Irina Holmes: Research

NSF Postdoctoral Fellow Department of Mathematics Michigan State University holmesir@math.msu.edu

Overview

Postdoctoral Research: My current research interests are in dyadic harmonic analysis, weighted inequalities, Bellman functions, and multiparameter harmonic analysis. A lot of my postdoctoral work has been in two-weight inequalities for commutators with Calderón-Zygmund operators. For more details, see my Research Statement.

Graduate Research: My graduate research interests were infinite-dimensional analysis, probability, functional analysis and machine learning. Together with my adviser, I worked on developing the Gaussian Radon transform for Banach spaces, an infinite-dimensional generalization of the classical Radon transform, and applications of this transform to machine learning. For more details, see my Graduate Research Statement, or my thesis.

Undergraduate Research: As an undergraduate at LSU, I worked as a research student at CCT. In particular, I worked with Horst Beyer on the Kerr metric of rotating black holes.

Publications

• Bi-parameter embedding and measures with restriction energy condition - joint with Nicola Arcozzi, Pavel Mozolyako, and Alexander Volberg.
• Obstacle problems generated by the estimates of square function - joint with Alexander Volberg.
• Bellman function sitting on a tree - joint with Nicola Arcozzi, Pavel Mozolyako, and Alexander Volberg.
• The Sharp Constant in the Weak (1,1) Inequality for the Square Function: A New Proof - joint with Paata Ivanisvili and Alexander Volberg.
• Two weight Commutators in the Dirichlet and Neumann Laplacian settings - joint with Xuan Thinh Duong, Ji Li, Brett D. Wick, and Dongyong Yang (Submitted).
• Weighted little bmo and two-weight inequalities for Journé commutators - joint with Stefanie Petermichl and Brett D. Wick (Analysis & PDE 11, No 7, 2018).
• Commutators with Fractional Integral Operators - joint with Robert Rahm and Scott Spencer (Studia Mathematica 233, no. 3, 2016).
• Two Weight Inequalities for Iterated Commutators with Calderón-Zygmund Operators - joint with Brett D. Wick (Journal of Operator Theory 79, No. 1, 2018.).
• Commutators in the Two-Weight Setting - joint with Michael T. Lacey and Brett D. Wick (Mathematische Annalen 376, Issue 1-2, 2017).
• Bloom's Inequality: Commutators in a Two-Weight Setting - joint with Michael T. Lacey and Brett D. Wick (Archiv der Mathematik 106, no. 1, 2016).
• The Gaussian Radon Transform in Classical Wiener Space - joint with Ambar Sengupta (Communications on Stochastic Analysis 8, no. 2, 2014).
• The Gaussian Radon Transform and Machine Learning - joint with Ambar Sengupta (Infinite Dimensional Analysis, Quantum Probability and Related Topics, 18, no. 3, 2015).
• An Inversion Formula for the Gaussian Radon Transform for Banach Spaces (Infinite Dimensional Analysis, Quantum Probability and Related Topics, 16, no. 4, 2013).
• A Gaussian Radon Transform for Banach Spaces - joint with Ambar Sengupta (Journal of Functional Analysis 263, no. 11, 2012).
• On a new symmetry of the solutions of the wave equation in the background of a Kerr black hole - joint with Horst R. Beyer (Classical and Quantum Gravity, 25, no. 13, 2008)
• On a Problem in the Stability Discussion of Rotating Black Holes (Proceedings of The National Conference On Undergraduate Research (NCUR) 2006)