
Combinatorics/Algebra Seminar
The seminar meets Mondays at 4.10 pm in Pearson 2158.
If you or a guest wish to talk, please visit the Post Office.
October 1
Hal Schenck: Wachspress varieties: algebraic geometry meets numerical analysis.
Abstract:
Let P_{d} be a convex polygon with d vertices. The associated Wachspress surface W_{d} is a fundamental object in approximation theory, defined as the image of the rational map w_{d} from P^{2} to P^{{d1}}, determined by the Wachspress barycentric coordinates for P_{d}. We show w_{d} is a regular map on a blowup X_{d} of P^{2}, and if d > 4 is given by a very ample divisor on X_{d}, so has a smooth image W_{d}. We determine generators for the ideal of W_{d}, and prove that in graded lex order, the initial ideal of I(W_{d}) is given by a StanleyReisner ideal. As a consequence, we show that the associated surface is arithmetically CohenMacaulay, of CastelnuovoMumford regularity two, and determine all the graded Betti numbers of I(W_{d}).

October 15, 22

Joey Iverson: tba.

October 8

Alex Nowak: tba.

October 1

Hal Schenck: Wachspress varieties: algebraic geometry meets numerical analysis.
Abstract:
Let P_{d} be a convex polygon with d vertices. The associated Wachspress surface W_{d} is a fundamental object in approximation theory, defined as the image of the rational map w_{d} from P^{2} to P^{{d1}}, determined by the Wachspress barycentric coordinates for P_{d}. We show w_{d} is a regular map on a blowup X_{d} of P^{2}, and if d > 4 is given by a very ample divisor on X_{d}, so has a smooth image W_{d}. We determine generators for the ideal of W_{d}, and prove that in graded lex order, the initial ideal of I(W_{d}) is given by a StanleyReisner ideal. As a consequence, we show that the associated surface is arithmetically CohenMacaulay, of CastelnuovoMumford regularity two, and determine all the graded Betti numbers of I(W_{d}).

September 24

Anna Romanowska (Warsaw University of Technology): Extensions of convex sets.
Abstract:
Convex sets may be viewed as abstract algebras, sets closed under the set of binary convex combinations indexed by the open unit interval of real numbers. Such algebras appear inside affine real spaces viewed as abstract algebras, sets equipped with binary affine combinations.
In this talk, I will discuss certain algebraic extensions of the
concept of a convex set. Such extensions take (ordered) subfields
(or subrings) of the field of real numbers, and consider different intervals within such rings.

September 10, 17

Jonathan Smith: Character groups and quasigroups.
Abstract:
Duality for finite abelian groups, which includes discrete and fast Fourier transforms, provides a group structure for the multiplication of characters of a finite abelian group. In these talks, we will discuss a new approach to aspects of the representation theory of finite nonabelian groups, introducing character groups and quasigroups which encode the decomposition of the product of irreducible characters as sums of irreducible characters.
Archive of earlier seminars
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