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Marco Bertola, PhD

Full Professor (on Sabbatical leave until July 2020), Mathematics and Statistics


Marco Bertola, PhD
Phone: (514) 848-2424 ext. 3228
Email: marco.bertola@concordia.ca

Education

Ph.D.:  SISSA-ISAS, Trieste, Italy 1999

Research interests

Mathematical Physics; Integrable systems; Inverse problems; Asymptotics in nonlinear integrable equations; Moduli spaces.

Representation of the theoretical behaviour of Nonlinear Schroedinger equation (representing amplitude of pulsed lasers in optical fibres, for example) near the ``gradient catastrophe''. The maximal gain of the chirps relative to the background is 3. This also is used to describe qualitatively the occurrence of solitary "monster waves" in the ocean.
Photo credit: M. Bertola, A. Tovbis, ”Universality for the focusing nonlinear Schr¨odinger equation at the gradient catastrophe point: rational breathers and poles of the Tritronqu´ee solution to Painlev´e I ”, Comm. Pure Appl. Math. 66 (2013) no.5, 678–752. doi:10.1002/cpa.21445
The dots are the (very regular) pattern of zeros of a "generalized Vorob'ev Yablonski polynomial" and also represents the set of poles of a special rational solution of the second Painlevé hierarchy of equations. The triangular array of dots near the corners is related to the position of the spikes in the other figure.
Photo credit: F. Balogh, M. Bertola, T. Bothner, ”Hankel Determinant Approach to Generalized Vorob’ev–Yablonski Polynomials and Their Roots”, Constr. Approx.,44 (2016), no 3, 417–453 doi:10.1007/s00365-016-9328-4
Amplitude of the quasi-periodic structures in certain solution of the Nonlinear Schrödinger equations, used in modelling rogue waves.
Photo credit: M. B. G. A. El, A. Tovbis, ``Rogue waves in multiples solutions of the focusing nonlinear Schrödinger equation", Proceedings A, 473, o, 2194 (2016)

Publications

2022

  1. M. Bertola, S. Tarricone, “Stokes’ manifolds and Cluster Algebras”, accepted, Comm. Math. Phys. doi.org/10.1007/s00220-021-04293-7

  2. M. Bertola, “Nonlinear steepest descent approach to orthogonality on elliptic curves”, Jour. Approx. Theory, (accepted) 2022.

2021

1.     M. Bertola, D. Korotkin, “Tau functions and Monodromy Symplectomorphisms” accepted Comm. Math. Phys. doi.org/10.1007/s00220-021-04224-6

2.  M.Bertola, D. Korotkin, “Extended Goldman symplectic structure in Fock-Goncharov coordinates”, Journal of Differential Geometry, accepted March 24 2021.

3.     M. Bertola, A. Katsevich, A. Tovbis, “On the spectral properties of the Hilbert transform operator on multi-intervals”, Journal of Spectral Theory, (2021), accepted September 24, 2021.

4.     A. Katsevich, M. Bertola, A. Tovbis, “Inversion formula and range conditions for a linear system related with the multi-interval finite Hilbert transform in L2”, Mathematische Nachtrischten, 8, no. 294 (2021), 1523–1546. doi.org/10.1002/mana.201800567

5.     F. Mobasheramini, M. Bertola, “Quantization of Calogero-Painlev´e system and multi-particle quantum Painlev´e equations II-VI. SIGMA Symmetry Integrability Geom. Methods Appl. 17 (2021), Paper No. 081. doi.org/10.3842/SIGMA.2021.081

6.     M. Bertola; “Padé approximants on Riemann surfaces and KP tau functions”. Anal. Math. Phys. 11 (2021), no. 4, Paper No. 149. doi.org/10.1007/s13324-021-00585-2

7.     M. Bertola, D. Korotkin; “Spaces of Abelian Differentials and Hitchin’s Spectral Covers”. Int. Math. Res. Not. IMRN 2021, no. 15, 11246–11269. doi.org/10.1093/imrn/rnz142

8.     M. Bertola, B. Dubrovin, Di Yang; “Simple Lie algebras, Drinfeld-Sokolov hierarchies, and multi-point correlation functions”. Mosc. Math. J. 21 (2021), no. 2, 233–270.

