Alex Kasman
Professor and Co-Director, Mathematics Graduate Program
Phone: 843.953.8018
E-mail: kasmana@cofc.edu
Personal Website: http://kasmana.people.cofc.edu/
Research Interests
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:
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The particle-like waves called `solitons'.
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The symmetry of linear wave equations known as `bispectrality'
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Algebro-geometric constructions for producing quantum integrable systems
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Rank one perturbations of operator identities and their connections to discrete integrable systems including the Bethe Ansatz and the Hirota Bilinear Difference Equation.
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The geometry of Grassmannian manifolds and associated functions which satisfy nonlinear partial differential equations of mathematical physics
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Classical and quantum integrable particle systems
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The geometry (spectral varieties) of commutative rings of partial differential operators.
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Solutions to the KP hierarchy associated to higher rank vector bundles over singular rational curves