picture of Alex on a background of mathematical notation taken by CofC PR department Alex Kasman
Department of Mathematics
College of Charleston
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Contact Information for Alex Kasman

Mailing Address:
Department of Mathematics
College of Charleston
Robert Scott Small Building / Room 339
Charleston, SC 29424

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 Nuilding, directly across from Maybank Hall.)

Office Hours: This semester I can be found in my office at the following times: Mon 10-11, Wed 11-12 and Fri 12-1.

(If you would like to meet with me at another time, please write or call 953-8018 and I would be happy to set up an appointment.)


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Alex Kasman's Teaching

In Fall 2014, I will be teaching twi sections of Math 203 (Linear Algebra) and one First Year Seminar (FYSM) entitled "Math in Fiction".

Office Hours: This semester I can be found in my office at the following times: Mon 10-11, Wed 11-12 and Fri 12-1.

(If you would like to meet with me at another time, please write or call 953-8018 and I would be happy to set up an appointment.)


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Alex Kasman's Research

Algebraic-geometry is an area of pure mathematics that is concerned with the geometric structure of objects determined by algebraic equations.  Mathematical physics is an applied area of mathematics that seeks to use the structures and tools of mathematics to understand things in the real world such as light waves, elementary particles, supernovas and rocket ships.  Until recently, these were two unrelated subjects because most of the equations in mathematical physics were differential equations, not algebraic equations.  However, as a result of recent advances, we now know that there is a large and important intersection between these two subjects.  In fact, in the last thirty years we have learned a great deal about each of these from studying the other. 

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.

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Alex Kasman's Publications

 

Research Articles:

Most of the articles listed below are available online as PDF files or on the arXiv of math/physics preprints. If you would like a printed copy of any of these articles, please write to me at the address above and I would be happy to send you one.
  1. Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems was published in Communications in Mathematical Physics 172 (1995). 
    Journal or 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
  25. Spin Calogero Particles and Bispectral Solutions of the Matrix KP Hierarchy with Maarten Bergvelt and Michael Gekhtman was published as Mathematical Physics, Analysis and Geometry 12 (2009) 181-200.
    arXiv or Journal
  26. Factorization and Resultants of Partial Differential Operators with Emma Previato was published as Mathematics in Computer Science 4 (2010) 169-184.
    (The Mathematica notebook containing the commands for multiplying, conjugating and taking resultants of partial differential operators described in the paper is available here for download.)
  27. Duality and Collisions of Harmonically Constrained Calogero Particles was published as Contemporary Mathematics (2013) 593 pp. 109--123.
    PDF

Books

Other Mathematics Related Publications:

  • 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.
  • Book Review: "PopCo" published in the AMS Notices, February 2006.
  • Book Review: "Pythagorean Crimes" published in the AMS Notices, January 2009.
  • My short story On the Quantum Theoretic Implications of Newton's Alchemy has been published in Analog (October 2007), Esli (February 2008) and the book The Shape of Content (AK Peters Publishing, 2008).
  • Book Review: "After Genius" in the December 2010 issue of the Mathematical Intelligencer.
  • My essay "Fictional Women in Mathematics" appeared in the January 2011 newletter of the AWM.
  • Bispectrality and Duality: A PDF copy of the talk I gave at the 2012 Joint Mathematics Meeting in Boston reviewing progress on my "research program" aiming to demonstrate a correspondence between the action-angle duality of integrable particle systems and the bispectral property. The talk mentions a bit about the quantum correspondence, but focuses especially on the classical case.
  • "A Survey of Fictional Mathematics in Literature" in Mathematics and Popular Culture: Essays on Appearances in Film, Fiction, Games, Television and Other Media (Sklar and Sklar, Eds.), McFarland 2012.

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Alex Kasman's Links

 

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


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