9.     M. Bertola, D. Korotkin; “WKB expansion for a Yang-Yang generating function and the Bergman tau function”. (Russian) Teoret. Mat. Fiz. 206 (2021), no. 3, 295–338. doi.org/10.4213/tmf9834

10.   M. Bertola, G. Ruzza; “Matrix models for stationary Gromov-Witten invariants of the Riemann sphere”. Nonlinearity 34 (2021), no. 2, 1168–1196. doi.org/10.1088/1361-6544/abd85e

11.   M. Bertola, “Correction to: The dependence on the monodromy data of the isomonodromic tau function”. Comm. Math. Phys. 381 (2021), no. 3, 1445–1461. doi.org/10.1007/s00220-009-0961-7


2020

1.     M. Bertola, D. Korotkin; “Hodge and Prym tau functions, Strebel differentials and combinatorial model of Mg,n”. Comm. Math. Phys. 378 (2020), no. 2, 1279–1341. doi.org/10.1007/s00220-020-03819-9

2.     M. Bertola, E. Blackstone, A. Katsevich, A. Tovbis; “Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach”. Anal. Math. Phys. 10 (2020), no. 3, Paper No. 27, 59 pp. doi.org/10.1007/s13324-020-00371-6

3.     M. Bertola, J. Harnad, B. Runov; “Generating weighted Hurwitz numbers”. J. Math. Phys. 61(2020), no. 1, 013506, 16 pp. doi.org/10.1063/1.5130554

2019

1.     M. Bertola, G. Ruzza; “Brezin-Gross-Witten tau function and isomonodromic deformations”. Commun. Number Theory Phys. 13 (2019), no. 4, 827–883.

dx.doi.org/10.4310/CNTP.2019.v13.n4.a4

2.     M. Bertola, A. Minakov; “Laguerre polynomials and transitional asymptotics of the modified Korteweg– de Vries equation for step-like initial data”. Anal. Math. Phys. 9 (2019), no. 4, 1761–1818. https://doi.org/10.1007/s13324-018-0273-1

3.     M. Bertola, J. Harnad; “Rationally weighted Hurwitz numbers, Meijer G-functions and matrix integrals”. J. Math. Phys. 60 (2019), no. 10, 103504, 15 pp. https://doi.org/10.1063/1.5099239

4.     M. Bertola, G. Ruzza, “The Kontsevich-Penner matrix integral, isomonodromic tau functions and open intersection numbers”. Ann. Henri Poincar´e 20 (2019), no. 2, 393–443.

doi-org/10.1007/s00023-018-0737-8

5.     M.Bertola, D.Korotkin; “Spaces of abelian differentials and Hitchin’s spectral covers.” International Mathematics Research Notices (2019) doi.org/10.1093/imrn/rnz142

6.     A. Katsevich, M. Bertola, A. Tovbis, “Spectral analysis and decomposition of normal operators related with the multi–interval finite Hilbert transform”, Contemporary Mathematics, 734 (2019), 171–192.

doi.org/10.1090/conm/734/14771

2018

1.     M. Bertola, M. Cafasso, V. Rubstov, “Noncommutative Painlev´e Equations and Systems of Calogero Type”, Comm. Math. Phys, 363, no.2, (2018) 503–530. doi.org/10.1007/s00220-018-3210-0

2.     M. Bertola, J. G. Elias Rebelo, T. Grava, “Painlev´e IV critical asymptotics for orthogonal polynomials in the complex plane”. SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), 091, 34 pages.

3.     M. Bertola, B. Dubrovin, D. Yang, “Simple Lie algebras and topological ODEs”. Int. Math. Res. Not. IMRN 2018, no. 5, 1368-1410, doi.org/10.1093/imrn/rnw285

4.     M. Bertola, D. Korotkin, “Discriminant circle bundles over local models of Strebel graphs and Boutroux curves” Teoret. Mat. Fiz. 197 (2018), no. 2, 163–207. DOI: 10.1134/S0040577918110016

2017

  1.             “Universal Peregrine soliton structure in nonlinear pulse compression in optical fiber” A. Tikan, C. Billet, G. El, A. Tovbis, M. Bertola, T. Sylvestre, F. Gustave, S. Randoux, G. Genty, P. Suret, J. M. Dudley, Phys. Rev. Lett. 119, 033901 (2017) doi.org/10.1103/PhysRevLett.119.033901.

