Michael Usher

I am an Assistant Professor in the Mathematics Department at the University of Georgia. My e-mail address is [my surname] at math dot uga dot edu.


Teaching

In Fall 2009, I am teaching Math 8230, a graduate course in symplectic geometry.
In Spring 2009, I taught two sections of Math 2250 (Calculus I for Science and Engineering). Here is the syllabus (pdf), and here are solutions to the first, second, and third exams.
In Fall 2008, I taught Math 4200/6200 (Point Set Topology).
Here is a fairly thorough set of lecture notes that I created for a real analysis course (mostly metric spaces and measure theory) in Fall 2007.

Research

My research has generally dealt with symplectic topology, and more specifically with symplectic four-manifolds, Hamiltonian dynamics, and symplectic Floer homology.

Here is the Symplectic Geometry wiki.

I coorganized the 2009 Georgia International Topology Conference, and will be editing its proceedings; participants in the conference who might be interested in submitting a paper should contact me.

Research Papers*

  • Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds, to appear in the Israel Journal of Mathematics.
  • The sharp energy-capacity inequality, to appear in Communications in Contemporary Mathematics.
  • Floer homology in disc bundles and symplectically twisted geodesic flows, Journal of Modern Dynamics, 3 (2009), no. 1, 61-101.
  • Kodaira dimension and symplectic sums, Commentarii Mathematici Helvetici, 84 (2009), no. 1, 57-85.
  • Spectral numbers in Floer theories, Compositio Mathematica, 144 (2008), no. 6, 1581-1592.
  • Standard surfaces and nodal curves in symplectic 4-manifolds, Journal of Differential Geometry 77 (2007), no. 2, 237-290.
  • Minimality and symplectic sums, International Mathematics Research Notices 2006 (2006), article ID 49857.
  • Vortices and a TQFT for Lefschetz fibrations on 4-manifolds, Algebraic and Geometric Topology 6 (2006), 1677-1743.
  • Symplectic forms and surfaces of negative square, with Tian-Jun Li, Journal of Symplectic Geometry 4 (2006), no. 1, 71-91.
  • The Gromov invariant and the Donaldson-Smith standard surface count, Geometry and Topology 8 (2004), 565-610.

    • *may differ somewhat from published versions

    Surveys and talks

  • Slides from a talk on C^0 stability in Morse theory and symplectic topology, given at the Georgia Topology Conference in May 2009.
  • Slides from a talk about applications of the filtration in Hamiltonian Floer theory, given at the Third Illinois-Indiana Symplectic Geometry Conference in Spring 2009.
  • Slides from a talk about the non-squeezing theorem and the energy-capacity inequality, given at the joint UGA-GaTech-Emory topology seminar in Fall 2008.
  • Slides from a talk about twisted geodesic flows, given at the UGA geometry seminar in Fall 2008.
  • Slides from a talk about my proof of the spectrality axiom for Floer homology theories, given at various locations in Fall 2007 and Spring 2008.
  • Slides from a survey talk about symplectic four-manifolds, given at various locations in January 2007
  • A survey article, Lefschetz fibrations and pseudoholomorphic curves, written for the Proceedings of the 2004 McMaster Conference on the Geometry and Topology of Manifolds, summarizing the results of the G&T and JDG papers listed above.
  • Slides from a November 2004 talk at the MIT differential geometry seminar about Lefschetz fibrations and nodal curves.
  • Slides from a March 2004 talk at MSRI about the Gromov and Donaldson-Smith invariants.

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