Mark Adler

Professor of Mathematics

Education:

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Although I was trained in the general area of differential equations and analysis, I have mostly worked as a mathematical physicist, involved in mathematical problems related to physics. This has lead me to study integrable systems, matrix models and finally random matrices, the latter which is part of statistical mechanics. In order to do research in these topics, I needed to learn to some extent, differential geometry, Lie algebra, algebraic geometry,combinatorics and finally probability theory. So I am a jack of all trades, mathematically speaking.

 

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Selected Publications

"Algebraic Integrability, Painleve' Geometry and Lie Algebras",  M. Adler, P. van Moerebeke, P. Vanhaecke, Volume 47, Ergebnisse der Mathematik und iher Grenzgebiete  Springer-Verlag 2004

Adler, M., Johansson, K., van Moerbeke P., "Lozenge tililngs of hexagons with cuts and asymptotic fluctuations: a new universality class". Math. Phys Anal Geom (2018) 21:9 (pp 1-53).

Adler, M., Johansson, K., van Moerbeke P.,"Tilings of non-convex Polygons, skew-Young Tableaux and determinantal Processes". Commun. Math. Phys. 364, 287-342 (2018).

Adler, M., Cafasso, M., van Moerbeke P., "Nonlinear PDEs for Fredholm determinants arising from string equations". AMS Contemporary Mathematics: volume 593, 2013.

Adler, M., Cafasso, M., van Moerbeke P., "Nonlinear PDEs for gap probabilities in random matrices and KP theory". Physica D, 241, 2012, 2265-2284.

Adler, M., Ferrari, P.L., van Moerbeke P., "Nonintersecting random walks in the neigborhood of a symmetric tacnode". Annals of Probability 2013, Vol. 41, No. 4, 2599-2647.

Adler, M., van Moerbeke P., Vanhaecke, P., "Moment matrices and multi-component KP, with applications to random matrix theory".  Comm. Math. Phys. 286 (2009), no. 1, 1–38.

Adler, M., van Moerbeke P., Vanhaecke, P., "Singularity confinement for a class of m-th order difference equations of combinatorics". Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 366 (2008), no. 1867, 877–922.

Adler, M., van Moerbeke P., "Recursion relations for Unitary integrals, Combinatorics and the Toeplitz Lattice".  Comm. Math. Phys. 237 (2003), no. 3, 397–440.

Adler, M., van Moerbeke P., "Hermitian, symmetric and symplectic random ensembles:PDEs for the distribution of the spectrum". Ann. of Math. (2) 153 (2001), no. 1, 149-189.

Adler, M., van Moerbeke P., "Coupled GUE-minor processes and domino tilings". Int. Math. Res. Not. IMRN 2015, no. 21, 10987–11044.

Adler, M., Chhita, S., Johansson, K., van Moerbeke, P., "Tacnode GUE-minor Processes and Double Aztec Diamonds".  Probab. Theory Related Fields 162 (2015), no. 1-2, 275–325.

Adler, M., van Moerbeke P., Wang, D., "Random matrix minor processes related to percolation theory". Random Matrices Theory Appl. 2 (2013), no. 4, 1350008, 72 pp.

Adler, M., Johansson, K., van Moerbeke, P., "Double Aztec Diamonds and the Tacnode Process". Adv. Math. 252 (2014), 518–571.

Adler, M., Cafasso, M., van Moerbeke, P., "From the Pearcey to the Airy process". Electron J Probab., Vol 16, 2011, no 36, 1048-1064.

Adler, M., Nordenstam, E.,van Moerbeke, P., "Consecutive Minors for Dyson's Brownian Motions". Stochastic Process. Appl. 124 (2014), no. 6, 2023–2051.

Adler, M., Nordenstam, E.,van Moerbeke, P, "The Dyson Brownian minor process". Ann. Inst. Fourier (Grenoble) 64 (2014), no. 3, 971–1009.

