Neel Patel

Research

My research is in the analysis of partial differential equations.  Below, are a list of my publications, grants and awards.

Links to papers by clicking on underlined titles.

Papers

• On Nonlinear Stability of Muskat Bubbles, with F. Gancedo, E. Garcia-Juarez and R. M. Strain;  accepted for publication in Journal de Mathematique Pures et Appliquees (2025)

• Bounds on the Aspect Ratio of the Momentum Support of a 2D Collisionless Plasma, with M. Hernandez and E. Salguero; preprint (2024)

• Rigidity of Acute Angled Corners For One Phase Muskat Interfaces, with S. Agrawal and S. Wu;  Advances in Mathematics (2023)

• Well-posedness for SQG Sharp Fronts with Unbounded Curvature, with F. Gancedo and H. Nguyen; Math. Models and Methods in App. Sci. (2022)

• Global Results for the Inhomogeneous Muskat Problem, with N. Shankar; preprint (2021)

• Global Regularity for Gravity Unstable Muskat Bubbles, with F. Gancedo, E. Garcia-Juarez and R. M. Strain; Memoirs of the AMS (2023)

• On the local existence and blow-up for generalized SQG patches, with F. Gancedo;  Annals of PDE;  (2020)

• On The Muskat Problem With Viscosity Jump: Global In Time Results, with F. Gancedo, E. Garcia-Juarez and R. M. Strain; Advances in Mathematics (2019).

• Large Time Decay Estimates for the Muskat Equation, with R. M. Strain; Communications in PDE (2017)

• Three New Results on Continuation Criteria for the 3D Relativistic Vlasov-Maxwell System, Journal of Differential Equations (2018)

• On Fiber Diameters of Continuous Maps, with P. S. Landweber and E. A. Lazar; The American Mathematical Monthly (2015)

Grants/Awards:

• AMS-Simons PUI Faculty Grant (July 2023-July 2026)

• AMS-Simons Travel Grant (July 1, 2019- June 30, 2021)

• The Carlitz-Zippin Prize for best PhD Thesis (2017)

Mathematical Activities

Seminar/Conference Talks Given

• Mathematical Research Institute of Oberwolfach; Deterministic Dynamics and Randomness in PDE (May 2022)

• One World Meeting of Young Mathematicians in Fluid Dynamics (June 2021)

• UC Davis; Partial Differential Equations Seminar (January 2021)

• Invited Talk at SIAM Conference on Nonlinear Waves and Coherent Structures; Stability/Instability in Hydrodynamics Mini-symposium; Bremen, Germany (July 2020) CANCELLED

• Brown University; Partial Differential Equations Seminar (February 2020)

• University of Massachusetts - Amherst; Analysis and PDE Seminar (February 2020)

• Invited Talk at SIAM PDE Conference 2019 - Recent Progress in Fluid Mechanics: Classical Flows, Geophysical Models and Complex Fluids Mini-symposium; La Quinta, California (December 2019)

• Michigan State University; Applied Mathematics Seminar (December 2019)

• CMSA (Center for Mathematical Sciences and Applications) Harvard University; Fluid Dynamics Seminars (April 2019)

• University of Pennsylvania; Analysis Seminar (February 2019)

• Princeton University; Analysis of Fluids and Related Topics Seminar (November 2018)

• University of Michigan; Differential Equations Seminar (September 2018)

• University of Pennsylvania; Analysis Seminar (May 2018)

• Imperial College London; Analysis and PDE Seminar (January 2018)

• Institute of Mathematical Sciences ICMAT, Madrid; Working Seminar on PDE and Fluid Mechanics (October 2017)

• Instituto de Matematicas Universidad de Sevilla (IMUS University of Seville); Conversaciones Fluidas (Conversations about Fluids) (October 2017)

Conferences

In collaboration with Francisco Gancedo and Eduardo Garcia-Juarez, I was an organizer of the MathFluids "Mathematical Analysis of Incompressible Fluids" conference in Sevilla, Spain during July 12-15, 2018.

In collaboration with Eduardo Garcia-Juarez and Annalaura Stingo, I was an organizer of the minisymposium "Recent Results in Incompressible Fluid Mechanics" at SIAM PDE 2019 in La Quinta, California during December 11-14, 2019.

Outreach

• FEMMES Saturday Sciences Capstone (Fall 2018): co-organized group activity for 4th-6th grade girls to learn about the Stable Matching Problem.

• REU 2019: Advised Nikhil Shankar on a project on porous media flows with permeability jump

• REU 2020: Advised Advika Rajapakse on a project on velocity growth bounds for collisionless plasmas

Teaching

My primary goal in teaching is to empower students to be self-sufficient learners and to foster interest in mathematics for students of all backgrounds. In Spring 2020, I was nominated for the Golden Apple Award at University of Michigan. Previously, I was awarded the Moez Alimohamed Graduate Student Award for Distinguished Teaching in Mathematics (2014-2015), given to one graduate student annually, and two Good Teaching Awards (Fall 2013 and Spring 2014) during my time as a graduate student at UPenn.

Courses (UMaine):

• MAT 453: Partial Differential Equations I

Courses (at UMichigan)

• Math 451: Advanced Calculus I (Winter 2021)

• Math 289: Problem Solving (Putnam Preparation) (Winter 2021)

• Math 555: Introduction to Complex Variables (Fall 2020)

• Math 289: Problem Solving (Putnam Preparation) (Fall 2020)

• Math 451: Advanced Calculus I (Winter 2020)

• Math 286: Honors Differential Equations (Fall 2019)

• Math 115: Calculus I (Fall 2019)

• Math 555: Introduction to Complex Variables (Winter 2019)

• Math 115: Calculus I (Fall 2018)