Metric spaces -- a set together with a distance that satisfies the triangle inequality.

• Lipschitz analysis: Differentiability of Lipschitz maps, area formula, co-area inequality

• Analysis on metric spaces: study of Sobolev functions and Poincare inequalities on doubling spaces

• Geometric Function Theory: quasisymmetric and quasiconformal mappings/parameterizations, conformal dimensions

• Uniformization: Given a space, is there a biLipschitz/quasi-symmetric/quasiconformal/conformal map from it to a better understood (familiar) space, e.g. a Euclidean space or a sphere?

Publication:

1. E. B., Mishra Riddhi: On removable sets for weighted Sobolev functions (Submitted; arXiv)

2. E. B., Rajala Kai:  Conformal uniformization of domains bounded by quasitripods (accepted to Duke Mathematical Journal; arXiv)

3. E. B., Schardine Ryan, and Shanmugalingam Nageswari: On Carrasco Piaggio's theorem connecting combinatorial modulus and Ahlfors regular conformal dimension (Accepted for publication; arXiv:2412.05549)

4. E. B., Ikonen Toni, Rajala Kai:  Coarea inequality for monotone functions on metric surfaces Trans. Amer. Math. Soc. 376 (2023), 7377-7406

5. E. B., Hajlasz Piotr: Lipschitz mappings, metric differentiability, and factorization through metric trees, J. Lond. Math. Soc. (2) 106 (2022), no. 3, 2663– 2695

6. E. B., Hajlasz Piotr: Coarea Inequality, Ann. Fenn. Math. 46 (2021), no. 2, 965–991

7. Geometric Function Theory in Metric Spaces, PhD Thesis, 2021 (ProQuest LLC, Ann Arbor, MI, 2021, 112 pp.ISBN: 979-8762-12188-0, PDF here: http://d-scholarship.pitt.edu/41368/)

Work in progress:

1. E. B., Koskela Pekka, Nguyen Khanh: Necessity of sharp Poincare inequality for regularity of quasiconformal maps

2. E. B., Hajlasz Piotr: Metric area and coarea formula

Conference Attendance:

1. Fourteenth Ohio River Analysis Meeting (ORAM 14), March 29-30, 2025, University of Cincinnati, OH

2. AMS Sectional Meeting in Albany, NY, Oct 19-20, 2024. I gave a 20-min talk.

3. Buckeye Metric Geometry Workshop. August 16-18, 2024, Ohio State University, Columbus, OH

4. Thirteenth Ohio River Analysis Meeting (ORAM 13), March 16-17, 2024, University of Kentucky in Lexington, KY

5. Analysis and PDEs on Fractals: A Conference in Memory of Professor Ka-Sing Lau, October 27-29, 2023, University of Pittsburgh

6. Potential Theory and Random Walks in Metric Spaces, Okinawa Institute of Science and Technology, Okinawa, Japan. May 30-June 2, 2023

7. Metric Analysis, Parametrizations, and Surfaces (MAPS), University of Fribourg, Switzerland, 13-15, April 2023

8. Smooth Functions on Rough Spaces and Fractals, with Connections to Curvature Functional Inequalities, Banff International Research Station (hybrid), Nov 20-25, 2022

9. Special Session on Quasiconformal Analysis and Geometry on Metric Spaces at the spring 2023 AMS Eastern Sectional Meeting, April 2023

10. Agent Forum 2021, Aalto University (Finland), Oct 27-29

11. Mathematical Research Community, MRC: Analysis on Metric Spaces, (online), Feb 10, 2020

12. Mid Atlantic Analysis Meeting (MAAM2), Online, October 16-18, 2020

13. The 3rd Annual Northeastern Analysis Meeting, the State University of New York at New Paltz, October 19-21, 2018

14. AMS Sectional Meeting, Northeastern University (Boston) April 21-22, 2018

15. Analysis on Metric Spaces, Department of Mathematics, University of Pittsburgh, March 10-11, 2017

16. Second Annual Temple Graduate Student Conference in Algebra, Geometry, and Topology, Temple University, Philadelphia, May 13-15, 2016

Talks and Presentations:

1. I gave two virtual talks at COMPASS SEMINAR SERIES:
   Sobolev Functions on Weighted R^n,  4th June, 2025 (YouTube link, Slides)
   Removable Sets for Sobolev Functions on Weighted R^n, 11th June, 2025 (YouTube Link) 

2. Analysis Seminar, University of Cincinnati, Feb 17, 2025, On motivations for measures on the Euclidean space other than the Lebesgue.

3. Analysis Seminar, University of Cincinnati Nov 2024, Surjective Lipschitz maps with low-rank derivative everywhere. 

4. AMS Sectional Meeting in Albany, NY, Oct 19-20, 2024. I gave a 20-min talk.

5. Bowling Green State University on Friday Oct 11, 2024:. Colloquium talk. 

6. Buckeye Metric Geometry Workshop. August 16-18, 2024, Ohio State University, Columbus, OH, Surjective Lipschitz Maps with Singular Derivative

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

8. 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

9. 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

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

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

12. University of Tennessee, Knoxville, Analysis Seminar, November 2021,  Spring Central Sectional Meeting (formerly at the University of Cincinnati), online, April 22, 2021, the co-area inequality

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

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

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

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

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

18. 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