Adler, M., van Moerbeke, P.,Vanderstichelen, D., "Non-intersecting Brownian motions leaving from and going to several points". Phys. D 241 (2012), no. 5, 443–460. 

Adler, M., Delepine, J.,van Moerbeke, P.,Vanhaecke, P., "A PDE for Nonintersecting Brownian Motions and Applications". Adv. Math. 226 (2011), no. 2, 1715–1755.

Adler, M., Orantin, M.,van Moerbeke,P., "Universality of the Pearcey process". Phys. D 239 (2010), no. 12, 924–941.

Adler, M., Ferrari, P.L., van Moerbeke P., "Airy processes with wanderers and new universality classes". Annals of Probability 2010, Vol. 38, No. 2, 714-769.

Adler, M., Delepine, J.,van Moerbeke, P., "Dyson's non-intersecting Brownian motions with a few outliers." Communication on Pure and Applied Mathematics, 62, issue 3, 334-395 (2009)

Adler, M., van Moerbeke P., "Joint Probability for the Pearcey process".

Adler, M., van Moerbeke P., "PDE's for the Gaussian ensemble with external source and the Pearcey distribution".

Adler, M., Borodin, A., van Moerbeke, P.,"Expectations of hook products on large partitions". Comm. Pure Appl. Math. 60 (2007), no. 9, 1261–1292

Adler, M., van Moerbeke, P., "PDEs for the joint distributions of the Dyson, Airy and Sine processes." Annals of Probability 2005, Vol. 33, No. 4, 1326-1361.

Adler, M., van Moerbeke, P., "Virasoro action on Schur function expansions, skew Young tableaux and random walks". Comm. Pure Appl. Math. 58 (2005), no. 3, 362–408. 

Adler, M., van Moerbeke, P., "PDEs for the joint distributions of the Dyson, Airy and sine processes". Ann. Probab. 33 (2005), no. 4, 1326–1361. 

Adler, M., van Moerbeke, P., "Integrals over Grassmannians and Random permutations". Adv. Math. 181 (2004), no. 1, 190–249.

Adler, M., van Moerbeke, P., "Integrals over classical Groups, Random permutations, Toda and Toeplitz lattices." Comm. Pure Appl. Math. 54 (2001), no. 2, 153–205. 

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Publications

Adler,M., van Moerbeke P. "Matrix integrals, Toda symmetries, Virasoro constrains and orthogonal polynominals"

Adler,M., van Moerbeke P. "Group factorization, moment matrices and Toda lattices"

Adler,M., van Moerbeke P. "String-orthogonal polynomials, String Equations and Two - Toda Symmetries". Comm. Pure and Appl. Math., 50 241-290 (1997).

van Moerbeke P. "Algèbres W et équations non-linéaries"

Adler,M., Horozov, E., van Moerbeke P. "The solution to the q-KdV equation". Physics letters A, 242, 1390151, (1998).

Adler,M., van Moerbeke P. "Toda- Darboux maps and vertex operators". "International Mathematics Research Notices", 10, 489-511 (1998).

Adler,M.,  Shiota,T., van Moerbeke P. "Random matrices, Virasor algebras and non-commutative KP".

Adler,M., van Moerbeke P. "Vertex operator solutions to the discrete KP-hierarchy". Comm. Math. Phys., 203, 185-210 (1999).

Adler,M., van Moerbeke P. "Generalized orthogonal polynominals, discrete KP and Riemann-Hilbert problems". Comm. Math. Phys., 207, 589-620 (1999).

Adler,M., Horozov, E., van Moerbeke P. "The Pfaff lattice and skew-orthogonal polynominals". Intern. Math. Research Notice 1999.

Adler,M., van Moerbeke P. "The spectrum of coupled random matrices". 

Adler,M., Forrester,P.J., Nagao,N, van Moerbeke P. "Classical skew orthogonal polynominals and random matrices".

van Moerbeke P. "Random matrices and random permutations".

