research

Broadly speaking, I am interested in asymptotic analysis, approximation theory, special function theory, and their application to problems in mathematical physics and random matrix theory. One of the main tools I use is the Riemann-Hilbert problem; for a nice introduction to the method as it applies to integrable systems, see Alexander Its' Notices article. 

1. A. Barhoumi. Special Function Solutions of Painlevé-IV and the Quartic Model. To appear in Stud. Appl. Math. [48 pages] arXiv:2312.11294 

2. A. Barhoumi, P. Bleher, A. Deaño, and M. Yattselev. On Airy solutions of PII and complex cubic ensemble of random matrices, II. To appear in: Contemp. Math. [26 pages]  arXiv:2403.03023

3. A. Barhoumi, O. Lisovyy, P. Miller, and A. Prokhorov. Painlevé-III Monodromy Maps Under the D6→D8 Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions. SIGMA 20 (2024), 019, [77 pages]. https://doi.org/10.3842/SIGMA.2024.019.

4. A. Barhoumi, P. Bleher, A. Deaño, and M. Yattselev. On Airy solutions of PII and complex cubic ensemble of random matrices, I. To appear in: Proceedings of the 16th International Symposium, Montreal, Canada, In honor to Richard Askey. [12 pages] arXiv:2310.14898

5. A. B. Barhoumi and M. L. Yattselev. Non-Hermitian Orthogonal Polynomials on a Trefoil. Constr Approx (2023). [61 pages] https://doi.org/10.1007/s00365-023-09640-6.

6. A. Barhoumi, P. Bleher, A. Deaño, and M. Yattselev. Investigation of the two-cut phase region in the complex cubic ensemble of random matrices. J. Math. Phys. (2022); 63, 063303 [40 pages] https://doi.org/10.1063/5.0086911

7. A. B. Barhoumi. Strong asymptotics of Jacobi-type kissing polynomials. Integral Transforms Spec. Funct. (2021); 32:5-8, 377-394 [18 pages], doi: 10.1080/10652469.2021.1923707

8. A. Barhoumi, A. F. Celsus, and A. Deaño. Global-phase portrait and large-degree asymptotics for the Kissing polynomials. Stud Appl Math. (2021); 147: 448– 526 [79 pages]. https://doi.org/10.1111/sapm.12387

9. A. Barhoumi and M. L. Yattselev. Asymptotics of polynomials orthogonal on a cross with a Jacobi-type weight. Complex Anal. Oper. Theory (2020); 14, 9 [44 pages] doi:10.1007/s11785-019-00962-7 

• A. Barhoumi. Orthogonal Polynomials on S-Curves Associated with Genus One Surfaces. Diss. 2020. http://hdl.handle.net/1805/23029