The Research of Thomas P. Branson

Michael G. Eastwood ; A. Rod Gover

Symmetry, integrability and geometry: methods and applications, Tome 4 (2008)

Résumé

The Midwest Geometry Conference 2007 was devoted to the substantial mathematical legacy of Thomas P. Branson who passed away unexpectedly the previous year. This contribution to the Proceedings briefly introduces this legacy. We also take the opportunity of recording his bibliography. Thomas Branson was on the Editorial Board of SIGMA and we are pleased that SIGMA is able to publish the Proceedings.

Détail

Keywords: conformal differential geometry; $Q$-curvature; spectrum geometry; intertwining operators; heat kernel asymptotics; global geometric invariants.

Bibliographie

[1] Branson T. P., The Yang–Mills equations: quasi-invariance, special solutions, and Banach manifold geometry, PhD dissertation, MIT, 1979

[2] Branson T. P., “Quasi-invariance of the Yang–Mills equations under conformal transformations and conformal vector fields”, J. Differential Geom., 16 (1981), 195–203 | MR

[3] Branson T. P., “Conformally covariant equations on differential forms”, Comm. Partial Differential Equations, 7 (1982), 393–431 | DOI | MR | Zbl

[4] Branson T. P., “Symplectic structure and conserved quantities for some new conformally covariant systems”, J. Differential Equations, 48 (1983), 35–59 | DOI | MR | Zbl

[5] Branson T. P., “Eventual partition of conserved quantities in wave motion”, J. Math. Anal. Appl., 96 (1983), 54–62 | DOI | MR | Zbl

[6] Branson T. P., Steeb W.-H., “Symmetries of nonlinear diffusion equations”, J. Phys. A: Math. Gen., 16 (1983), 469–472 | DOI | MR | Zbl

[7] Branson T. P., Kosmann-Schwarzbach Y., “Conformally covariant nonlinear equations on tensor-spinors”, Lett. Math. Phys., 7 (1983), 63–73 | DOI | MR | Zbl

[8] Branson T. P., “Conserved quantity partition for Dirac's equation”, Quart. Appl. Math., 42 (1984), 179–191 | MR | Zbl

[9] Branson T. P., “Intertwining differential operators for spinor-form representations of the conformal group”, Adv. in Math., 54 (1984), 1–21 | DOI | MR | Zbl

[10] Branson T. P., “Eventual partition of conserved quantities for Maxwell's equations”, Arch. Rational Mech. Anal., 86 (1984), 383–394 | DOI | MR | Zbl

[11] Branson T. P., “Differential operators canonically associated to a conformal structure”, Math. Scand., 57 (1985), 293–345 | MR | Zbl

[12] Branson T. P., Ørsted B., “Conformal indices of Riemannian manifolds”, Compositio Math., 60 (1986), 261–293 | MR | Zbl

[13] Branson T. P., “Group representations arising from Lorentz conformal geometry”, J. Funct. Anal., 74 (1987), 199–291 | DOI | MR | Zbl

[14] Branson T. P., Østed B., “Conformal deformation and the heat operator”, Indiana Univ. Math. J., 37 (1988), 83–110 | DOI | MR | Zbl

[15] Branson T. P., Østed B., “Generalized gradients and asymptotics of the functional trace”, Deformations of Mathematical Structures (Łódź/Lublin, 1985/1987), Kluwer Acad. Publ., Dordrecht, 1989, 247–262 | MR

[16] Branson T. P., “Conformal transformation, conformal change, and conformal covariants”, Proceedings of the $8^{\mathrm{th}}$ Winter School on Geometry and Physics (Srní, 1988), Rend. Circ. Mat. Palermo Suppl., 21, 1989, 115–134 | MR | Zbl

[17] Branson T. P., Gilkey P. B., “The asymptotics of the Laplacian on a manifold with boundary”, Comm. Partial Differential Equations, 15 (1990), 245–272 | DOI | MR | Zbl

[18] Branson T. P., Gilkey P. B., Ørsted B., “Leading terms in the heat invariants”, Proc. Amer. Math. Soc., 109 (1990), 437–450 | DOI | MR | Zbl

[19] Branson T. P., Gilkey P. B., Ørsted B., “Leading terms in the heat invariants for the Laplacians of the de Rham, signature, and spin complexes”, Math. Scand., 66 (1990), 307–319 | MR

[20] Branson T. P., Ólafsson G., “Equipartition of energy for waves in symmetric space”, J. Funct. Anal., 97 (1991), 403–416 | DOI | MR | Zbl

