Patrick McDonald
Professor of Mathematics
Data Science | Mathematics | Applied Mathematics
Education
Ph.D., Massachusetts Institute of Technology
B.S., M.S., The Ohio State University
Professor McDonald’s mathematical research centers on partial differential equations, microlocal analysis, and geometry. His published work includes results concerning analytic surgery, analytic torsion, infinite dimensional Morse theory, statistics, geometric aspects of Brownian motion, spectral geometry and over determined boundary value problems. He has published extensively in physics where his interests lie primarily in understanding the effects of symmetry breaking on the Standard Model. He is currently engaged in several projects involving health records.
He enjoys teaching analysis, probability, data visualization and Brazilian jiu-jitsu.
Recent Courses
Partial Differential Equations
Real Analysis II
Probability
Visualization, Reporting and Reproducible Research
Selected Publications
1. The Laplacian on Spaces With Cone-Like Singularities, MIT Thesis, (1990), 71 pages.
2. On the functional logdet and related flows on the space of closed embedded curves on S2 -with D. Burghelea, L. Friedlander, T. Kappeler, Jour. Func. Anal. 120 (1994), pages 440-466.
3. Analytic and Reidemeister torsion for representations in finite type Hilbert modules -with D. Burghelea, L. Friedlander, T. Kappeler, GAFA, 6 (1996), pages 751-859.
4. The extended generalized lambda distribution system for fitting distribution to data: history, completion (theory and tables) and applications – the final word on moment fits -with Z. Karian, E. Dudewicz, Comm. Stat.: Simulation and Computation, 25 (1996), pages 611-642.
5. Brownian functionals on hypersurfaces in Rn -with K. Kinateder, Proc. AMS, 125 (1997), pages 1815-1822.
6. Hypersurfaces in Rn and the variance of exit times for Brownian motion -with K. Kinateder, Proc. AMS 125 (1997), pages 2453-2462.
7. Symmetry problems arising in probability -with S. Fromm, Proc. AMS, 125 (1997), pages 3293-3298
8. Exit times for diffusions in Euclidean Space -with K. Kinateder and D. Miller, Probability Theory and Rel., 111 (1998), pages 469-487.
9. Variational principles for average exit time moments for diffusions in Euclidean space -with K. Kinateder, Proc. AMS, 127 (1999), pages 2767-2772.
10. Isoperimetric conditions, Poisson problems, and diffusions in Riemannian manifolds, Potential Analysis (2002) 16 pages 115-138.
11. Diffusions on graphs, Poisson problems and spectral geometry -with R. Meyers, Trans. AMS, 354 (2002), pages 5111-5136.
12. An Ito formula for domain valued process driven by stochastic flows -with K. Kinateder, Probability Theory and Rel., (2002) 124 pages 73-99.
13. Redefining spinors in Lorentz violating QED -with D. Colladay, Jour. Math. Physics (2002) 43, pages 3554-3564.
14. Dirichlet spectrum and heat content -with R. Meyers, Jour. Funct. Analysis (2003) 200, pages 150-159.
15. Isospectral polygons, planar graphs and heat content -with R. Meyers, Proc. AMS (2003) 131, pages 3589–3599.
16. Moment problems and the causal set approach to quantum gravity -with A. Ash, Jour. Math. Physics (2003) 44, pages 1666-1678.
17. Brownian motion, exit time moments and Weierstrass theorems -with V. de la Pena, Proc. AMS (2004) 132, pages 2465-2474.
18. Yang-Mills instantons with Lorentz violations -with D. Colladay, Jour. Math. Physics (2004) 45, pages 3228-3238.
19. Statistical mechanics and Lorentz violation -with D. Colladay, Phys. Rev. D (2004) 70, 125007.
20. Recent results in geometric analysis involving probability, In: Recent Advances in Applied Proba- bility, ed. R. Baez-Yates, J. Glaz, H. Gzyl, J. Hussler, J.L. Palacio, Springer (2005) pp 351–395.