  2.               M. Bertola, C. Norton, D. Korotkin, ”Symplectic geometry of the moduli space of projective structures in homological coordinates”, Inventiones Mathematicæ, 210, pages 759-814 (2017). doi.org/10.1007/s00222-017-0739-z

  3.              M. Bertola, ”The Malgrange form and Fredholm determinants”, SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper No. 046, 12 pp. doi.org/10.3842/SIGMA.2017.046

  4.                M. Bertola, A. Tovbis, ”Maximal amplitudes of finite-gap solutions for the focusing Nonlinear Schr¨odinger Equation”, Comm. Math. Phys. 354 (2017), no. 2, 525–547. doi.org/10.1007/s00220-017-2895-9

  5.               M. Bertola, M. Cafasso, ”Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles”, Random Matrices: Theory and Applications, 06, (2017) no. 04, 22 pp

  6.              M. Bertola, M. Cafasso, ”The Kontsevich matrix integral: convergence to the Painlev´e hierarchy and Stokes’ phenomenon”, Comm. Math. Phys. 352 (2017), no. 2, 585–619.  doi.org/10.1007/s00220-017-2856-3

2016

1.     M. Bertola, G.A. El, A. Tovbis, ”Rogue waves in multiphase solutions of the focusing nonlinear Schr¨odinger equation”, Proc. A. 472 (2016), no. 2194. DOI: 10.1098/rspa.2016.0340

2.     M. Bertola, A. Tovbis, ”On asymptotic regimes of orthogonal polynomials with complex varying quartic exponential weight”, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 118, 50 pages. doi.org/10.3842/SIGMA.2016.118

3.     M. Bertola, B. Dubrovin, D. Yang, ”Correlation functions of the KdV hierarchy and applications to

Intersection numbers on Mg,n Physica D, Nonlinear Phenomena 327 (2016), 30–57.

doi:10.1016/j.physd.2016.04.008

4.     M. Bertola, A. Katsevich, A. Tovbis, ”Singular value decomposition of a finite Hilbert transform defined on several intervals and the interior problem of tomography: the Riemann-Hilbert problem approach”, Comm. Pure Appl. Math. 69, (2016), no.3, 407–477. doi:10.1002/cpa.21547

5.     F. Balogh, M. Bertola, T. Bothner, ”Hankel Determinant Approach to Generalized Vorob’ev–Yablonski Polynomials and Their Roots”, Constr. Approx.,44 (2016), no 3, 417–453 doi:10.1007/s00365-016-9328-4

6.     M. Bertola, A. Katsevich, A. Tovbis, ”On Sobolev instability of the interior problem of tomography”, Journal of Mathematical Analysis and Applications, 438, (2016), 962–990.

doi:10.1016/j.jmaa.2015.12.062

2015

1.     M. Bertola, M. Cafasso, ”Darboux Transformations and Random Point Processes”, IMRN, 2015, no. 15, 6211–6266. doi:10.1093/imrn/rnu122

2.     M. Bertola, P. Giavedoni, ”A degeneration of two-phase solutions of focusing NLS via Riemann-Hilbert problems ”, J. Math. Phys., 56, no. 6, (2015), 17pp. doi:10.1063/1.4922362

3.     M. Bertola, D. Yang, ”The partition function of the extended r-reduced Kadomtsev-Petviashvili hierarchy”, J. Phys A, 48 (2015, May), no. 19, 20pp.

4.     M. Bertola, T. Bothner, ”Universality Conjecture and Results for a Model of Several Coupled PositiveDefinite Matrices ”, Comm. Math. Phys. 337, no.3, 1077–1141 doi:10.1007/s00220-015-2327-7.

5.     M. Bertola, A. Tovbis, ”Meromorphic differentials with imaginary periods on degenerating hyperelliptic curves”, Anal. Math. Phys., 5, no.1, 2015, 1–22. doi:10.1007/s13324-014-0088-7.

6.     M. Bertola, A. Tovbis, ”Asymptotics of orthogonal polynomials with complex varying quartic weight: global structure, critical point behaviour and the first Painlev´e equation”, Constr. Approx. 41, no. 3, 529–587, 2015, doi:10.1007/s00365-015-9288-0.