Adler,M., Shiota,T., van Moerbeke P. "Pfaff τ-functions".

van Moerbeke P. "Random matrices and permutations, matrix integrals and integrable systems".

Adler,M., Kuznetsov,V.B., van Moerbeke P. "Rational solutions to the Pfaff lattice and Jack polynominals".

Adler,M., van Moerbeke P. "Recursion eelations for unitary integrals, combinatorics and the toeplitz lattice".

Adler,M., van Moerbeke P. "Integrals over Grassmannians and random permutations".

Adler,M. "Virasoro action on schur function expansions, skew young tableaux, and random walks".

Adler,M., van Moerbeke P. "PDE's for the joint distributions of teh Dyson, Airy and Sine processes".

Adler,M., Borodin,A., van Moerbeke P. "Expectations of hook products on large partitions".

van Moerbeke P. "Combinatorics and integrable Geometry".

Adler,M., van Moerbeke P. "PDE's for the Gaussian ensemble with external souce and the Pearcey distribution".

Adler,M. "Dyson's Nonintersecting Brownian Motions with a Few Outliers". Comm. Pure and Appl. Math., Vol. ooo, 0001 -0059 (2000).

Adler,M., Ferrari, Patrik L.,van Moerbeke P. "Airy process with wanderers and new universality classes".

Adler,M., Orantin,N.,van Moerbeke P. "University of the Pearcey process". 

Adler,M., Delélpine,J., van Moerbeke P., Vanhaecke, P. "A PDE for non-intersecting Brownian motions and applications".  

Adler,M., van Moerbeke P., Vanhaecke,P. "Moment matrices and multi-component KP, with applications to random matrix theory"

Adler,M., van Moerbeke P. "Joint probability for the Pearcey process". 

Adler,M. "Linearization of Hamiltonian Systems, Jacobi Varieties and Pepresentation Theory"

Adler,M. "Completely Integrable Systems, Euclidean Lie Algebras, and Curves". 

Adler,M. "Birkhoff Strata, Bäcklund Transformations, and Regularization of Isospectral Operators"

Adler,M. "On the Bäcklund Transformation for the Gel'fand-Dickey Equations".

Adler,M. "Spectral statistics of orthogonal and symplectic ensembles".

Adler,M., Moser,J. "On a Class of Polynomials Connected with the Korteweg-de Vries Equation".

Adler,M. "PDE's for the Dyson, Airy and Sine processes".

Adler,M., van Moerbeke P. "Kowalewski's Asymptotic Method, Kac-moody Lie Algebras and Regularization".

Adler,M., van Moerbeke P. "The Kowalewski and Hénon-Heiles Motions as Manakov Geodesic Flows on SO(4) - a Two-Dimensional Family of Lax Pairs".

Adler,M., Haine,L., van Moerbeke P. "Limit matrices for the Toda flow and periodic flags for loop groups". 

Adler,M. "Integrable Systems, Random Matrices and Random Processes". 

Adler,M., van Moerbeke P. "The Intersection of Four Quadrics in IP, Abelian Surfaces and their Moduli".

Adler,M. "Some Finite Dimensional Integrable Systems and Their Scattering Behavior". 

Adler,M. "Completely integrable systems and sympletic actions".

Adler,M., van Moerbeke P. "Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum."

Adler,M., van Moerbeke P. "Darboux Transforms on Band Matrices, Weights, and Associated Polynomials".

Adler,M., van Moerbeke P. "Toda versus Pfaff lattice and related polynomials".

Adler,M., van Moerbeke P. "Integerals over Classical Groups, Random Permutation, Toda and Toeplitz Lattices".

Adler,M., van Moerbeke P., Vanhaecke,P. "Moment matrices and multi-component KP, with applications to random matrix theory".

Adler,M., van Moerbeke P., Vanhaecke,P. "Singularity confinement for a class of m-th order difference equations of combinatorics".

Adler,M., van Moerbeke P. "A PED for the joint distributions of the Airy Process".