The School of Data, Mathematical, and Statistical Sciences (SDMSS) Colloquium series offers a diverse platform for research scholars, faculty, students, and industry experts to share and exchange ideas, fostering discussion and networking across mathematics, statistics, and data science.
Professor Annalisa Calini from College of Charleston will speak at this week's colloquium on "Soliton equations and geometric flows for Legendrian and transverse curves in the 3-sphere."
Abstract: Many classical objects in differential geometry are described by soliton equations: nonlinear PDEs with infinitely many conserved quantities that are (in some sense) solvable. Since the 1980s, studies of curve evolutions that are invariant under a group of transformations have revealed more connections with well-known integrable PDEs, such as the KdV, mKdV, sine-Gordon, and NLS equations.
I will begin with two canonical examples of integrable geometric flows for curves—the vortex filament flow in Euclidean geometry and the NLS equation, and Pinkall’s flow in centro-affine geometry and the KdV equation—highlighting the role of moving frames in integrability.
In the rest of the talk, I will focus on Legendrian and transverse curves in the 3-sphere endowed with the standard contact structure, and describe how a well-known integrable model of short wave-long wave interaction arises from a simple geometric flow on transverse curves. Curves associated with periodic plane wave solutions provide a wealth of examples of closed transversed curves with non-trivial topology: I will conclude with discussing their construction and visualization, and how their knot type and geometry relate to the parameters in the solutions.
This talk is based on recent work with Tom Ivey (College of Charleston) and a long-standing collaboration with Emilio Musso (Politecnico di Torino, Italy).
Speaker Bio: Annalisa Calini received her Laurea in Physics from the Uni- versity of Milano, Italy, and her PhD in Applied Mathematics from the University of Arizona. She is a Professor of Mathe- matics at the College of Charleston, where she has served on the faculty for many years. She has held appointments as Pro- gram Officer in Applied Mathematics at the National Science Foundation and currently serves as Deputy Editor of Studies in Applied Mathematics.
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