Abstract
In this article we want to present a meeting ground between rather different mathematical topics from numerical analysis to index theory and symplectic geometry. The unifying idea is that of linear representation of tori inside which the combinatorics, analysis and geometry is developed.
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Atiyah M.: Elliptic operators and compact groups. In: Lecture Notes in Mathematics, vol 201. Springer, Berlin (1974)
Atiyah, M.F.: K-theory. Advanced Book Classics, 2nd edn. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989). Notes by D.W. Anderson
Atiyah M.F., Singer I.M.: The index of elliptic operators. I. Ann. Math. 87, 484–530 (1968)
Atiyah M.F., Singer I.M.: The index of elliptic operators. III. Ann. Math. (2) 87 (2), 546–604 (1968)
Atiyah M.F., Singer I.M.: Index theory for skew-adjoint Fredholm operators. Inst. Hautes Études Sci. Publ. Math. 37, 5–26 (1969)
Atiyah M.F., Singer I.M.: The index of elliptic operators. IV . Ann. Math. (2) 93 (2), 119–138 (1971)
Atiyah M.F., Singer I.M.: The index of elliptic operators. V. Ann. Math. (2) 93 (2), 139–149 (1971)
Berline N., Vergne M.: The Chern character of a transversally elliptic symbol and the equivariant index. Invent. Math. 124(1–3), 11–49 (1996)
Berline N., Vergne M.: L’indice équivariant des opérateurs transversalement elliptiques. Invent. Math. 124(1–3), 51–101 (1996)
Brion M., Vergne M.: Residue formulae, vector partition functions and lattice points in rational polytopes. J. Am. Math. Soc. 10(4), 797–833 (1997)
Brion M., Vergne M.: Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue. Ann. Sci. École Norm. Sup. (4) 32(5), 715–741 (1999)
Coutinho S.C.: A primer of algebraic D-modules. In: London Mathematical Society Student Texts, vol. 33. Cambridge University Press, Cambridge (1995)
Dahmen W., Micchelli C.A.: Translates of multivariate splines. Linear Algebra Appl. 52/53, 217–234 (1983)
Dahmen W., Micchelli C.A.: The number of solutions to linear Diophantine equations and multivariate splines. Trans. Am. Math. Soc. 308(2), 509–532 (1988)
de Boor C., Höllig K., Riemenschneider S.: Box splines. In: Applied Mathematical Sciences, vol. 98. Springer, New York (1993)
De Concini C., Procesi C.: Wonderful models of subspace arrangements. Selecta Math. (N.S.) 1(3), 459–494 (1995)
De Concini, C., Procesi, C.: Nested sets and Jeffrey-Kirwan residues. In: Geometric Methods in Algebra and Number Theory. Progr. Math., vol. 235, pp. 139–149. Birkhäuser Boston, Boston (2005)
De Concini, C., Procesi, C., Vergne, M.: The infinitesimal index. arXiv:1003.3525
De Concini C., Procesi C., Vergne M.: Partition function and generalized Dahmen-Micchelli spaces. Transf. Groups 15(4), 751–773 (2010)
De Concini C., Procesi C., Vergne M.: Vector partition functions and index of transversally elliptic operators. Transf. Groups 15(4), 775–811 (2010)
De Concini, C., Procesi, C., Vergne, M.: Box splines and the equivariant index theorem. arXiv:1012. 1049
De Concini, C., Procesi, C., Vergne, M.: Infinitesimal index: cohomology computations. arXiv:1005. 0128
De Concini C., Procesi C.: Topics in Hyperplane Arrangements, Polytopes and Box-Splines. Universitext. Springer, New York (2011)
Guillemin, V.W., Sternberg, S.: Supersymmetry and equivariant de Rham theory. In: Mathematics Past and Present. Springer, Berlin (1999). With an appendix containing two reprints by Henri Cartan [MR0042426 (13,107e); MR0042427 (13,107f)]
Hirzebruch, F.: Topological methods in algebraic geometry. In: Classics in Mathematics. Springer, Berlin (1995). Translated from the German and Appendix One by R.L.E. Schwarzenberger, With a preface to the third English edition by the author and Schwarzenberger, Appendix Two by A. Borel, Reprint of the 1978 edition
Jeffrey L.C., Kirwan F.C.: Localization for nonabelian group actions. Topology 34(2), 291–327 (1995)
Paradan, P.-E., Vergne, M.: The index of transversally elliptic operators. Bull. Soc. Math. France. (to appear). arXiv:0804.1225
Paradan, P.-É.: Jump formulas in Hamiltonian geometry. arXiv:math/0411306
Paradan P.-É.: Localization of the Riemann–Roch character. J. Funct. Anal. 187, 442–509 (2001)
Paradan, P.-É., Vergne, M.: Index of transversally elliptic operators. arXiv:math 0805.1225
Schoenberg, I.J.: Cardinal Spline Interpolation. Society for Industrial and Applied Mathematics, Philadelphia (1973). Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12
Segal G.: Equivariant K-theory. Inst. Hautes Études Sci. Publ. Math. 34, 129–151 (1968)
Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall Inc., Englewood Cliffs (1973). Prentice-Hall Series in Automatic Computation
Szenes, A., Vergne, M.: Residue formulae for vector partitions and Euler-MacLaurin sums. Adv. Appl. Math. 30(1–2), 295–342 (2003). Formal Power Series and Algebraic Combinatorics (Scottsdale, AZ, 2001)
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Communicated by E. Zelmanov.
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Procesi, C. Splines and index theorem. Bull. Math. Sci. 2, 57–123 (2012). https://doi.org/10.1007/s13373-011-0013-4
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DOI: https://doi.org/10.1007/s13373-011-0013-4