7.     F. Balogh, M. Bertola, S. Y. Lee, K. T-R. McLaughlin, ”Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane”, Comm. Pure Appl. Math., 68, no. 1, 112–172 (2015), doi:10.1002/cpa.21541

8.     M. Bertola, T. Bothner, ”Zeros of Large Degree Vorob’ev-Yablonski Polynomials via a Hankel Determinant Identity”, IMRN, 2015, no. 19, 9330–9399. doi:10.1093/imrn/rnu239

2014

1. M. Bertola, M. Gekhtman, J. Szmigielski, ”Cauchy–Laguerre Two-Matrix Model and the Meijer-G Random Point Field”, Comm. Math. Phys. 326 (2014), no. 1, 111–144, doi:10.1007/s00220-013-1833-8

2013

1.      M. Bertola, A. Tovbis, ”Universality for the focusing nonlinear Schr¨odinger equation at the gradient catastrophe point: rational breathers and poles of the Tritronqu´ee solution to Painlev´e I ”, Comm. Pure Appl. Math. 66 (2013) no.5, 678–752. doi:10.1002/cpa.21445

2.     M. Bertola, M. Cafasso ”The gap probabilities of the tacnode, Pearcey and Airy point processes, their mutual relationship and evaluation”, Rand. Matr. Th. Appl. 02 (2013), no 2, 13500003. doi:10.1142/S2010326313500032

3.     M. Bertola, M. Gekhtman, J. Szmigielski, ”Strong asymptotics for Cauchy biorthogonal polynomials with application to the Cauchy two-matrix model”, J. Math. Phys. 54, (2013) no 4, 043517, 25. doi:10.1063/1.4802455

4.     M. Bertola, R. Buckingham, S. Y. Lee, V. Pierce, ”Spectra of random Hermitian matrices with a small-rank external source: the supercritical and subcritical regimes” J. Stat. Phys. 153 (2013), no. 4, 654–697. doi:10.1007/s10955-013-0845-2  

5.     M. Bertola, A. Katsevich, A. Tovbis, ”Inversion formulae for the cosh-weighted Hilbert transform” Proc. Amer. Math. Soc. 141 (2013) no. 8, 2703–2718.  doi:10.1090/S0002-9939-2013-11642-4

2012

1.     M. Bertola, R. Buckingham, S. Y. Lee, V. Pierce, ”Spectra of random Hermitian matrices with a small-rank external source; the critical and near-critical regimes”, J. Stat. Phys. 146 (2012), no. 3, 475–518. doi:10.1007/s10955-011-0409-2

2.     M. Bertola, “On the location of poles for the Ablowitz-Segur family of solutions to the second Painlev´e equation”, Nonlinearity, 25 (2012), no. 4, 1179–1185. doi:10.1088/0951-7715/25/4/1179

3.     M. Bertola, M. Cafasso, “Riemann–Hilbert approach to multi-time processes: the Airy and the Pearcey cases”, Physica D: Nonlinear Phenomena, 241, no. 23-24, (Dec. 2012), 2237–2245 doi:10.1016/j.physd.2012.01.003

4.     M. Bertola, M. Cafasso, ”Fredholm determinants and pole–free solutions to the noncommutative Painlev´e II equation”, Comm. Math. Phys. 309 (2012, Feb), no. 3, 793–833. doi:10.1007/s00220- 011-1383-x


2011

1.     M. Bertola, “Boutroux curves with external field: equilibrium measures without a minimization problem”, (37 pages), Analysis and Mathematical Physics, 1, no. 2 (2011, Sep), 167–211. doi:10.1007/s13324-011-0012-3

2.     M. Bertola, M. Cafasso, “The Transition between the Gap Probabilities from the Pearcey to the Airy

Process: a Riemann–Hilbert Approach”, IMRN, 2011, 1-50. doi:10.1093/imrn/rnr066

2010

1.     M. Bertola, A. Tovbis, “Universality in the Profile of the Semiclassical Limit Solutions to the Focusing Nonlinear Schrdinger Equation at the First Breaking Curve”, IMRN, 11, 2119–2167, 2010, doi:10.1093/imrn/rnp196.

2.     M. Bertola, M. Gekhtman, J. Szmigielski, “Cauchy Biorthogonal Polynomials”, Journal of Approximation Theory 162 (2010), pp 832–867. doi:10.1016/j.jat.2009.09.008

3.     M. Bertola, S. Y. Lee, “First Colonization of a Hard-Edge in Random Matrix Theory”, Constr. Approx. 31 (2010 Apr.) no.2, 231–257. doi:10.1007/s00365-009-9052-4

4.     M. Bertola, “The dependence on the monodromy data of the isomonodromic tau function”, Comm. Math. Phys. 294 , no. 2, (2010, Mar.), pag 539–579. doi:10.1007/s00220-009-0961-7


