Abdelmalek Abdesselam, University of Virginia Research Page


Areas of interest :

I am a mathematical physicist working on:

  • Constructive quantum field theory.
  • Rigorous renormalization group methods.
  • Mathematical statistical mechanics.
  • Combinatorics related to Feynman diagrams and cluster expansions.
  • Classical invariant theory, and applications of the classical symbolic method to problems in algebraic geometry and representation theory.

  • Editorial activities :

    Annales Henri Poincaré.

    Journal of Mathematical Physics.

    Annales de l'Institut Henri Poincaré D, Combinatorics, Physics and their Interactions.


    Membership in professional societies:

    International Association of Mathematical Physics .

    Americal Mathematical Society.


    Organizer for the Mathematical Physics Seminar and co-organizer for the Mathematics Department Colloquium, at the University of Virginia.

    Organizer for the workshop "Harmonic Analysis and the Renormalization Group", April 18-21 2014, to be held at the University of Virginia, Charlottesville.

    Co-organizer for the upcoming thematic program "Combinatorics, geometry and physics", June 2-July 31 2014, to be held at the Erwin Schrödinger International Institute for Mathematical Physics, Vienna.

    Co-organizer for the conference "Recent Mathematical Advances in Classical, Quantum, and Statistical Mechanics" held at the University of Virginia, March 8-9 2013, on the occasion of the retirement of Larry Thomas.

    Co-organizer for the Quantum Field Theory Session at the XVIIth International Congress on Mathematical Physics held in Aalborg, Denmark, August 6-11 2012.

    April 2008 Cambridge conference I co-organized : Combinatorial Identities and Their Applications in Statistical Mechanics.

    Some links and references on forest and tree formulae in constructive field theory.


    Preprints :

    - A. A., C. Ikenmeyer and G. Royle, "16,051 formulas for Ottaviani's invariant of cubic threefolds", arXiv:1402.2669[math.AG].

    - A. A., "QFT, RG, and all that, for mathematicians, in eleven pages", arXiv:1311.4897[math.PR].

    - A. A., A. Chandra and G. Guadagni, "Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions", arXiv:1302.5971[math.PR].

    - A. A., A. Chandra and G. Guadagni, "Rigorous quantum field theory functional integrals over the p-adics: research announcement", arXiv:1210.7717[math.PR].

    - A. A. and G. W. Anderson, "Counting colored planar maps free-probabilistically", arXiv:1203.3185v2[math.PR].

    - A. A., and D. C. Brydges, "Cramer's rule and loop ensembles", preliminary version, Oct 2006, (.pdf) . This is a review of a result of G. X. Viennot which we think is important for statistical mechanics.


    Slides of talks :

    - A. A., "On the ground state energy for massless Nelson models", slides of talk given at the conference "QMath11 Mathematical Results in Quantum Physics", Hradec Kralove, Czech Republic, September 2010.

    - A. A., "Renormalization group trajectories between two fixed points", slides of invited lecture at the International Congress on Mathematical Physics, Prague, August 2009.

    - A. A., "Field theoretic cluster expansions and the Brydges-Kennedy forest sum formula", slides of my April 2008 talk for the conference "Combinatorial Identities and Their Applications in Statistical Mechanics", at the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK.

    - A. A., "The combinatorics of classical invariant theory revisited by modern physics", slides of my Feb 2007 talk at the Montreal CRM workshop "Combinatorial Problems Raised by Statistical Mechanics".

    - A. A., "On the Hadamard-Foulkes-Howe conjecture and multidimensional resultants", slides of my Sept 2005 talk at the 55th Séminaire Lotharingien de Combinatoire in Bertinoro.

    - A. A., "Combinatoire du logarithme et renormalisation en physique", slides of my May 2003 talk at the Journées du LIPN Combinatoire, Informatique er Physique, at Université Paris 13.

    - A. A., "Multivariable power series, reversion and Lagrange-Good inversion via quantum field theory", slides of my Oct 2002 talk at the 49th Séminaire Lotharingien de Combinatoire in Ellwangen.


    Published articles :

    1 - A. A. and J. Chipalkatti, "On Hilbert covariants". Canadian J. Math. 66 (2014), no 1, 3--30. Preprint version. Journal version.

    2 - A. A. and J. Chipalkatti, "Quadratic involutions on binary forms". Michigan Math. J. 61 (2012), no. 2, 279--296. Preprint version. Journal version.

    3 - A. A., "On the volume conjecture for classical spin networks". J. Knot Theory Ramifications 21 (2012), no. 3, 1250022 (62 pages). Preprint version. Journal version.

    4 - A. A. and D. Hasler, "Analyticity of the ground state energy for massless Nelson models". Comm. Math. Phys. 310 (2012), no. 2, 511--536. Preprint version. Journal version.