21. Random partial orders, posts and discrete quantum gravity -with A. Ash, Jour. Math. Physics (2005) 46, 062502.
22. Deformed instanton -with D. Colladay, In: Proceeding of the Third Meeting on CPT and Lorentz Sym- metry, ed. V. Alan Kostelecky, World Scientific (2005) pp 312–318.
23. Nonrelativistic ideal gasses and Lorentz violation -with D. Colladay, In: Proceeding of the Third Meeting on CPT and Lorentz Symmetry, ed. V. Alan Kostelecky, World Scientific (2005) pp 264–269.
24. Zeta regularized determinants for conic manifolds -with. P. Loya and J. Park, Jour. Func. Analysis (2007) 242, pp 195–229.
25. Hitting times and inverse problems for Markov Chains -with V. de la Pena and H. Gzyl, Jour App. Prob. (2008) 45 pp 640–649.
26. Bose-Einstein condensation and Lorentz violation -with D. Colladay, Phys. rev. D. (2006) 73 105006.
27. One loop renormalization of pure Yang-Mills theory with Lorentz violation -with D. Colladay, Phys Rev. D. (2007) 75 105002.
28. Inverse problems for random walks on trees: network tomography, -with V. de la Pen˜a and H. Gzyl, Stat. and Prob. Letters (2008) 78 pp 3176–3183.
29. One loop renormalization of QCD with Lorentz violation -with D. Colladay, Phys Rev. D. (2008) 77 085006.
30. Renormalization of gauge theories with Lorentz violation -with D. Colladay, In: Proceeding of the Fourth Meeting on CPT and Lorentz Symmetry, ed. V. Alan Kostelecky, World Scientific (2008) pp 224–229.
31. Bose Einstein condensates and Lorentz violation -with D. Colladay, In: Proceeding of the Fourth Meeting on CPT and Lorentz Symmetry, ed. V. Alan Kostelecky, World Scientific (2008) pp 255–260.
32. One loop renormalization of the electroweak sector with Lorentz violation -with D. Colladay, Phys Rev. D. (2009) 79 125019.
33. Factoring the dispersion relation in the presence of Lorentz violation -with D. Colladay and D. Mullins, J. Phys A. (2010) 43 275202.
34. Quaternion formulation of the Lorentz-violating Dirac equation -with D. Colladay and D. Mullins, In: Proceeding of the Fifth Meeting on CPT and Lorentz Symmetry, ed. V. Alan Kostelecky, World Scientific (2011) pp 199–204.
35. Lorentz violation and extended supersymmetry -with D. Colladay, In: Proceeding of the Fifth Meeting on CPT and Lorentz Symmetry, ed. V. Alan Kostelecky, World Scientific (2011) pp 234–239.
36. Vector superfields and Lorentz violation -with D. Colladay, Phys Rev. D. (2011) 83 025021.
37. Classical lagrangians for momentum dependent Lorentz violation -with D. Colladay, Phys. Rev. D (2012) 83 044042.
38. Exit time, moment problems and comparison geometry Potential Analysis (2012) DOI 10.1007/s11118-012-9318-5.
39. Gupta-Bleuler photon quantization in the Standard Model Extension -with D. Colladay and R. Potting, Phys. Rev. D (2014) 89 085014.
40. Singular Lorentz-violating Lagrangians and associated Finsler structures -with D. Colladay, Phys. Rev. D92 (2015) no.8, 085031 arXiv:1507.01018 [hep-ph].
41. Cherenkov Radiation with Massive, CPT-violating Photons -with D. Colladay and R. Potting, Phys. Rev. D (2016) 93 125007.
42. Quantum graphs, torsional rigidity and isospectrality -with D. Colladay and L. Kaganovskiy (submitted).
43. Heat content determines planar triangles -with R. Meyerson (submitted).