[21] Branson T. P., Østed B., “Conformal geometry and global invariants”, Differential Geom. Appl., 1 (1991), 279–308 | DOI | MR | Zbl

[22] Branson T. P., Østed B., “Explicit functional determinants in four dimensions”, Proc. Amer. Math. Soc., 113 (1991), 669–682 | DOI | MR | Zbl

[23] Gilkey P. B., Branson T. P., Fulling S. A., “Heat equation asymptotics of “nonminimal” operators on differential forms”, J. Math. Phys., 32 (1991), 2089–2091 | DOI | MR | Zbl

[24] Branson T. P., “Harmonic analysis in vector bundles associated to the rotation and spin groups”, J. Funct. Anal., 106 (1992), 314–328 | DOI | MR | Zbl

[25] Branson T. P., Gilkey P. B., “Residues of the eta function for an operator of Dirac type”, J. Funct. Anal., 108 (1992), 47–87 | DOI | MR | Zbl

[26] Branson T. P., Gilkey P. B., Ørsted B., Pierzchalski A., “Heat equation asymptotics of a generalized Ahlfors Laplacian on a manifold with boundary”, Operator Calculus and Spectral Theory (Lambrecht, 1991), Birkhäuser, Basel, 1992, 1–13 | MR | Zbl

[27] Branson T. P., Chang S.-Y. A., Yang P. C., “Estimates and extremals for zeta function determinants on four-manifolds”, Comm. Math. Phys., 149 (1992), 241–262 | DOI | MR | Zbl

[28] Branson T. P., Gilkey P. B., “Residues of the eta function for an operator of Dirac type with local boundary conditions”, Differential Geom. Appl., 2 (1992), 249–267 | DOI | MR

[29] Branson T. P., The functional determinant, Global Analysis Research Center Lecture Notes Series, 4, Seoul National University, 1993 | MR | Zbl

[30] Branson T. P., Gilkey P. B., Pierzchalski A., “Heat equation asymptotics of elliptic operators with nonscalar leading symbol”, Math. Nachr., 166 (1994), 207–215 | DOI | MR | Zbl

[31] Branson T. P., Gilkey P. B., “The functional determinant of a four-dimensional boundary value problem”, Trans. Amer. Math. Soc., 344 (1994), 479–531 | DOI | MR | Zbl

[32] Branson T. P., Ólafsson G., Schlichtkrull H., “A bundle valued Radon transform, with applications to invariant wave equations”, Quart. J. Math. Oxford Ser. (2), 45 (1994), 429–461 | DOI | MR | Zbl

[33] Branson T. P., Gilkey P. B., Pohjanpelto J., “Invariants of locally conformally flat manifolds”, Trans. Amer. Math. Soc., 347 (1995), 939–953 | DOI | MR | Zbl

[34] Branson T. P., Ólafsson G., Schlichtkrull H., “Huygens' principle in Riemannian symmetric spaces”, Math. Ann., 301 (1995), 445–462 | DOI | MR | Zbl

[35] Blažić N., Bokan N., Branson T. P., Gilkey P. B., “When the leading terms in the heat equation asymptotics are coercive”, Houston J. Math., 21 (1995), 75–82 | MR | Zbl

[36] Branson T. P., “Sharp inequalities, the functional determinant, and the complementary series”, Trans. Amer. Math. Soc., 347 (1995), 3671–3742 | DOI | MR | Zbl

[37] Branson T. P., Ólafsson G., Ørsted B., “Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup”, J. Funct. Anal., 135 (1996), 163–205 | DOI | MR | Zbl

[38] Branson T. P., “An anomaly associated with $4$-dimensional quantum gravity”, Comm. Math. Phys., 178 (1996), 301–309 | DOI | MR | Zbl

[39] Branson T. P., “Nonlinear phenomena in the spectral theory of geometric linear differential operators”, Quantization, Nonlinear Partial Differential Equations, and Operator Algebra (Cambridge, MA, 1994), Proc. Sympos. Pure Math., 59, Amer. Math. Soc., 1996, 27–65 | MR | Zbl

[40] Branson T. P., Gilkey P. B., “The functional determinant in the standard conformal class and four dimensional balls and spherical shells”, Proceedings of the $24^{\mathrm{th}}$ National Conference on Geometry and Topology (Timisoara, 1994), Editura Mirton, 1999, 59–80 | MR

