Differential Geometry
Mathematical Physics
WHY DO I STUDY DIFFERENTIAL GEOMETRY ?
Geometry is one of the oldest disciplines of Science and dates back to ancient times. The Egyptians used geometric formulas to measure the land, whence the greek word: geometry. The Greeks themselves adored the subject and cultivated it into a beautiful abstract piece of Mathematics, as epitomized in Euclid's books, which still remain for most of us the first, and unfortunately, for many, the last encounter with the subject.
Modern Differential Geometry began with Gauss' work on the Theory of Surfaces in the early 19th century. Gauss was again partly motivated by practical surveying problems about mapping the surface of the earth. By that time it was well known that the earth is not flat and hence that Euclidean Geometry is not adequate. Gauss applied the powerful method of the Infinitesimal Calculus, invented about a century earlier by Newton and Leibniz, in his investigations on the curvature of surfaces. A few decades later this Gaussian theory was generalized to a higher level of abstraction and dimension by Riemann, who introduced the notion of a manifold as an appropriate form of space where one can study geometries. Euclidean geometry then became just one very special case among an infinity of possible geometries and the laws of Euclidean geometry were postulated to be true only for measurements at a very small scale, to be precise, at the infinitesimal level. These Euclidean measurements, in other words, the metric, is now allowed to vary from one point to the next. The idea of space itself as a dynamic entity was born.
It was the genius of Einstein who realized that these new geometric ideas should be the basis for understanding not just the shape of the earth but that of the while universe of space and time. His revolutionary General Theory of Relativity is a masterpiece of Geometric Physics explaining that mysterious fundamental force of Nature, Gravitation, which holds the universe together on a large scale, as a manifestation of the curvature of space-time itself. In order to formulate this theory, Einstein relied on the differential-geometric calculus developed by Gauss, Riemann, Christoffel, Ricci and Levi-Civita. In his later years Einstein dreamed of generalizing his theory to encompass all the other known forces of Nature. This dream, known as the GUT (Grand Unified Theory), is still pursued by theoretical physicists and over the last three decades, there has been some spectacular new theoretical advances, which might ultimately become important stepping stones to a GUT. Most of these theories such as Gauge Theory, String Theory, M-theory etc., are in fact very geometrical and have strong dialectical interactions with recent advances in Pure Mathematics, especially in Geometry and Topology. One basic idea of these new physical theories is to extend the arena of physics from the 4-dimensional space-time of Relativity to a higher dimensional manifold (say of dimension 10 or 11), to incorporate all the degrees of internal freedoms and symmetries that are needed to explain all the other forces of nature. The other fundamentally new point of view is to treat particles and the forces between them not just through points and lines but by using higher dimensional strings and menbranes in order to resolve the basic contradiction between General Relativity and Quantum Mechanics. The laws of nature should then be described (at least at an approximate level) by field equations involving some form of curvature and reflecting all the inherent (super-)symmetries. Curvature, in its various manifestations, is the fundamental invariant of Differential Geometry. It is calculated locally by means of the Infinitesimal Calculus but it governs, on a broader scale, the global shape and size of the whole space leading to a fascinating interplay between geometry, topology and analysis. For example Einstein's theory is governed by a single field equation relating the Ricci curvature to the stress-energy tensor.
My own research in this vast field of Differential Geometry is centered around the problem of deforming curvature and investigating its stability properties under perturbations. I have worked on deformation problems associated to solving the Riemannian version of Einstein's equation and also on rigidity problems for the scalar curvature arising from the spinorial proof given by E.Witten for the Positive Mass Theorem in General Relativity. Recently, there has been a lot of activity in theoretical physics on the geometry of black holes, which bears some relation to the mathematical results that I was investigating. I have also became interested in using differential geometric methods in problems arising in probability and statistics, in particular, understanding statistical and quantum features of mechanical and geometrical problems in the large dimensional limit.