2009

1.     M. Bertola, M. Gekhtman, J. Szmigielski, “Cubic String Boundary Value Problems and Cauchy Biorthogonal polynomials”, J. Phys. A: Math. Theor. 42 (2009) 454006, 13pp. doi:10.1088/17518113/42/45/454006

2.     M. Bertola, A. Prats Ferrer, “Topological expansion for the Cauchy two–matrix model”, J. Phys. A: Math. Theor. 42 (2009) 335201. doi:10.1088/1751-8113/42/33/335201

3.     M. Bertola, M. Gekhtman, J. Szmigielski, “The Cauchy two-matrix model”, Comm. Math. Phys. 287 (2009, May), no. 3, 983– 1014. doi:10.1007/s00220-009-0739-y

4.     M. Bertola, O. Marchal, “The partition function of the two–matrix model as an isomonodromic tau–function”, J. Math. Phys. 50 (2009), no. 1, 013529, 17pp. doi:10.1063/1.3054865

5.     M. Bertola, S. Y. Lee, M. Y. Mo, “Mesoscopic colonization of a spectral band”, J. Phys. A, 42, (2009), no. 41, 415204, 17pp. doi:10.1088/1751-8113/42/41/415204

6.     M. Bertola, “Moment determinants as isomonodromic tau functions”, Nonlinearity 22 (2009), no. 1, 29–50. doi:10.1088/0951-7715/22/1/003

7.     M. Bertola, M. Y. Mo, “Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights”, Adv. Math. 22 (2009, Jan.), no. 1, 154–218. doi:10.1016/j.aim.2008.09.001

8.     F. Balogh, M. Bertola, “ Regularity of a vector potential problem and its spectral curve”, J. Approx. Theory. 161 (2009, Nov), no.1, 353–370. doi:10.1016/j.jat.2008.10.010

9.     M. Bertola, S. Y. Lee, “First colonization of a Spectral Outpost in Random Matrix Theory”, Constr. Approx. 30 (2009, Oct.) no. 2, 225–263. doi:10.1007/s00365-008-9026-y


2008

1.     M. Bertola, A. Prats-Ferrer, “Harish-Chandra integrals as nilpotent integrals”, International Mathematics Research Notices 20082008:rnn062-15 doi:10.1093/imrn/rnn062

2007

1.     M. Bertola, M. Gekhtman, “Effective inverse spectral problem for rational Lax matrices and applications”, arXiv:0705.0120, International Mathematical Research Notices (IMRN), 2007 Volume 2007:

article ID rnm103, 39 pages. doi:10.1093/imrn/rnm103

2.     M. Bertola, M. Gekhtman, “Biorthogonal Laurent polynomials, Toeplitz determinants, minimal Toda orbits and isomonodromic tau functions”, Constr. Approx. 26, no.3 (2007), 383–430. doi:10.1007/s00365006-0656-1

3.     M. Bertola, F. Corbetta, U. Moschella, ”Massless scalar field in two–dimensional de Sitter universe”, in Rigorous Quantum Field Theory: a Festshrift for Jacques Bros, Progress in Mathematics, Birkh¨auser, 251 (2007), pag. 27–38.

4.     M. Bertola, “Biorthogonal polynomials for 2–matrix models with semiclassical potentials”, J. Approx. Theory. 144, no.2, (2007), 162–212. doi:10.1016/j.jat.2006.05.006


2006

1.     M. Bertola, “Two matrix model with semiclassical potentials and extended Whitham hierarchy”, J. Phys. A, 28, (2006), 8823–8855. doi:10.1088/0305-4470/39/28/S05

2.     M. Bertola, B. Eynard, “The PDEs of biorthogonal polynomials arising in the two–matrix model”, Mathematical Physics, Analysis and Geometry, 9 (2006), no.1, pag. 23–52. doi:10.1007/s11040005-9000-x

3.     M. Bertola, B. Eynard, J. Harnad, “Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions”, Commun. Math. Phys. 263 (2006) no.2, pag. 401–437. doi:10.1007/s00220005-1505-4

2005

1.     M. Bertola, M. Y. Mo, “Isomonodromic deformation of resonant rational connections”, International Mathematical Research Papers (IMRP) Volume 2005 (2005), Issue 11, pag. 565–635.

doi:10.1155/IMRP.2005.565

2003

1.     M. Bertola, “ Free Energy of the two–matrix model/dToda tau–function”, Nucl. Phys B 669, no. 3 (2003), pag. 435–461 doi:10.1016/j.nuclphysb.2003.07.029

