• Contact Information
• About the classes I teach
• About my Area of Research
• Publications
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Contact Information for Alex Kasman

Phone: 843-953-8018
Fax: 843-953-1410

e-mail: kasmana@cofc.edu

Office: My office is in room 336 RSS. (That's the Robert Scott Small building, sometimes called "the old library". It is directly across from Maybank Hall.)

Faculty List for Math Department
Department of Mathematics Homepage
College of Charleston Homepage

Alex Kasman's Teaching

For Spring 2008, I will be teaching MATH 495, the capstone course for math majors. It will be taught as a special topics course focusing on "Soliton Theory". This is a branch of mathematics that combines diverse "abstract" topics (algebraic geometry, linear algebra, and differential equations) and also has important applications (in communications, particle physics, engineering, and even in biology). Although it is too large a subject to really cover in a single undergraduate course, I hope that we will be able to get a good sense of the sorts of results that make it up. For more information, please read about my research, my papers and the references that they cite.

Faculty List for Math Department
Department of Mathematics Homepage
College of Charleston Homepage

Alex Kasman's Research

My own research is done on this `boundary' between mathematical physics and algebraic geometry.   Among the things I have written about are:

• The particle-like waves called `solitons'.
• The symmetry of linear wave equations known as `bispectrality'
• Algebro-geometric constructions for producing quantum integrable systems
• Rank one perturbations of operator identities and their connections to discrete integrable systems including the Bethe Ansatz and the Hirota Bilinear Difference Equation.
• The geometry of Grassmannian manifolds and associated functions which satisfy nonlinear partial differential equations of mathematical physics
• Classical and quantum integrable particle systems
• The geometry (spectral varieties) of commutative rings of partial differential operators.
• Solutions to the KP hierarchy associated to higher rank vector bundles over singular rational curves.

Faculty List for Math Department
Department of Mathematics Homepage
College of Charleston Homepage

Alex Kasman's Publications

Research Articles:

1. Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems was published in Communications in Mathematical Physics 172 (1995). 
   PDF or arXiv. 
2. Orthogonal Polynomials and Finite Toda Lattices appeared in The Journal of Mathematical Physics, 38 (1) 1997 pp. 247-254.
   PDF. 
3. Bispectral Darboux Transformations: the Generalized Airy Case by Mitchell Rothstein and myself was published in Physica D 102 (1997), no. 3-4 pp. 159-176.
   PDF or arXiv. 
4. Darboux Transformations from n-KdV to KP appeared in Acta Applicandae Mathematicae 49 no. 2 (1997) pp. 179-197.
   PDF. 
5. The Bispectral Involution as a Linearizing Map in "Caloger-Moser-Sutherland-Models", CRM Series in Mathematical Physics, JF van Diejen and L Vinet, eds. Springer (2000) pp. 221--230. 
6. Darboux Transformations and the Bispectral Problem in CRM Proceedings and Lecture Notes 14 American Mathematical Society (1998) pp.81-91. 
7. Integrable Systems with Singular Rational Spectral Varieties was my talk at the Workshop on Supersymmetry and Integrable Models, University of Illinois - Chicago, April 1997. Published as Lecture Notes in Physics502 Springer-Verlag (1998) pp. 77-90
8. D-modules and Darboux Transformations written jointly with Yuri Berest appeared in Letters in Mathematical Physics 43 (1998) pp. 279-294.
   PDF or arXiv.
9. Duality and Construction of Quantum Integrable Systems (with E. Horozov) Trends in mathematical physics (Knoxville, TN, 1998), 289--297, AMS/IP Stud. Adv. Math., 13, Amer. Math. Soc., Providence, RI, 1999
10. Spectral Difference Equations Satisfied by KP Soliton Wavefunctions was published as Inverse Problems 14 (1998) pp.1481-1487.
    PDF or arXiv. 
11. Kernel Inspired Factorizations of Partial Differential Operators appeared as The Journal of Mathematical Analysis and Applications 234/2 (1999) pp. 580-591.
    PDF. 
12. Grassmannians, Nonlinear Waves and Generalized Schur Functions was published in Contemporary Mathematics 246 (1999) pp. 163-174.
    PDF.
    Note: I liked my first draft of this article better than the one which was published. That version was longer and did a better job of introducing ideas (like Grasmmannian manifolds) which were cut out of the final version at the editor's request. It is available from the arXiv as http://arXiv.org/abs/math-ph/9811008v2.
13. Darboux Transformations of Bispectral Quantum Integrable Systems written jointly with Emil Horozov, was published as Letters in Mathematical Physics 49: 131-143, 1999.
    PDF or arXiv.
14. Finite Canonical Commutation Relations and Rational Nested Bethe Ansatz was published as Regular and Chaotic Dynamics 6 (2), 211-214 (2001).
    PDF or arXiv or journal. 
15. Commutative Partial Differential Operators written with Emma Previato was published as Physica D 152-153 (2001) 66-77
    PDF. 
16. Solitons and Almost-Intertwining Matrices with Michael Gekhtman appeared in the Journal of Mathematical Physics 42 8 (2001) 3540-3553.
    PDF or arXiv or journal. 
17. On the Quantization of a Self-Dual Integrable System was published as Journal of Physics A 23 (2001) 6307-6312.
    PDF.
18. Overcoming the Phage Replication Threshold: A mathematical model with implications for phage therapy with Laura M. Kasman, Caroline Westwater, Joseph Dolan, Michael G. Schmidt, and James S. Norris was published in The Journal of Virology. 76 11, (2002) 557-5564.
    PDF.
19. Integrable Systems and Rank One Conditions for Rectangular Matrices with Michael Gekhtman was published as Theoretical and Mathematical Physics 133(2): 1498-1503 (2002) [and in Russian as Teoreticheskaya i Matematicheskaya Fizika Vol. 133 No. 2 pp. 211-217, November 2002].
    arXiv or PDF
    (Funny Note: I wrote this paper in English and submitted it to this Russian journal which translated it into Russian for the original edition of the journal. Interestingly, when the American Mathematical Society summarized the paper in Math Reviews, they translated the Russian title back into English and ended up with something just slightly different than our original title!)
20. On KP Generators and the Geometry of the HBDE with Michael Gekhtman was published in The Journal of Geometry and Physics 56 (2006) 282-309.
    Journal or arXiv
21. On Decompositions of the KdV 2-Soliton with Nicholas Benes and Kevin Young was published as The Journal of Nonlinear Science Volume 16 Number 2 (2006) pages 179-200.
    PDF or Journal or Slides from a Talk
22. When is negativity not a problem for the ultra-discrete limit? with Stéphane Lafortune was published as Journal of Mathematical Physics 47, 103510 (2006).
    Journal or arXiv
23. Tau-Functions, Grassmannians and Rank One Conditions with Michael Gekhtman was published as The Journal of Computational and Applied Mathematics 202 (2007) 80--87.
    Journal
24. Universality of Rank 6 Plücker Relations and Grassmann Cone Preserving Maps with Takahiro Shiota, Kathryn Pedings and Amy Reiszl was published as The Proceedings of the American Mathematical Society 136 (2008) 77-87
    arXiv or Journal

