Dan Knopf

Dan Knopf is a professor in the Department of Mathematics at The University of Texas at Austin. He earned his Ph.D. in mathematics from the University of Wisconsin-Milwaukee, under the supervision of Kevin McLeod, with mentorship from Bennett Chow. Prior to his doctoral studies, he completed his B.A. in mathematics at the same institution.

Before joining UT Austin in 2004, Knopf held academic positions as a visiting assistant professor at The University of Iowa and as a VIGRE Van Vleck Visiting Assistant Professor at the University of Wisconsin-Madison.

Knopf's research focuses on geometric analysis, particularly geometric heat flows, which are nonlinear partial differential equations related to the heat equation. These flows aim to evolve geometric objects toward optimal structures but often develop singularities. His work involves analyzing singularity formation and studying the dynamical stability of special solutions.

Throughout his career, Knopf has received several grants and fellowships, including a National Science Foundation (NSF) CAREER Award, several standard NSF grants, co-leadership of a $2.5M NSF Research Training Group (RTG) grant, and a Simons Foundation Collaboration Grant. He has also been recognized for his teaching excellence, receiving the College of Natural Sciences Teaching Excellence Award and the John R. Durbin Teaching Excellence Award in Mathematics.

In addition to his research and teaching, Knopf served as the Associate Dean for Graduate Education in the College of Natural Sciences at UT Austin from 2014 to 2024.

As a geometric analyst, Knopf primarily studies geometric heat flows. These are partial differential equations and systems that are nonlinear relatives of the heat equation. Intuitively, one expects such flows to improve a given geometric object, evolving it towards an optimal or canonical structure. But because geometric flows have a diffusion-reaction structure, their solutions often develop singularities. For these flows to have successful geometric applications requires a deep understanding of how singularities form and of how solutions can be continued past them. So research in the Knopf group includes extensive asymptotic analysis of singularity formation as well as detailed studies of dynamical stability of special solutions.

Areas of focus include: Geometric analysis, Differential geometry, and Geometric Partial Differential Equations.

Knopf is a member of the Geometry research group in the UT Austin Department of Mathematics, interacting with the research groups in Partial Differential Equations and Topology.