Vlassis Mastrantonis | University of Maryland

About

Welcome to my webpage! I am a Ph.D. candidate at the University of Maryland, College Park, advised by Yanir A. Rubinstein. I am currently in my sixth year, and plan to graduate Spring 2025. I am interested in geometric analysis, convex geometry, several complex variables, and geometric flows.

I received my undergraduate degree from the University of Athens in 2017, and completed the Part III of the Mathematical Tripos at Cambridge University in 2018.

Books

An introduction to convex, complex, and differential geoemtry via the Mahler conjectures (with Y.A. Rubinstein),

solicited by the Amer. Math. Soc. (manuscript in preparation)

Publications and Preprints

\(L^p\)-Legendre transforms and Mahler integrals\(:\) Asymptotics and the Fokker-Planck heat flow

in preparation, 2024

Two-dimensional Błocki, \(L^p\)-Mahler, and Bourgain conjectures (with Y.A. Rubinstein),

preprint, 2024, arxiv: 2401:10992, to appear in Indiana Univ. Math. J.

A Santaló inquality for the \(L^p\)-polar body

preprint, 2024, arxiv: 2401:10836, to appear in Contemp. Math., AMS.

\(L^p\)-polarity, Mahler volumes, and the isotropic constant (with B. Berndtsson and Y.A. Rubinstein),

Analysis & PDE 17-6 (2024), 2179-2245.

The Nazarov proof of the non-symmetric Bourgain-Milman inequality (with Y.A. Rubinstein),

Indiana Univ. Math. J. 73 (2024), 911-953.

Relations of the spaces \(A^p(\Omega)\) and \(C^p(\partial\Omega)\) (with N. Georgakopoulos and V. Nestoridis),

Results Math. 73, 86 (2018).

Nowhere differentiable functions of analytic type on products of finitely connected planar domains (with C. Panagiotis),

Monatsh. Math. 187 (2018), 327-341.

Invited Conference Talks

Midwestern Workshop on Asymptotic Analysis, Indiana University Bloomington, 11-13 October 2024.

Gothenburg Complex Geometry Conference, Gothenburg, Sweeden, 26-30 August 2024.

Recent Developments in Geometric Analysis, AMS Spring Eastern Sectional Meeting, Howard University, 6-7 April 2024.

Convex and Complex: Perspectives on Positivity in Geometry, Cetraro, Italy, October 2022.

(You can watch the talk here)

Invited Seminar Talks

Analysis Seminar SAMPS, Athens, Greece, January 13, 2025.

Measure Theory Seminar, Kent State University, March 6, 2024.

Outreach

Over the past three years, I have mentored five undergraduate projects for the Directed Reading Program at the University of Maryland:

Alejandro Escoto, Geometric Flows in Python, Fall 2023
Brooke Guo, The isoperimetric inequality from antiquity to Steiner, Spring 2023
Alejandro Escoto, The de Rham cohomology, Fall 2022
Abdulrahman Alenazi, The 2-dimensional Mahler conjecture, Spring 2022
Aryan Kaul, The 2-dimensional Brouwer fixed point theorem, Spring 2021

I also gave a talk at the Interaction between Convex Geometry and Complex Geometry REU at the University of Maryland on July 1, 2024.

Code

magicMahler is a Python library designed for geometric computations involving polytopes in the plane. Key features include calculating essential notions from my research:

\(L^p\)-Mahler volumes
The isotropic constant
The B-constant
Implementation of the Mahler algorithm
Implementation of the Graham scan for finding the extreme points in \(nlogn\) time.

Here is an example of a Graham scan implementation:

Graham Scan