2.     M. Bertola, B. Eynard, “ Mixed Correlation Functions of the Two-Matrix Model” J.Phys.              A 36 (2003), pag. 7733-7750. doi:10.1088/0305-4470/36/28/304

3.     M. Bertola, B. Eynard, J. Harnad, “ Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann–Hilbert problem”, Comm. Math. Phys. 243 (2003), no.2, pag. 193–240. doi:10.1007/s00220-003-0934-1

4.     M. Bertola, “Second and third order observables of the two–matrix model”, J. High Energy Phys. 2003, no. 11, 062, 30pages. (electronic). doi:10.1088/1126-6708/2003/11/062

5.     M. Bertola, B. Eynard, J. Harnad, “Partition functions for matrix models and isomonodromic tau functions” Random matrix theory. J. Phys. A 36 (2003), no. 12, 3067–3083.

doi:10.1088/0305-4470/36/12/313

6.     M. Bertola, “Bilinear semiclassical moment functionals and their integral representation” J. Approx.

Theory 121 (2003), no. 1, pag. 71–99. doi:10.1016/S0021-9045(02)00044-8

7.     M. Bertola, B. Eynard, J. Harnad, “The duality of spectral curves that arises in two–matrix models”, Theor. Math. Phys. 134 (2003), no. 1, pag. 27–38.

doi:10.1023/A:1021811505196

2002

1.     M. Bertola, B. Eynard, J. Harnad, “ Duality, Biorthogonal Polynomials and Multi-Matrix Models” , Commun. Math. Phys. 229 (2002) 1, pag. 73-120. doi:10.1007/s002200200663

2.     H. De Guise, M. Bertola, “Coherent-state realization of su(n 1) on the n-torus”, J. Math. Phys. 43

(2002), 7, pag. 3425–3444. doi:10.1063/1.1479301

2001

1.     M. Bertola, D. Gouthier, “Lie Triple Systems and Warped Products”, Rend. Mat. Appl. (7) 21 (2001), no. 1-4, pag. 275–293, Roma.

2.     M. Bertola, D. Gouthier, “Warped products with special Riemannian curvature”, Bol. Soc. Brasil. Mat. (N.S.) 32 (2001), no. 1, pag. 45–62. doi:10.1007/BF01238957

2000

1.     M. Bertola, “Frobenius manifold structure on orbit space of Jacobi group; Part II”, Differential Geom. Appl. 13 (3) (2000), pag. 213–233 doi:10.1016/S0926-2245(00)00027-9

2.     M. Bertola, “Frobenius manifold structure on orbit space of Jacobi group; Part I”, Differential Geom. Appl. 13 (2000), pag. 19–41. doi:10.1016/S0926-2245(00)00026-7

3.     M. Bertola, J. Bros, U. Moschella and R. Schaeffer, “A general construction of conformal field theories from scalar anti-de Sitter quantum field theories”, Nuclear Phys. B 587 (2000), pag. 619–644. doi:10.1016/S0550-3213(00)00463-6

4.     M. Bertola, J. Bros, V. Gorini, U. Moschella, R. Schaeffer, “Decomposing Quantum Fields on Branes”,

Nuclear Phys. B 581 (2000), pag. 575-603. doi:10.1016/S0550-3213(00)00280-7

1999

1.     M. Bertola, V. Gorini, U. Moschella, R. Schaeffer, “Correspondence between Minkowski and de Sitter Quantum Field Theory”, Phys.             Lett.         B 462 (1999), pag.         249–253.                 doi:10.1016/S03702693(99)00927-2

Refereed Proceedings

  M. Bertola, A. Assra, “Random Matrix Approach for the Capacity of Large-Scale MIMO Systems using the Harish-Chandra Formula”, 2020 14th International Conference on Signal Processing and Communication Systems, ICSPCS 2020 - Proceedings, doi:10.1109/ICSPCS50536.2020.9310003

   S.T. Ali, M. Bertola, “Symplectic geometry of the Wigner transform on noncompact symmetric spaces”, Inst. Phys. Conf. Ser. No. 173: Section 7 (2003), pag. 847–854.

   H. De Guise, M. Bertola, “Coherent-state realizations of su( 1) in terms of subgroup functions”, Inst. Phys. Conf. Ser. No. 173, Section 7 (2003), pag. 523–526.

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