is central as we used it to prove in [2] a conjecture of David and Schul about when a map factors through a metric tree. In yet unpublished work [4], we prove that (almost all) the level sets of Lipschitz maps into arbitrary metric spaces are rectifiable.

Uniformization. This umbrella term refers to attempts at deforming a given space within a certain class, e.g. biLipschitz, quasi-symmetrically, conformally, etc. to a better understood (simpler) space. In my work with Kai Rajala, we proved that certain domains in the Riemann sphere can be conformally mapped to a circle domain -- a domain whose boundary components are all circles or points.

Research Papers:

Preprint(s):

Published:

In progress...

My PhD thesis (2021) is freely available here: http://d-scholarship.pitt.edu/41368/ 

Conference Participations

Talks and Presentations 

1.   Analysis Seminar, University of Cincinnati 2024, Conformal uniformization of quasitripod domains

2.  Potential Theory and Random Walks in Metric Spaces, Okinawa Institute of Science and Technology, Okinawa, Japan. May 30-June 2, 2023, Poincare Inequality on a Square Pinched on a Product Cantor Set

3.      Special session on Quasiconformal Analysis and Geometry on Metric Spaces at the spring 2023 AMS Eastern Sectional Meeting, April 2023, Coarea Inequality for Monotone Functions on Metric Surfaces

4.      University of Jyväskylä, Analysis Seminar, Oct 2022, Coarea Inequality for Monotone Functions on Metric Surfaces

5.      University of Jyväskylä, Weighted coverings and the Hausdorff measure, Analysis Seminar, Dec 2021

6.      University of Tennessee, Knoxville, Analysis Seminar, November 2021, On equivalent assumptions in the (metric) implicit function theorem

7.      Spring Central Sectional Meeting (formerly at the University of Cincinnati), online, April 22, 2021, the co-area inequality

8.      Mathematical Research Community, MRC: Analysis on Metric Spaces, (online), Feb 10, 2020, area and Co-area formula for maps into metric spaces

9.      Bilkent University, Ankara, Turkey, (online) Feb 10, 2020, area and Co-area formula for maps into metric spaces

10.  International Prague Seminar on Function Spaces (online), Dec 10, 2020, co-area formula for maps into metric spaces

11.  Geometric Analysis (Organized online by: Simon Blatt, Philipp Reiter, Armin Schikorra, Guofang Wang), Nov 17, 2020, co-area formula for maps into metric spaces

12.  Mid Atlantic Analysis meeting (MAAM2), Online, October 16-18, 2020, co-area inequality

13.  The 3rd Annual Northeastern Analysis Meeting, the State University of New York at New Paltz, October 19-21, 2018, Holder extension of maps between Heisenberg groups