Department of Mathematics: Alex Kasman's Homepage

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Web Name: Department of Mathematics: Alex Kasman's Homepage

WebSite: http://kasmana.people.cofc.edu

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Office: My office is in room 336 RSS. (That's the Robert Scott Small Building, directly across from Maybank Hall.) Office Hours: This semester I can be found in my officeat the following times: Tuesday 4:30-6:00PM and Thursday 10:00-11:30AM.(If you would like to meet with me at another time, please write or call 843-953-8018 and I would be happy to set up an appointment.) (Return to the top of the page) Department of Mathematics Homepage College of Charleston Homepage Office Hours: This semester I can be found in my officeat the following times: Tuesday 4:30-6:00 PM and Thursday 10:00-11:30 AM.(If you would like to meet with me at another time, please write or call 843-953-8018 and I would be happy to set up an appointment.) (Return to the top of the page) Faculty List for Math Department Department of Mathematics Homepage College of Charleston Homepage Algebraic-geometry is an area of pure mathematics that is concerned withthe geometric structure of objects determined by algebraic equations.Mathematical physics is an applied area of mathematics that seeks to usethe structures and tools of mathematics to understand things in the realworld such as light waves, elementary particles, supernovas and rocketships. Until recently, these were two unrelated subjects becausemost 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 thesetwo subjects. In fact, in the last thirty years we have learned agreat deal about each of these from studying the other. My own research is done on this `boundary' between mathematical physicsand algebraic geometry. Among the things I have written about The symmetry of linear wave equations known as `bispectrality'Rank one perturbations of operator identities and their connections todiscrete integrable systems including the Bethe Ansatz and the Hirota BilinearDifference Equation.The geometry of Grassmannian manifolds and associated functions which satisfynonlinear partial differential equations of mathematical physicsThe geometry (spectral varieties) of commutative rings of partial differentialoperators.Solutions to the KP hierarchy associated to higher rank vector bundlesover singular rational curves.Most of the articles listed below are available online as PDF files oron the arXiv of math/physics preprints. If you would like a printed copyof any of these articles, please write to me at the addressabove and I would be happy to send you one. Orthogonal Polynomials and Finite Toda Lattices appearedin The Journal of Mathematical Physics, 38 (1) 1997 pp. 247-254. PDF. Bispectral Darboux Transformations: the Generalized Airy Caseby Mitchell Rothstein and myself was published in Physica D 102(1997), no. 3-4 pp. 159-176. PDF arXiv. Darboux Transformations from n-KdV to KP appeared in ActaApplicandae Mathematicae 49 no. 2 (1997) pp. 179-197. PDF. The Bispectral Involution as a Linearizing Map in "Caloger-Moser-Sutherland-Models",Series in Mathematical Physics, JF van Diejen and L Vinet, eds. Springer(2000) pp. 221--230. Darboux Transformations and the Bispectral Problem in CRMProceedings and Lecture Notes 14 American Mathematical Society(1998) pp.81-91. Integrable Systems with Singular Rational Spectral Varietieswas my talk at the Workshop on Supersymmetry and Integrable Models, Universityof Illinois - Chicago, April 1997. Published as Lecture Notes in Physics502Springer-Verlag (1998) pp. 77-90 D-modules and Darboux Transformations written jointly withYuri Berest appeared in Letters in Mathematical Physics 43 (1998)pp. 279-294. PDF or arXiv. Trends in mathematical physics (Knoxville, TN, 1998),289--297, AMS/IP Stud. Adv. Math., 13, Amer. Math. Soc., Providence, RI, Spectral Difference Equations Satisfied by KP Soliton Wavefunctionswas published as Inverse Problems 14 (1998) pp.1481-1487. PDF arXiv. Kernel Inspired Factorizations of Partial Differential Operators appeared as TheJournal of Mathematical Analysis and Applications 234/2 (1999) pp.580-591. PDF. Grassmannians, Nonlinear Waves and Generalized Schur Functions was published in ContemporaryMathematics 246 (1999) pp. 163-174. PDF. Note: I liked my first draft of this article better than the one whichwas published. That version was longer and did a better job of introducingideas (like Grasmmannian manifolds) which were cut out of the final versionat the editor's request. It is available from the arXiv as http://arXiv.org/abs/math-ph/9811008v2. Darboux Transformations of Bispectral Quantum Integrable Systemswritten jointly with Emil Horozov, was published as Letters in MathematicalPhysics 49: 131-143, 1999. PDF or arXiv. Commutative Partial Differential Operators written with EmmaPreviato was published as Physica D 152-153 (2001) 66-77 PDF. On the Quantization of a Self-Dual Integrable System was published as Journal of Physics A 23 (2001) 6307-6312. PDF. Overcoming the Phage Replication Threshold: A mathematical modelwith implications for phage therapy with Laura M. Kasman, CarolineWestwater, Joseph Dolan, Michael G. Schmidt, and James S. Norris waspublished in The Journal of Virology. 