Ben McMillan

Office: 739 Ingkarni Wardli
Email:benjamin.mcmillan@adelaide.edu.au

I am Ben McMillan, a postdoc at the University of Adelaide. My current research interests are differential geometry, particularly the geometric structure of differential equations, but also Cartan's method of equivalence, exterior differential systems, conservation laws, parabolic geometries, singular foliations, and quasi-conformal geometry.

A photo of Ben

Research

I have two papers extending work from my thesis, with the capstone result stating that the only evolutionary scalar parabolic equations that admit more than zero non-trivial conservation laws are the parabolic Monge-Ampère equations.

• B. McMillan, Geometry and conservation laws for a class of second-order parabolic equations I: geometry, J Geom. Physics, 157:103824, 29, 2020. ArXiv preprint here.

• B. McMillan, Geometry and Conservation Laws for a Class of Second-Order Parabolic Equations II: Conservation Laws, SIGMA, 17:Paper No. 047, 24, 2021. ArXiv preprint here.

While at Adelaide, I have collaborated with Lachlan MacDonald, to extended Bott's Chern-Weil style construction of the secondary characteristic classes to Haefliger singular foliations.

• L. MacDonald, B. McMillan, Chern-Weil theory for singular foliations, ArXiv Preprint, 2021.

I have also been learning parabolic geometric constructions from Mike Eastwood and Thomas Leistner while at Adelaide. With them, and Federico Costanza, we have an extension of a construction of Calabi to characterize the image of the Killing operator in Riemannian and Lorentzian signature. This is of recent real world relevance, due to applications to gravitational waves, as seen by LIGO.

• F. Costanza, M. Eastwood, T. Leistner, B. McMillan, A Calabi operator for Riemannian locally symmetric spaces, ArXiv Preprint, 2021.

While at Stony Brook, Dennis Sullivan suggested a strategy for applying a result of mine to resolve a longstanding question in high dimensional quasiconformal geometry: I can demonstrate quasiconformal mappings (on manifolds) that cannot be decomposed as a composition of almost-conformal mappings.

• The arXiv link will be here shortly.

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.

Seminars

I am currently co-organizing the University of Adelaide Differential Geometry seminar.

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, with links to notes for some of the talks.

Talks and Presentations

Here is a list of talks I've given, some with links to the notes.
  1. A Calabi operator for locally symmetric spaces, QMAP group, UC Davis, 1/28/22. [Notes]
  2. The secondary characteristic classes of singular foliations, AustMS Meeting 2021, Newcastle, 12/8/2021. [Notes]
  3. Conservation laws and parabolic Monge-Ampère equations AMS JMM Special Session 2020, Colorado, 1/17/2020. [Notes]
  4. Conservation laws and parabolic Monge-Ampère equations, AustMS Meeting 2019, Monash University, 12/5/2019.
  5. Conservation laws and the moduli space of solutions to PDE, Topology Seminar, Ohio State University, 3/7/2019.
  6. Conservation laws and parabolic Monge-Ampère equations, Geometry/Topology Seminar, Stony Brook, 9/25/2018. [Notes]
  7. Obstructions to flatness via the equivalence method, Equivalence and EDS Seminar, Stony Brook, 2/15/2017. [Notes]
  8. Exterior Differential Systems, definitions and examples, Equivalence and EDS Seminar, Stony Brook, 2/8/2017.
  9. The Geometry and Conservation Laws of Parabolic Equations, Topology Seminar, OSU, 10/25/2016. [Notes]
  10. The Geometry and Conservation Laws of Parabolic Equations, Geometry/Topology seminar, Stony Brook, 10/4/2016. [Notes]
  11. The Geometry and Conservation Laws of Parabolic Equations, Geometric Analysis seminar, Rutgers, 9/27/2016. [Notes]
  12. The Geometry and Conservation Laws of Parabolic Equations, Student Differential Geometry Seminar, Berkeley, 9/24/2015.
  13. The Geometry of Conservation Laws, Student Differential Geometry Seminar, Berkeley, 12/5/2014.
  14. Moving Frames and Geometric Invariants, Graduate Student Topology & Geometry Conference, UT Austin, 4/5/2014. [Notes]
  15. The Newlander-Nirenberg Theorem, The Cartan Seminar, Stanford, 2/27/2014. [Notes]
  16. Spencer Cohomology and Geometric Invariants, Student Differential Geometry Seminar, Berkeley, 2/10/2014. [Notes]
  17. Euler-Lagrange systems of EDS, Student Differential Geometry Seminar, Berkeley, 11/14/2013 and 11/21/2013. [Notes]
  18. The Newlander-Nirenberg Theorem, Student Differential Geometry Seminar, Berkeley, 4/10/2013. [Notes]
  19. Calibrated Geometries, Student Differential Geomrety Seminar, Berkeley, 2/6/2013.
  20. Holonomy on Riemannian Manifolds, Student Differential Geometry Seminar, Berkeley, 11/13/2012. [Notes]
  21. EDS - Riemannian Surfaces Isometrically Embed in Space, Student Differential Geometry Seminar, Berkeley, 9/25/2012. [Notes]
  22. Exterior Differential Systems, Harmonic Analysis and PDE Seminar, Berkeley, 4/3/2012.
  23. Curvature and Local/Global Results in Geometry, Many Cheerful Facts seminar, Berkeley, 4/8/2011.
  24. Stochastic Loewner Evolutions and the Ginibre-Girko Ensemble, Summer Undergraduate Research Fellowship presentation, Caltech, 10/2009.
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