Nirenberg Lectures in Geometric Analysis
Rohil Prasad
Rohil Prasad is emerging as an international leader in the application of modern analytical methods to problems in low-dimensional symplectic and contact dynamics. After receiving his PhD from Princeton University in 2023 under the supervision of Helmut Hofer, he was awarded the prestigious Miller Fellowship at the University of California, Berkeley. In 2024, he received the Brin Prize in Dynamical Systems for Young Mathematicians in recognition of his contributions to symplectic topology and its applications to the dynamics of area-preserving maps, contact flows, and Hamiltonian systems.
Lecture 1 – This lecture is intended for a general mathematical audience.
Monday, November 16, 2026, at 3:30 p.m. - Room 6214 (CRM).
A wine and cheese reception will follow.
Title: From Holomorphic Curves to Invariant Measures
Abstract: The theory of holomorphic curves provides a remarkable bridge from the symplectic geometry of a space to the dynamics of Hamiltonian flows on that space. The greatest successes from this connection have been in the detection of periodic orbits. This talk, and the next, will explore some ideas using holomorphic curves to detect dynamical objects that are more general than periodic orbits. To start, we will discuss a method for finding invariant probability measures of Hamiltonian flows. Under some simple topological conditions, we can show that these measures are "interesting"; they are not the standard symplectic volume measure. However, many questions remain open.
Lecture 2
Tuesday, November 17, 2026, at 2:30 p.m. - Room 6214 (CRM).
A coffee gathering will follow.
Title: From Holomorphic Curves to Invariant Sets
Abstract: This talk continues our discussion on using holomorphic curves to find dynamical objects that are more general than periodic orbits. We will turn our attention from invariant measures to invariant sets. In this talk, I will present some recent work on detecting invariant sets of Hamiltonian flows on $\mathbb{R}^4$. I will show that, for any Hamiltonian and any compact regular level set $Y$, there exist many nontrivial invariant subsets. More precisely, there exists an infinite collection of distinct closed subsets of $Y$, invariant under the Hamiltonian flow, whose union is dense in $Y$. I will also discuss related joint work with Dan Cristofaro-Gardiner concerning Reeb flows in three dimensions and area-preserving surface diffeomorphisms.
Michael Hutchings (University of California, Berkeley)
Michael Hutchings is a professor of mathematics at the University of California, Berkeley. He is a world leader in homological methods in low-dimensional symplectic and contact topology, having notably introduced and developed the theory of embedded contact homology. He is also recognized for his work on the double bubble conjecture in geometric analysis. He received his PhD from Harvard University in 1998 under the supervision of Clifford Taubes. He held postdoctoral research and visiting professorship positions at Stanford University, the Max Planck Institute for Mathematics in Bonn, and the Institute for Advanced Study in Princeton before joining the faculty of the University of California, Berkeley in 2001. He was awarded a Sloan Research Fellowship in 2003 and delivered an invited lecture at the International Congress of Mathematicians in 2010. In 2012, he was named a Fellow of the American Mathematical Society.
Lecture 1
Thursday, November 19, 2026, at 2:30 p.m. - Room 6214 (CRM).
A coffee gathering will follow the lecture.
Title: Weyl Laws for Min-Max Spectral Invariants of Three-Dimensional Contact Manifolds
Abstract: We define "elementary spectral invariants" of a closed three-dimensional manifold equipped with a contact form as the min-max energy of certain pseudoholomorphic curves subject to point constraints. It follows from this definition that each of these spectral invariants, when finite, is equal to the sum of the periods of a finite collection of periodic orbits of the Reeb vector field. Much less obviously, these spectral invariants satisfy a "Weyl law" relating their asymptotic behavior to the contact volume. We will present the main ingredients involved in the proof of this Weyl law.
Lecture 2 – This lecture is intended for a general mathematical audience.
Friday, November 20, 2026, at 3:30 p.m. - Room 6214 (CRM).
A wine and cheese reception will follow the lecture.
Title: Periodic Orbits of Reeb Vector Fields in Three Dimensions
Abstract: Reeb vector fields on odd-dimensional manifolds generalize the geodesic flow on a Riemannian manifold as well as Hamiltonian vector fields on certain level sets. We will present several important recent results concerning the existence and properties of periodic orbits of Reeb vector fields in three dimensions. The proofs rely on the spectral invariants discussed in the previous lecture (but attendance at that lecture is not required to follow this presentation).