[41] Branson T. P., “Spectral theory of invariant operators, sharp inequalities, and representation theory”, Proceedings of the $16^{\mathrm{th}}$ Winter School on Geometry and Physics (Srní, 1996), Rend. Circ. Mat. Palermo Suppl., 46, 1997, 29–54 | MR | Zbl

[42] Branson T. P., Ólafsson G., “Helmholtz operators and symmetric space duality”, Invent. Math., 129 (1997), 63–74 | DOI | MR | Zbl

[43] Branson T. P., Gilkey P. B., Vassilevich D. V., “The asymptotics of the Laplacian on a manifold with boundary. II”, Boll. Un. Mat. Ital. B (7), 11, no. 2, suppl. (1997), 39–67 ; hep-th/9504029 | MR | Zbl

[44] Branson T. P., Lano R. P., Rodgers V. G. J., “Yang–Mills, gravity, and string symmetries”, Phys. Lett. B, 412 (1997), 253–258 ; hep-th/9610023 | DOI | MR

[45] Branson T. P., Hijazi O., “Vanishing theorems and eigenvalue estimates in Riemannian spin geometry”, Internat. J. Math., 8 (1997), 921–934 | DOI | MR | Zbl

[46] Branson T. P., “Stein–Weiss operators and ellipticity”, J. Funct. Anal., 151 (1997), 334–383 | DOI | MR | Zbl

[47] Branson T. P., Gilkey P. B., Vassilevich D. V., “Vacuum expectation value asymptotics for second order differential operators on manifolds with boundary”, J. Math. Phys., 39 (1998), 1040–1049 ; Erratum J. Math. Phys., 41 (2000), 3301 ; hep-th/9702178 | DOI | MR | Zbl | DOI | MR

[48] Branson T. P., “Second order conformal covariants”, Proc. Amer. Math. Soc., 126 (1998), 1031–1042 | DOI | MR | Zbl

[49] Branson T. P., Ólafsson G., “Asymptotics of the d'Alembertian with potential on a pseudo-Riemannian manifold”, Proc. Amer. Math. Soc., 127 (1999), 1339–1345 | DOI | MR | Zbl

[50] Branson T. P., “Spectra of self-gradients on spheres”, J. Lie Theory, 9 (1999), 491–506 | MR | Zbl

[51] Branson T. P., Gilkey P. B., Kirsten K., Vassilevich D. V., “Heat kernel asymptotics with mixed boundary conditions”, Nuclear Phys. B, 563 (1999), 603–626 ; hep-th/9906144 | DOI | MR | Zbl

[52] Branson T. P., Spectral problems in geometry and arithmetic: preface Spectral Problems in Geometry and Arithmetic (Iowa City, 1997), Contemp. Math., 237, 1999, ix–x | MR | Zbl

[53] Branson T. P., “Kato constants in Riemannian geometry”, Math. Res. Lett., 7 (2000), 245–261 | MR | Zbl

[54] Branson T. P., Hijazi O., “Improved forms of some vanishing theorems in Riemannian spin geometry”, Internat. J. Math., 11 (2000), 291–304 | MR | Zbl

[55] Branson T. P., Rodgers V. G. J., Yasuda T., “Interactions of a string inspired graviton field”, Internat. J. Modern Phys. A, 15 (2000), 3549–3562 ; hep-th/9812098 | Zbl

[56] Branson T. P., “Automated symbolic computation in spin geometry”, Clifford Analysis and Its Applications (Prague, 2000), NATO Sci. Ser. II Math. Phys. Chem., 25, Kluwer, 2001, 27–38 | MR | Zbl

[57] Avramidi I. G., Branson T. P., “Heat kernel asymptotics of operators with non-Laplace principal part”, Rev. Math. Phys., 13 (2001), 847–890 ; math-ph/9905001 | DOI | MR | Zbl

[58] Branson T. P., Gover A. R., “Conformally invariant non-local operators”, Pacific J. Math., 201 (2001), 19–60 | DOI | MR | Zbl

[59] Bennett C., Branson T. P., “Curvature actions on $\mathrm{Spin}(n)$ bundles”, Adv. Appl. Clifford Algebras, 11 (2001), 93–120 ; hep-th/0109188 | DOI | MR

[60] Branson T. P., Hijazi O., “Bochner–Weitzenböck formulas associated with the Rarita–Schwinger operator”, Internat. J. Math., 13 (2002), 137–182 ; hep-th/0110014 | DOI | MR | Zbl

