Splines at the crossroads of algebra, combinatorics, geometry and numerical analysis, or Introduction to zonotopal algebra I
speaker: Amos Ron (University of Wisconsin - Madison)abstract: Splines are compactly supported piecewise analytic functions defined on the Euclidean domain ℝn, n ≥ 1. In my talk, I will describe connections between spline theory on the one hand, and various topics, setups and problems in algebra, in combinatorics, in geometry and in numerical analysis.
A partial list of areas and topics that are pertinent (not all discussed...) is the following:
1. Numerical analysis (e.g., multivariate polynomial interpolation)
2. Graph theory (e.g., Tutte polynomial and parking functions)
3. Enumerative combinatorics (e.g., f-vectors of a matroid or magic square problems)
4. Convex geometry and lattice geometry (e.g., hyperplane arrangement and zonotopes).
5. Non-commutative algebra and group representation (e.g., partition functions, Weyl’s formula and Macdonald polynomials)
6. Commutative algebra, algebraic geometry and homological algebra (e.g., power ideals, toric varieties, and (equivariant) cohomology of hyperplane arrangements)
The actual talk contains fours parts. The 1st is an elboration on specific nature of the interface between spline theory and algebra, combinatorics and geoemtry. The 2nd is quick histrorical survey of zonotopal algebra. The 3rd introduces the the least map and its basic properties. This map was borne out of the attempt to provide a universal and canonical solution to the problem of polynomial interpolation at arbitrary finite pointset of ℝn. However, I will use it in my talk for a different purpose: the map connects between geometry and algebra, hence can be used to validate the coherence in our constructions and theories. The 4th part consists of examples within the zonotopal algebra framework that illustrates this point.
timetable:
Wed 23 Jun, 9:00 - 10:00, Aula Dini
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