In conclusion, I would like to point out that in spite of its fascinating and intriguing relationship to Physics, Differential Geometry is a purely mathematical discipline, which can be pursued in its own right. It is certainly a very active and promising area of mathematical research with many important and interesting developments during the last 50 years. It is also a subject which interacts with many other branches of Mathematics. At an elementary level, it is based only on Calculus and Linear Algebra, the 2 basic Mathematics courses taught during the first 2 years at any University.
Mathematical Physics
I began to become interested in mathematics at a young age when I discovered that this was the language of the universe. My "Jugendtraum" was really to understand physics and astronomy. I feel lucky that my main research area, differential geometry is the main "lingua franca" of physics. Presently, I am working on calibrated cycles in manifolds with special holonomy, problems that show up in string theory and M-theory in theoretical physics. My "Alterstraum" is now to understand the AdS/CFT correspondence, in particular mirror symmetry.
FOR A SHORT INFORMAL INTRODUCTION TO GENERAL RELATIVITY BY JOHN (not Joan) BAEZ CLICK HERE
LIST OF PUBLICATIONS
Min-Oo: Kruemmung und differenzierbare Struktur auf reell-projektiven Raeumen; Diplom Arbeit, Bonn (1973)
Min-Oo: Kruemmung und differenzierbare Struktur auf komplex-projektiven Raeumen; Bonner Math. Schriften, vol.93, (1977).
Min-Oo and E.A.Ruh: Comparison theorems for compact symmetric spaces ; Ann. Sci. Ec. Norm. Sup. t.12 (1979), p.335-353.. pdf-file
Min-Oo and E.A.Ruh: Vanishing theorems and almost symmetric spaces of non-compact type ; Math. Ann. 257 (1981), p. 419-443. pdf-file
Min-Oo: An L2-Isolation Theorem for Yang-Mill fields ; Comp. Math. 47.2 (1982), p.153-163.
J.Dodziuk and Min-Oo: An L 2-Isolation Theorem for Yang-Mills fields over complete manifolds ; Comp. Math 47.2 (1982), p.165-16.
Min-Oo: Yang-Mills fields and the isoperimetric inequality; in Global Riemannian Geometry (Durham), E.Horwood Ltd.(1984).
Min-Oo: Maps of minimum energy from compact simply-connected Lie groups ; Ann.Global Anal. & Geom. 2(1984), p.119-128.
J.Bemelmans, Min-Oo and E.A.Ruh: Smoothing Riemannian metrics ; Math. Z. 188 (1984), p.69-74. pdf-file
Min-Oo: Integrabilitaet geometrischer Strukturen vom halb-einfachen Typ ; Habilitationsschrift, Bonn(1984).
Min-Oo and E.A.Ruh: An integrability condition for semi-simple Lie groups ; in Diff.Geom. & Complex Analysis, (Rauch memorial vol.) Springer (1985).
M.Min-Oo: Spectral rigidity for manifolds wit negative curvature operator ; Contemp. Math., vol.51 (1985) AMS, Providence
M.Min-Oo and E.A.Ruh: Curvature deformations ; Proc.Katata(Japan), Lect.Notes in Math. vol.1201, Springer (1985).
M.Min-Oo: Almost symmetric spaces ; Asterisque, vol. 163-164 (1988), p.221-246.
M.Min-Oo: Scalar curvature rigidity of asymptotically hyperbolic spin manifolds ; Math. Ann. vol.285 (1989), p.527-539. pdf-file
M.Min-Oo and E.A.Ruh: L2 -curvature pinching ;Comment. Helv. Math. vol.65 (1990), p.36-51. pdf-file
M.Min-Oo: Almost Einstein manifolds of negative Ricci curvature ; J. Diff. Geom. vol. 32 (1990), p.457-472.