    5 - A. A., "A massless quantum field theory over the p-adics", contribution to the Oberwolfach Report for the 2011 meeting "The Renormalization Group" organized by: M. Disertori, J. Feldman and M. Salmhofer. Oberwolfach Rep. 8 (2011), no. 1, pp. 785--788. Journal version. Oberwolfach version. Preprint version.

    6 - A. A., "The ground state energy of the massless spin-Boson model". Ann. Henri Poincaré 12 (2011), no. 7, 1321--1347. Preprint version. Journal version.

    7 - A. A., "Renormalization group trajectories between two fixed points". In: "XVIth International Congress on Mathematical Physics, Prague, Czech Republic, 3 -- 8 August 2009", Ed.: Pavel Exner, 346--351, World Scientific, Singapore, 2010. Preprint version. Publisher page.

    8 - A. A., and J. Chipalkatti, "The Higher Transvectants are Redundant". Ann. Inst. Fourier (Grenoble) 59 (2009), no. 5, 1671--1713. Preprint version . Journal version.

    9 - A. A., A. Procacci, and B. Scoppola, "Clustering bounds on n-point correlations for unbounded spin systems". J. Stat. Phys. 136 (2009), no. 3, 405--452. Preprint version . Journal version.

    10 - A. A., and J. Chipalkatti, "On the Linear Combinants of a Binary Pencil". Glasgow Math. J. 51 (2009), no. 3, 481--498. Preprint version . Journal version.

    11 - A. A., "A complete renormalization group trajectory between two fixed points". Comm. Math. Phys. 276 (2007), no. 3, 727--772. Preprint version . Journal version.

    12 - A. A., "Towards a complete renormalization group trajectory between two fixed points", contribution to the Oberwolfach Report for the 2006 meeting "The Rigorous Renormalization Group" organized by: C. Kopper, V. Rivasseau and M. Salmhofer. Oberwolfach Rep. 3 (2006), no. 2, 1061--1063. Journal version. Oberwolfach version.

    13 - A. A., Short appendix for the article "On the Jacobian ideal of the binary discriminant" by C. D'Andrea and J. Chipalkatti. Collect. Math. 58 (2007), no. 2, 155--180. Preprint version . Journal version.

    14 - A. A., and J. Chipalkatti, "On the Wronskian combinants of binary forms". J. Pure Appl. Algebra 210 (2007), no. 1, 43--61. Preprint version . Journal version.

    15 - A. A., and J. Chipalkatti, "Brill-Gordan Loci, transvectants and an analogue of the Foulkes conjecture". Adv. Math. 208 (2007), no. 2, 491--520. Preprint version . Journal version.

    16 - A. A., and J. Chipalkatti, "The bipartite Brill-Gordan locus and angular momentum". Transform. Groups 11 (2006), no. 3, 341--370. Preprint version . Journal version.

    17 - A. A., "A computational solution to a question by Beauville on the invariants of the binary quintic". J. Algebra 303 (2006), no. 2 (Computational Algebra), 771--788. Preprint version . Journal version.

    18 - A. A., "The Grassmann-Berezin calculus and theorems of the matrix-tree type". Adv. in Appl. Math. 33 (2004), no. 1, 51--70. Preprint version . Journal version.

    19 - A. A., "Feynman diagrams in algebraic combinatorics". Sém. Lothar. Combin. 49 (2002/04), Art. B49c, 45 pp. (electronic). Preprint version . Journal version.

    20 - A. A., "A physicist's proof of the Lagrange-Good multivariable inversion formula". J. Phys. A 36 (2003), no. 36, 9471--9477. Preprint version . Journal version.

    21 - A. A., "The Jacobian conjecture as a problem of perturbative quantum field theory". Ann. Henri Poincaré (2003), no. 2, 199--215. Preprint version . Journal version.

    22 - A. A., J. Magnen, and V. Rivasseau, "Bosonic monocluster expansion". Comm. Math. Phys. 229 (2002), no. 2, 183--207. Preprint version . Journal version.

    23 - A. A., and V. Rivasseau, "Explicit Fermionic tree expansions". Lett. Math. Phys. 44 (1998), no. 1, 77--88. Preprint version . Journal version.

    24 - A. A., and V. Rivasseau, "An explicit large versus small field multiscale cluster expansion". Rev. Math. Phys. 9 (1997), no. 2, 123--199. Preprint version . Journal version.

    25 - A. A., and V. Rivasseau, "Trees, forests and jungles: a botanical garden for cluster expansions". Constructive physics (Palaiseau, 1994), 7--36, Lecture Notes in Phys., 446, Springer, Berlin, 1995. Preprint version . Publisher page.

    Ph.D. Thesis :

    "Explicit constructive renormalization" june 1997, (in French). Chapter I has a quick introduction to BPHZ renormalization from a mathematical standpoint. The papers in Chapters II and III which have been published since, have been removed (see hep-th/9409094 for the preprint version of the paper in Chapter II and hep-th/9605094 for that in Chapter III). Chapter IV presents a nonperturbative analog of Zimmermann's forest formula.

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