76 11, (2002) 557-5564. PDF.Systems and Rank One Conditions for Rectangular Matrices withMichael Gekhtman was published as Theoretical and MathematicalPhysics 133(2): 1498-1503 (2002) [and in Russian as Teoreticheskaya i Matematicheskaya Fizika Vol. 133 No. 2pp. 211-217, November 2002]. arXiv or PDF(Funny Note: I wrote this paper in English and submitted it to thisRussian journal which translated it into Russian for the originaledition of the journal. Interestingly, when the American MathematicalSociety 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!) 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 When is negativity not a problem for the ultra-discrete limit? with St phane Lafortune was published as Journal of Mathematical Physics 47, 103510 (2006).Journal or arXiv Universality of Rank 6 Pl cker Relations and GrassmannCone Preserving Maps with Takahiro Shiota, Kathryn Pedings and AmyReiszl was published as The Proceedings of the American Mathematical Society 136 (2008) 77-87. arXivJournal 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 JournalFactorization 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.)Duality and Collisions of Harmonically Constrained Calogero Particles was published as Contemporary Mathematics (2013) 593 pp. 109--123.PDFBispectrality of N-Component KP Wave Functions: A Study in Non-Commutativitywas published as SIGMA 11 (2015), 087, 22 pagesJournal or arXivFactorization of a Matrix Differential Operator Using Functions in its Kernel was published as The American Mathematical Monthly Vol. 123, No. 7 (August-September 2016), pp. 704-709arXivJournalRecurrence Relations for Exceptional Hermite Polynomials with D. Gomez-Ullate, A. Kuijlaars and R. Milson The Journal of Approximation Theory (2016), pp. 1-16arXivJournalOn Factoring an Operator Using Elements of its Kernel Communications in Algebra (2017) 45 pp. 1443-1451.arXivJournalThe Duplexing of the Genetic Code and Sequence-Dependent DNA Geometry Bulletin of Mathematical Biology 80 (10), 2734-2760 (2018).Journal or Downloadable Mathematica Notebook or Springer Nature SharingQuaternion-Valued Breather Soliton, Rational, and Periodic KdV Solutions (with John Cobb, Albert Serna, and Monique Sparkman) The Journal of Nonlinear Mathematical Physics 27:3, 429-452 (2020).arXiv or JournalDid Sequence Dependent Geometry Influence the Evolution of the Genetic Code? (with Brenton LeMesurier) Contemporary Mathematics 746 pp.41-56 (2020).arXiv or JournalThe Adelic Grassmannian and Exceptional Hermite Polynomials (with Robert Milson) (submitted for publication)arXiv TheBispectral Problem CRM Proceedings and Lecture Notes Volume 14 (1999)Edited by John Harnad and Alex Kasman is available from the AmericanMathematical Society. Reality Conditions: Short Mathematical Fiction (2005), acollection of mathematically themed short stories was published by the Mathematical Association of America. Glimpses of Soliton Theory: The Algebra and Geometry of Nonlinear PDEs (2010) is part of the Student Mathematical Library Series of 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: ImaginaryNumbers in the Notices of the AMS, Volume 47 Number 7, August 2000.Book Review: Cryptonomiconin the Notices of the AMS, Volume 46 Number 11, December 1999. Letter to the Editorin the Notices of the AMS, Volume 44 Number 6. (On Calculus Education.) Letter to the Editorin the Notices of the AMS, Volume 42 Number 10. (On typesetting preferencesof mathematicians.) Letter to the Editorin the Notices of the AMS, Volume 43 Number 5. (On the need for mathematicalexposition to non-mathematicians.) Letter tothe Editor in the Notices of the AMS, Volume 49 Number 6. (On theconnection between mathematics and insanity implicit in works of fictionsuch as the film A Beautiful Mind.) UnreasonableEffectiveness, a short story was published in Math Horizonsmagazine. "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 MathematicalMonthly, October 2003. Book Review: "After Math, The Fractal Murders and OtherMathematical Murder Mysteries" in the AMS Notices, October 2003,1087-1092. "Mathematics in Science Fiction" in Math Horizons, April 2004, DNA Solitons and Codon Bias: A PDF copy of a poster that I presented at Mathematics of DNA Structure, Function and Interactions (Institute of Mathematics and its Applications, September 2007). 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. TheMathematical Fiction Homepage: A continuously updated, searchable, browsable list of all of the works of fiction (novels, comic books, TV Shows, films, plays, etc.) I know which involve mathematics and/or mathematicians. Each entry includes a brief review of the mathematical content, ratings and comments from site visitors, as well as links to similar works. TheAmanda Museum: This site was first created when my daughter was little and we wanted to share her drawings with relatives and friends far away. But, we stopped updating it when she finished high school. To see what she's doing now (as an art conservator) visit her Wix page. Laura Kasman's Website of Virus-Virus Interactions

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