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Register a new userSplines in geometry and topology
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Julianna Tymoczko
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This survey paper describes the role of splines in geometry and topology, emphasizing both similarities and differences from the classical treatment of splines. The exposition is non-technical and contains many examples, with references to more thorough treatments of the subject.
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In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billeys formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag variety and of the Schubert varieties, and the combinatorics of root systems (generalizing
Semialgebraic splines are functions that are piecewise polynomial with respect to a cell decomposition into sets defined by polynomial inequalities. We study bivariate semialgebraic splines, formulating spaces of semialgebraic splines in terms of graded modules. We compute the dimension of the space of splines with large degree in two extreme cases when the cell decomposition has a single interior vertex. First, when the forms defining the edges span a two-dimensional space of forms of degree n---then the curves they define meet in n^2 points in the complex projective plane. In the other extreme, the curves have distinct slopes at the vertex and do not simultaneously vanish at any other point. We also study examples of the Hilbert function and polynomial in cases of a single vertex where the curves do not satisfy either of these extremes.
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by Chekhov, Eynard and Orantin. We call a map fully simple if it has non self-intersecting disjoint boundaries, and ordinary if such a restriction is not imposed. We study the combinatorial relation between fully simple and ordinary maps with the topology of a disk or a cylinder, which reproduces relations between moments and free cumulants established in the context of free probability. We propose a combinatorial interpretation of the exchange symplectic transformation of TR. We provide a matrix model interpretation for fully simple maps via the formal hermitian matrix model with external field and deduce a universal relation between generating series of fully simple and ordinary maps, which involves double monotone Hurwitz numbers. In particular, we obtain an ELSV-like formula for double $2$-orbifold strictly monotone Hurwitz numbers. We consider ordinary maps endowed with an $O(mathsf{n})$ loop model, which is a classical model in statistical physics, and determine which shapes are more likely to occur by looking at the nesting properties of the loops decorating the maps. We want to study the limiting objects when the number of vertices becomes arbitrarily large, which can be done by studying the generating series at dominant singularities. We analyze the nesting statistics in the $O(mathsf{n})$ loop model on random maps of arbitrary topologies in the presence of large and small boundaries, relying on previous results for disks and cylinders and TR for this model. We study the generating series of maps which realize a fixed nesting graph and characterize their critical behavior in the dense and dilute phases.
Second order spiral splines are $C^2$ unit-speed planar curves that can be used to interpolate a list $Y$ of $n+1$ points in $R ^2$ at times specified in some list $T$, where $ngeq 2$. Asymptotic methods are used to develop a fast algorithm, based on a pair of tridiagonal linear systems and standard software. The algorithm constructs a second order spiral spline interpolant for data that is convex and sufficiently finely sampled.
We discuss the direct use of cubic-matrix splines to obtain continuous approximations to the unique solution of matrix models of the type $Y(x) = f(x,Y(x))$. For numerical illustration, an estimation of the approximation error, an algorithm for its implementation, and an example are given.
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