Ben McMillan

Office: Simons 508
Email:benjamin.mcmillan@stonybrook.edu

I am Ben McMillan, a postdoc with a joint appointment at Stony Brook math and the Simons Center for Geometry and Physics. My current research interests are differential geometry, particularly the geometry of PDE.


Research

For my thesis I studied the geometry of (scalar) parabolic partial differential equations, as well as the relation between this geometry and the existence of conservation laws. You can read Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations, or you can read a summary of the thesis, which you can find here.

I have two papers extending this work:

Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations I: Geometry

Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws

Although my current interests are in differential geometry, in the past I was very interested in graph theory. Work with Cheng Yeaw Ku resulted in the paper

C. Y. Ku and B. McMillan, Independent sets of maximal size in tensor powers of vertex transitive graphs, Journal of Graph Theory, 60 (4) (2009), 295-301. [PDF]


Seminars

In the Spring of 2017, I organized a seminar on Cartan's equivalence method and exterior differential systems. You can find a schedule of talks here.

Talks and Presentations

When I give a talk, I find it helpful to write out notes as I prepare. Since the notes often work without the talk, I post them here.
  1. Conservation laws and parabolic Monge-Ampere equations, Geometry/Topology Seminar, Stony Brook, 9/25/2018. [PDF]
  2. Obstructions to flatness via the equivalence method, Equivalence and EDS Seminar, Stony Brook, 2/15/2017. [PDF]
  3. Exterior Differential Systems, definitions and examples, Equivalence and EDS Seminar, Stony Brook, 2/8/2017.
  4. The Geometry and Conservation Laws of Parabolic Equations, Topology Seminar, OSU, 10/25/2016. [PDF]
  5. The Geometry and Conservation Laws of Parabolic Equations, Geometry/Topology seminar, Stony Brook, 10/4/2016. [PDF]
  6. The Geometry and Conservation Laws of Parabolic Equations, Geometric Analysis seminar, Rutgers, 9/27/2016. [PDF]
  7. The Geometry and Conservation Laws of Parabolic Equations, Student Differential Geometry Seminar, Berkeley, 9/24/2015.
  8. The Geometry of Conservation Laws, Student Differential Geometry Seminar, Berkeley, 12/5/2014.
  9. Moving Frames and Geometric Invariants, Graduate Student Topology & Geometry Conference, UT Austin, 4/5/2014. [PDF]
  10. The Newlander-Nirenberg Theorem, The Cartan Seminar, Stanford, 2/27/2014. [PDF]
  11. Spencer Cohomology and Geometric Invariants, Student Differential Geometry Seminar, Berkeley, 2/10/2014. [PDF]
  12. Euler-Lagrange systems of EDS, Student Differential Geometry Seminar, Berkeley, 11/14/2013 and 11/21/2013. [PDF]
  13. The Newlander-Nirenberg Theorem, Student Differential Geometry Seminar, Berkeley, 4/10/2013. [PDF]
  14. Calibrated Geometries, Student Differential Geomrety Seminar, Berkeley, 2/6/2013.
  15. Holonomy on Riemannian Manifolds, Student Differential Geometry Seminar, Berkeley, 11/13/2012. [PDF]
  16. EDS - Riemannian Surfaces Isometrically Embed in Space, Student Differential Geometry Seminar, Berkeley, 9/25/2012. [PDF]
  17. Exterior Differential Systems, Harmonic Analysis and PDE Seminar, Berkeley, 4/3/2012.
  18. Curvature and Local/Global Results in Geometry, Many Cheerful Facts seminar, Berkeley, 4/8/2011.
  19. Stochastic Loewner Evolutions and the Ginibre-Girko Ensemble, Summer Undergraduate Research Fellowship presentation, Caltech, 10/2009.

Teaching

This semester I am teaching Math 315, Advanced Linear Algebra. The course webpage is here.