Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBilleys formula in combinatorics, geometry, and topology
202
0
0.0
(
0
)
Authors
Julianna S. Tymoczko
Ask ChatGPT about the research
No Arabic abstract
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
rate research
Read More
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.
We explicate the combinatorial/geometric ingredients of Arthurs proof of the convergence and polynomiality, in a truncation parameter, of his non-invariant trace formula. Starting with a fan in a real, finite dimensional, vector space and a collection of functions, one for each cone in the fan, we introduce a combinatorial truncated function with respect to a polytope normal to the fan and prove the analogues of Arthurs results on the convergence and polynomiality of the integral of this truncated function over the vector space. The convergence statements clarify the important role of certain combinatorial subsets that appear in Arthurs work and provide a crucial partition that amounts to a so-called nearest face partition. The polynomiality statements can be thought of as far reaching extensions of the Ehrhart polynomial. Our proof of polynomiality relies on the Lawrence-Varchenko conical decomposition and readily implies an extension of the well-known combinatorial lemma of Langlands. The Khovanskii-Pukhlikov virtual polytopes are an important ingredient here. Finally, we give some geometric interpretations of our combinatorial truncation on toric varieties as a measure and a Lefschetz number.
Frieze patterns of numbers, introduced in the early 70s by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present paper aims to review the original work of Coxeter and the new developments around the notion of frieze, focusing on the representation theoretic, geometric and combinatorial approaches.
In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Wittens conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers.
The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define negative sets to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using negative posets, we establish Stanleys reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