P.Ghanaat, M.Min-Oo and E.A.Ruh: Local structure of Riemannian manifolds ; Indiana Univ. J. Math., vol.39.4 (1990) p.1305-1312. pdf-file
M.Min-Oo, E.A.Ruh and P.Tondeur: Vanishing theorems for the basic cohomology of Riemannian foliations ; J. reine u. ang. Math., vol. 415 (1991), p.167-174. pdf-file
M.Min-Oo, E.A.Ruh and P.Tondeur: A comparison theorem for almost Lie foliations ; Ann. Global Anal. & Geom., vol. 9.1 (1991), p.61-66.
M.Min-Oo, E.A.Ruh and P.Tondeur: Transversal curvature and tautness for Riemannian foliations; Lect. Notes in Math. vol.1481, p.145-146, Springer (1991).
J.Escobar, A.Freire and M.Min-Oo: L2 vanishing theorems in positive curvature ; Indiana Univ. J. Math, vol.42.4 (1993) p.1545-1554. pdf-file
P.March, M.Min-Oo, and E.A.Ruh: Mean Curvature of Riemannian Foliations ; Canadian Math. Bull. 39.1 (1996), p.95-105.
M. Min-Oo: Scalar curvature rigidity of certain symmetric spaces ; Geometry, topology, and dynamics (Montreal),p. 127--136, CRM Proc. Lecture Notes, 15, A. M. S.,1998. pdf-file
M.Lovric, M.Min-Oo and E.A.Ruh: Multivariate normal distributions parametrized as a Riemannian symmetric space; J. Multivariate Analysis, vol.74.1 (2000), pp. 36 - 48. pdf-file
M.Lovric, M.Min-Oo and E.A.Ruh: Deforming transversal Riemannian metrics of foliations ; Asian J. of Math., vol. 4.2 (2000), pp. 303 - 314. pdf-file
M.Min-Oo and J. A. Toth: The Levy concentration phenomenon for special functions on rank-one symmetric spaces ; Methods and Applications of Analysis, vol. 7.1 (2000), p. 151 - 164. pdf-file
A. Bourget, D. Jakobson, M.Min-Oo and J. A. Toth: A Law of large numbers for the zeros of Heine-Stieltjes polynomials ; Letters of Mathematical Physics, vol. 64.2 (2003), p. 105 -118. pdf-file
M. Min-Oo: Dirac Operator in Geometry and Physics in Global Riemannian Geometry: Curvature and Topology; Advanced courses in Mathematics, CRM Barcelona, Birkhauser (2003). pdf-file
M. Ionel, S. Karigiannis and M. Min-Oo: Bundle constructions of calibrated submanifolds in R^7 anf R^8; Mathematical Research Letters, vol. 12.4 (2005) pp. 493 - 512 . pdf-file
S. Karigiannis and M. Min-Oo: Calibrated sub-bundles in non-compact manifolds of special holonomy; Annals of Global Analysis and Geometry, vol. 28 (2005) pp. 371 - 394. pdf-file
H. Davaux and M. Min-Oo: Vafa-Witten bound on the complex-projective space, Annals of Global Analysis and Geometry, vol.30 (2006) pp. 29 - 36. pdf-file
M. Ionel and M. Min-Oo: Cohomogeneity one special lagrangian submanifolds in the deformed and resolved conifolds, Illinois J. Math., vol. 52.3 (2008), pp 839 - 865. pdf-file
G. Fan, M. Min-Oo and G.S.K. Wolkowicz: "Hopf bifurcation of delay differential equations with delay dependent parameters", Canadian Appl. Math. Quaterly, vol. 17.1 (2009), pp 37-60. pdf-file
PREPRINTS
A. Kolly, M. Min-Oo and E.A. Ruh: The Gauss-Bonnet-Chern Theorem (1999)
D. Egloff and M. Min-Oo: Convergence of Monte Carlo algorithms for pricing American Options (2002). pdf-file
M. Ionel and M. Min-Oo: Cohomogeneity one special lagrangian submanifolds in the deformed conifold (2005) pdf-file
M. Ionel and M. Min-Oo: Special Lagrangians of cohomogeneity one in the resolved conifold (2005) pdf-file