About

Ben Weinkove is associated with the Northwestern University Mathematics Department, which supports research and training activities in the field of dynamics. The research group, supported by the RTG grant titled 'Dynamics: Classical, Modern, and Quantum,' aims to expose young researchers to dynamical systems in the broad sense, including the study of the time-evolution of mechanical systems and abstract models of such systems. The group emphasizes increasing research strength in dynamics both at Northwestern and across the US, with a focus on training a broad group of researchers and fostering vertically integrated mentoring. The faculty involved in this research area cover a wide range of dynamical fields and interact with various other areas of mathematics, including number theory, combinatorics, partial differential equations, quantum mechanics, group theory, and representation theory.

Research topics

• Mathematical analysis
• Mathematics
• Combinatorics
• Geometry
• Applied mathematics
• Physics
• Mathematical optimization
• Quantum mechanics
• Pure mathematics

Selected publications

• Stochastic neighborhood embedding and the gradient flow of relative entropy

  Discrete and Continuous Dynamical Systems · 2025-01-01

  articleOpen access1st authorCorresponding

  Dimension reduction, widely used in science, maps high-dimensional data into low-dimensional space. We investigate a basic mathematical model underlying the techniques of stochastic neighborhood embedding (SNE) and its popular variant t-SNE. Distances between points in high dimensions are used to define a probability distribution on pairs of points, measuring how similar the points are. The aim is to map these points to low dimensions in an optimal way so that similar points are closer together. This is carried out by minimizing the relative entropy between two probability distributions.We consider the gradient flow of the relative entropy and analyze its long-time behavior. This is a self-contained mathematical problem about the behavior of a system of nonlinear ordinary differential equations. We find optimal bounds for the diameter of the evolving sets as time tends to infinity. In particular, the diameter may blow up for the t-SNE version, but remains bounded for SNE.

  PublisherDOI

• Composite media, almost touching disks and the maximum principle

  Pure and Applied Mathematics Quarterly · 2025-01-01

  articleSenior author
  PublisherDOI

• Stochastic neighborhood embedding and the gradient flow of relative entropy

  arXiv (Cornell University) · 2024-09-25

  preprintOpen access1st authorCorresponding

  Dimension reduction, widely used in science, maps high-dimensional data into low-dimensional space. We investigate a basic mathematical model underlying the techniques of stochastic neighborhood embedding (SNE) and its popular variant t-SNE. Distances between points in high dimensions are used to define a probability distribution on pairs of points, measuring how similar the points are. The aim is to map these points to low dimensions in an optimal way so that similar points are closer together. This is carried out by minimizing the relative entropy between two probability distributions. We consider the gradient flow of the relative entropy and analyze its long-time behavior. This is a self-contained mathematical problem about the behavior of a system of nonlinear ordinary differential equations. We find optimal bounds for the diameter of the evolving sets as time tends to infinity. In particular, the diameter may blow up for the t-SNE version, but remains bounded for SNE.

  PublisherOA PDFDOI

• Smoothness up to the free boundary for the $p$-Laplacian evolution equation and the $α$-Gauss curvature flow

  arXiv (Cornell University) · 2024-12-13

  preprintOpen accessSenior author

  The $p$-Laplacian evolution equation and the $α$-Gauss curvature flow with a flat side are degenerate parabolic equations with evolving free boundaries. We give proofs of smooth short-time existence, up to the free boundaries, using a result of the authors on linear degenerate equations on a fixed domain.

  PublisherOA PDFDOI

• Differentialgeometrie im Grossen

  Oberwolfach Reports · 2024-04-18

  articleOpen accessSenior author

  Over the past several decades, classical differential geometry has undergone a remarkable expansion, helped by the integration of tools and insights from neighboring fields like partial differential equations, complex analysis, and geometric topology. In keeping with the spirit of previous gatherings, this meeting aimed to bridge the gaps between researchers working in seemingly disparate subfields of differential geometry, illuminating the connections that unite them. Amongst other things, this meeting was centered around the theme of scalar curvature, which has recently emerged as a fundamental element across various fields, including differential geometry, metric geometry, topology, and complex geometry. This shared topic presented an ideal opportunity for scholars from these distinct areas to convene, discuss their individual progress, and foster a vibrant exchange of ideas.

  PublisherOA PDFDOI

• Non-preservation of α-concavity for the porous medium equation

  Advances in Mathematics · 2024-02-12 · 5 citations

  articleSenior authorCorresponding
  PublisherDOI

• The perfect conductivity problem with arbitrary vanishing orders and non-trivial topology

  Journal of Mathematical Analysis and Applications · 2023-06-05

  articleSenior authorCorresponding
  PublisherDOI

• Instantaneous convexity breaking for the quasi-static droplet model

  Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2023-08-15 · 2 citations

  articleOpen accessSenior author

  We consider a well-known quasi-static model for the shape of a liquid droplet. The solution can be described in terms of time-evolving domains in \mathbb{R}^n . We give an example to show that convexity of the domain can be instantaneously broken.

  PublisherDOI

• In Memory of Steve Zelditch

  Notices of the American Mathematical Society · 2023-10-01

  articleOpen access
  PublisherDOI

• The perfect conductivity problem with arbitrary vanishing orders and non-trivial topology

  arXiv (Cornell University) · 2023-01-09

  preprintOpen accessSenior author

  The perfect conductivity problem concerns optimal bounds for the magnitude of an electric field in the presence of almost touching perfect conductors. This reduces to obtaining gradient estimates for harmonic functions with Dirichlet boundary conditions in the narrow region between the conductors. In this paper we extend estimates of Bao-Li-Yin to deal with the case when the boundaries of the conductors are given by graphs with arbitrary vanishing orders. Our estimates allow us to deal with globally defined narrow regions with possibly non-trivial topology. We also prove the sharpness of our estimates in terms of the distance between the perfect conductors. The precise optimality statement we give is new even in the setting of Bao-Li-Yin.

  PublisherOA PDFDOI

Recent grants

• Elliptic and Parabolic Partial Differential Equations on Manifolds

  NSF · $207k · 2017–2020

• Parabolic flows and canonical metrics in Kahler geometry.

  NSF · $116k · 2005–2008

• PDE's in complex and symplectic geometry

  NSF · $136k · 2008–2011

• Nonlinear PDEs and complex geometry

  NSF · $185k · 2014–2018

• Nonlinear Partial Differential Equations and Geometry

  NSF · $248k · 2020–2023

Frequent coauthors

• Jian Song

  Rutgers, The State University of New Jersey

  31 shared

• Valentino Tosatti

  28 shared

• Albert Chau

  University of British Columbia

  16 shared

• Jacob Sturm

  Rutgers, The State University of New Jersey

  13 shared

• D. H. Phong

  7 shared

• Gábor Székelyhidi

  Northwestern University

  7 shared

• Morgan Sherman

  California Polytechnic State University

  7 shared

• Jianchun Chu

  Peking University

  5 shared

Labs

• RTG Dynamics: Classical, Modern and QuantumPI

Awards & honors

• Alfred P. Sloan Fellowship (2008)
• Fellow of the American Mathematical Society (2017)
• Madan Lal Puri Professor of Mathematics (2024)