UW Department of Mathematics
Office: 522 Van Vleck Hall
Email: aai (at) math (dot) wisc (dot) edu

I am currently an Analysis and PDE RTG Postdoc with the Department of Mathematics. My postdoctoral advisor is Mihaela Ifrim. Previously I was a graduate student in the math department at UC Berkeley, advised by Daniel Tataru.

Research Interests

My research interests lie primarily in the fields of nonlinear dispersive partial differential equations, fluid dynamics, and harmonic analysis. One focus of my current research is the question of low regularity well-posedness, which asks to what extent well-posedness holds even when the initial data lies in a very low regularity Sobolev space. My work has involved several models from dispersive PDE and fluids, including the nonlinear wave equation, water waves, the surface quasi-geostrophic (SQG) equation, and the generalized Benjamin-Ono equation.

Here is my CV.

Papers and Preprints

1. November 2023: Low regularity well-posedness for the generalized surface quasi-geostrophic front equation, joint with Ovidiu-Neculai Avadanei.
2. October 2023: Low regularity solutions for the surface quasi-geostrophic front equation, joint with Ovidiu-Neculai Avadanei.
3. August 2023: Improved low regularity theory for gravity-capillary waves.
4. November 2022: Well-posedness for the surface quasi-geostrophic front equation, joint with Ovidiu-Neculai Avadanei, under revision at Nonlinearity (2023).
5. October 2021: The time-like minimal surface equation in Minkowski space: low regularity solutions, joint with Mihaela Ifrim and Daniel Tataru, to appear in Inventiones mathematicae, (2023).
6. May 2021: Well-posedness for the dispersive Hunter-Saxton equation, joint with Ovidiu-Neculai Avadanei, Int. Math. Res. Not. IMRN, no. 9, 7883-7924 (2023).
7. September 2020: Two dimensional gravity waves at low regularity II: Global solutions, joint with Mihaela Ifrim and Daniel Tataru, Ann. Inst. H. Poincaré C Anal. Non Linéaire, Vol. 39, no. 4, 819-884 (2022).
8. October 2019: Two dimensional gravity waves at low regularity I: Energy estimates, joint with Mihaela Ifrim and Daniel Tataru
9. November 2018: Low regularity solutions for gravity water waves II: The 2D case, Ann. PDE, Vol. 6, no. 1, Paper No. 4, 117 pp. (2020). https://doi.org/10.1007/s40818-020-00081-z
10. December 2017: Low regularity solutions for gravity water waves, Water Waves, 1(1):145-215 (2019).
11. November 2012: Sylvester-Gallai type theorems for approximate collinearity, joint with Z. Dvir, S. Saraf, A. Wigderson, Forum Math. Sigma, Vol. 2 (2014).
12. August 2012: One-bit compressed sensing with non-gaussian measurements, joint with A. Lapanowski, Y. Plan, and R. Vershynin, Linear Algebra and its Applications, Vol. 441, p. 222-239 (2014).

Activities and Seminars

• July 17 - August 11, 2023: MSRI Mathematical Problems in Fluid Dynamics, Part II
• October 23 - October 29, 2022: Invited mini-series lecturer, Oberwolfach seminar
• Fall 2021: PDE and geometric analysis seminar (UW-Madison)
• Fall 2020 - Fall 2021: Berkeley-Madison virtual PDE seminar
• Spring 2021: Postdoctoral researcher with the MSRI program Mathematical problems in fluid dynamics (virtual)
• Summer 2020: TA for the MSRI summer school Introduction to water waves (virtual)

Teaching

• Fall 2023: Madison Experimental Mathematics Lab advisor: Aperiodic Monotiles
• Fall 2023: Instructor for Math 551: Elementary Topology
• Spring 2022: Instructor for Math 540: Linear Algebra II
• Fall 2021: Instructor for Math 431, Introduction to Probability
• Fall 2020: Instructor for Math 431, Introduction to Probability (virtual)

Updated: 11/27/2023