Other Mathematics Related Publications:

• The Bispectral Problem CRM Proceedings and Lecture Notes Volume 14 Edited by John Harnad and Alex Kasman is available from the American Mathematical Society.
• Student Answer Manual to accompany "Calculus" by Hughes-Hallett, Gleason, et al. John Wiley & Sons, 1993 (with S.A. Mallozzi, A.H. Wang and G. Zhang).
• Book Review: Imaginary Numbers in the Notices of the AMS, Volume 47 Number 7, August 2000.
• Book Review: Cryptonomicon in the Notices of the AMS, Volume 46 Number 11, December 1999.
• Letter to the Editor in the Notices of the AMS, Volume 44 Number 6. (On Calculus Education.)
• Letter to the Editor in the Notices of the AMS, Volume 42 Number 10. (On typesetting preferences of mathematicians.)
• Letter to the Editor in the Notices of the AMS, Volume 43 Number 5. (On the need for mathematical exposition to non-mathematicians.)
• Letter to the Editor in the Notices of the AMS, Volume 49 Number 6. (On the connection between mathematics and insanity implicit in works of fiction such as the film A Beautiful Mind.)
• Unreasonable Effectiveness, a short story was published in Math Horizons magazine.
• "Mathematics in Fiction: An Interdisciplinary Course" appeared in PRIMUS Volume XIII Number 1, March 2003, Pages 1-16.
• Book Review: "It Must Be Beautiful" in the American Mathematical Monthly, October 2003.
• Book Review: "After Math, The Fractal Murders and Other Mathematical Murder Mysteries" in the AMS Notices, October 2003, 1087-1092.
• "Mathematics in Science Fiction" in Math Horizons, April 2004, 5-7.
• Reality Conditions: Short Mathematical Fiction, a collection of mathematically themed short stories was published by the Mathematical Association of America and is available through Amazon.com.
• Book Review: "PopCo" published in the AMS Notices, February 2006.

Faculty List for Math Department
Department of Mathematics Homepage
College of Charleston Homepage

Alex Kasman's Links

Check out these sites that I (in theory) maintain:

• The Mathematical Fiction Homepage
• The Amanda Museum
• The Secular Humanists of the Lowcountry

Faculty List for Math Department
Department of Mathematics Homepage
College of Charleston Homepage