Michael Usher

I am Professor and Head in the Mathematics Department at the University of Georgia, affiliated with the topology and geometry groups.

My email address is [my surname]@uga.edu.

Research

My research generally deals with symplectic topology, the study of global questions relating to spaces equipped with the geometric structures that lie at the root of classical mechanics.

Over the years, I have worked on symplectic four-manifolds and Lefschetz fibrations, Hamiltonian dynamics, the geometry of symplectic diffeomorphisms and symplectic embeddings, and foundational issues related to filtered Floer and Morse homology.

My papers can be found on the arXiv here.

I created a computer program that facilitates experiments with some four-dimensional symplectic embedding problems here.

Here are slides from a May 2023 talk about interlevel persistence for Morse and Floer theory.

Teaching

Due to my administrative duties, I am not teaching in Spring 2026.  In the fall, I will teach MATH 4250/6250 (Differential Geometry).

Lecture Notes

  • Here are the combined lecture notes (110 pp.) for introductory symplectic geometry and topology courses that I taught in Spring 2019 and Spring 2024. These are intended to be accessible to readers with a basic background in smooth manifolds and differential forms.  Beyond standard introductory material (symplectic linear algebra, Hamiltonian flows, the Moser technique, etc.) they cover some more advanced topics like Weinstein handle attachment, Lefschetz fibrations and open books, and generating functions.
  • Here are lecture notes for some of the Topology of Manifolds course that I taught in Fall 2011 (54 pp. For most of the course we followed the start of Bott and Tu’s Differential Forms in Algebraic Topology; these notes concern background material not covered in detail in the book).
  • Here are the notes for the majority of my course on pseudoholomorphic curves (74 pp.) from Spring 2010; these are designed to give an accessible account of the basic analytic foundations of the theory.
  • Here are lecture notes for a topics course on elliptic PDE’s on manifolds (111 pp.) that I taught in Fall 2016. They give a through treatment of Sobolev spaces and the de Rham and Dolbeault versions of the Hodge theorem, and a brief introduction to pseudoholomorphic curves.
  • Here are lecture notes (50 pp.) for the first half of my Spring 2015 topics course, which used the theory of area-preserving maps as an entry point to symplectic geometry. (We used Cannas’ Lectures on Symplectic Geometry for the second half of the course.)
  • Here are lecture notes for a course on vector bundles and cohomology (99 pp.) from Fall 2012.
  • I created these notes (78 pp.) for an analysis course that I taught as a postdoc in Fall 2007.
  • Here are lecture notes for the start of my Spring 2014 algebraic topology course (30 pp. We used Hatcher’s book for the rest of the course).
  • Here are some brief notes on symplectic cutting and blow-ups (6 pp.) that I wrote while teaching symplectic geometry in Fall 2009.