Fri, March 10, 2017
Public Access Category: All |
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- Time:
- 10:30am
- Location:
- Ramanujan Hall
- Description:
- Title: The Robinson-Schensted-Knuth Algorithm for Real Matrices
Abstract: The Robinson-Schensted-Knuth (RSK) correspondence is a bijection
from the set of matrices with non-negative integer entries onto the set of
pairs of semistandard Young tableaux (SSYT) of the same shape. SSYT can be
expressed as integral Gelfand-Tsetlin patterns. We will show how Viennot's
light-and-shadows algorithm for computing the RSK correspondence can be
extended from matrices with non-negative integer entries to matrices with
non-negative real entries, giving rise to real Gelfand-Tsetlin patterns.
This real version of the RSK correspondence is piecewise-linear. Indeed,
interesting combinatorial problems count lattice points in polyhedra, and
interesting bijections are induced by volume-preserving piecewise-linear
maps.
- Time:
- 3:30pm-5:00pm
- Location:
- Ramanujan Hall
- Description:
- Title: Koszul algebras V
Abstract: In the first half of the talk, we shall recall Koszul filtation
and Grobner flag. Let R be a standard graded algebra. If R has a Koszul
filtation, then R is Koszul. If R has a Grobner flag, then R is
G-quadratic. I will mention an important result of Conca, Rossi, and
Valla: Let R be a quadratic Gorenstein algebra with Hilbert series 1 + nz
+ nz^2 + n^3. Then for n=3 and n=4, R is Koszul.
In the second half of the talk, we shall focus on class of strongly Koszul
algebras. If time permits, I will prove that Koszul algebras are preserved
under various classical constructions, in particular, under taking tensor
products, Segre products, fibre products and Veronese subrings.