Lyuboslav Panchev | Searcharxiv

arXiv2019

On the $v_1$ periodicity of the Moore space

We present progress in trying to verify a long-standing conjecture by Mark Mahowald on the $v_{1}$-periodic component of the classical Adams spectral sequence for a Moore space $M$. The approach we follow was proposed by John Palmieri in his work on the stable category of $A$-comodules. We improve on Palmieri's work by working with the endomorphism ring of $M$ - $End(M)$ thus resolving some of the initial difficulties of his approach and formulating a conjecture of our own that would lead to Mahowald's formulation.

math.AT↗

arXiv2014

On discontinuity of planar optimal transport maps

Consider two bounded domains $Ω$ and $Λ$ in $\mathbb{R}^{2}$, and two sufficiently regular probability measures $μ$ and $ν$ supported on them. By Brenier's theorem, there exists a unique transportation map $T$ satisfying $T_\#μ=ν$ and minimizing the quadratic cost $\int_{\mathbb{R}^{n}}|T(x)-x|^{2}dμ(x)$. Furthermore, by Caffarelli's regularity theory for the real Monge--Ampère equations, if $Λ$ is convex, $T$ is continuous. We study the reverse problem, namely, when is $T$ discontinuous if $Λ$ fails to be convex? We prove a result guaranteeing the discontinuity of $T$ in terms of the geometries of $Λ$ and $Ω$ in the two-dimensional case. The main idea is to use tools of convex analysis and the extrinsic geometry of $\partialΛ$ to distinguish between Brenier and Alexandrov weak solutions of the Monge--Ampère equation. We also use this approach to give a new proof of a result due to Wolfson and Urbas. We conclude by revisiting an example of Caffarelli, giving a detailed study of a discontinuous map between two explicit domains, and determining precisely where the discontinuities occur.

math.AP↗