[61] Avramidi I., Branson T. P., “A discrete leading symbol and spectral asymptotics for natural differential operators”, J. Funct. Anal., 190 (2002), 292–337 | DOI | MR | Zbl

[62] Branson T. P., Gover A. R., “A conformally invariant differential operator on Weyl tensor densities”, J. Geom. Phys., 42 (2002), 283–295 ; hep-th/0109210 | DOI | MR | Zbl

[63] Branson T. P., Gover A. R., “Electromagnetism, metric deformations, ellipticity and gauge operators on conformal $4$-manifolds”, Differential Geom. Appl., 17 (2002), 229–249 ; hep-th/0111003 | DOI | MR | Zbl

[64] Branson T. P., “Clifford bundles and Clifford algebras”, Lectures on Clifford (Geometric) Algebras and Applications, Birkhäuser, 2004, 157–188 | MR

[65] Branson T. P., “Conformal structure and spin geometry”, Dirac Operators: Yesterday and Today, Int. Press, 2005, 163–191 | MR | Zbl

[66] Branson T. P., “$Q$-curvature and spectral invariants”, Proceedings of the $24^{\mathrm{th}}$ Winter School on Geometry and Physics (Srní, 2004), Rend. Circ. Mat. Palermo Suppl., 75, 2005, 11–55 | MR | Zbl

[67] Branson T. P., Gover A. R., “Conformally invariant operators, differential forms, cohomology and a generalisation of $Q$-curvature”, Comm. Partial Differential Equations, 30 (2005), 1611–1669 ; math.DG/0309085 | DOI | MR | Zbl

[68] Branson T. P., Ólafsson G., Pasquale A., “The Paley–Wiener theorem and the local Huygens' principle for compact symmetric spaces: the even multiplicity case”, Indag. Math., 16 (2005), 393–428 ; math.AP/0411383 | DOI | MR | Zbl

[69] Branson T. P., Ólafsson G., Pasquale A., “The Paley–Wiener theorem for the Jacobi transform and the local Huygens' principle for root systems with even multiplicities”, Indag. Math., 16 (2005), 429–442 ; math.AP/0508234 | DOI | MR | Zbl

[70] Branson T. P., Čap A., Eastwood M. G., Gover A. R., “Prolongations of geometric overdetermined systems”, Internat. J. Math., 17 (2006), 641–664 ; math.DG/0402100 | DOI | MR | Zbl

[71] Branson T. P., Ørsted B., “Spontaneous generation of eigenvalues”, J. Geom. Phys., 56 (2006), 2261–2278 ; math.DG/0506047 | DOI | MR | Zbl

[72] Branson T. P., Choi Y. H., “Option pricing on multiple assets”, Acta Appl. Math., 94 (2006), 137–162 | DOI | MR | Zbl

[73] Branson T. P., Hong D., “Spectrum generating on twistor bundle”, Arch. Math. (Brno), 42, suppl. (2006), 169–183 ; math.DG/0606524 | MR | Zbl

[74] Branson T. P., Gover A. R., “The conformal deformation detour complex for the obstruction tensor”, Proc. Amer. Math. Soc., 135 (2007), 2961–2965 ; math.DG/0605192 | DOI | MR | Zbl

[75] Branson T. P., Villanueva A., “Symmetries in differential geometry: a computational approach to prolongations”, Acta Appl. Math., 98 (2007), 63–80 ; arXiv:0705.0764 | DOI | MR | Zbl

[76] Branson T. P., “$Q$-curvature, spectral invariants, and representation theory”, SIGMA, 3 (2007), 90, 31 pp., ages ; arXiv:0707.2471 | MR

[77] Branson T. P., Hong D., “Translation to bundle operators”, SIGMA, 3 (2007), 102, 14 pp., ages ; math.DG/0606552 | MR | Zbl

[78] Branson T. P., Gover A. R., “Pontrjagin forms and invariant objects related to the $Q$-curvature”, Commun. Contemp. Math., 9 (2007), 335–358 ; math.DG/0511311 | DOI | MR | Zbl

[79] Branson T. P., Gover A. R., “Variational status of a class of fully nonlinear curvature prescription problems”, Calc. Var. Partial Differential Equations, 32 (2008), 253–262 ; math.DG/0610773 | DOI | MR | Zbl

[80] Branson T. P., Fontana L., Morpurgo C., Moser–Trudinger and Beckner–Onofri's inequalities on the CR sphere, arXiv:0712.3905