In a landmark paper in 1992, Stanley developed the foundations
of what is now known as the Kazhdan--Lusztig--Stanley (KLS) theory.
To each kernel in a graded poset, he associates special functions
called KLS polynomials. This unifies and puts a...
We heard last week in Daoji's talk about positroid varieties,
which are subvarieties in the Grassmannian defined by cyclic rank
conditions, and which are related to Schubert varieties. In this
talk, we will provide a criterion for whether positroid...
A valuation is a finitely additive measure on the class of all
convex compact subsets of Rn. Over the past two decades, a number
of structures has been discovered on the space of translation
invariant smooth valuations. Recently, these findings...
Influential work of Hodge from the 1940s led the way in using
Gröbner bases to combinatorially study the Grassmannian. We follow
Hodge's approach to investigate certain subvarieties of the
Grassmannian, called positroid varieties. Positroid...
The theme of the lecture is the notion of points over F1, the
field with one element. Several heuristic computations led to
certain expectations on the set of F1-points: for example the Euler
characteristic of a smooth projective complex variety X...
Schubert Calculus studies cohomology rings in (generalized) flag
varieties, equipped with a distinguished basis - the fundamental
classes of Schubert varieties - with structure constants satisfying
many desirable properties. Cotangent Schubert...
Vertex decomposition, introduced by Provan and Billera in 1980,
is an inductive strategy for breaking down and understanding
simplicial complexes. A simplicial complex that is vertex
decomposable is shellable, hence Cohen--Macaulay. Through
the...
A class of tensors, called "concise (m,m,m)-tensors of
minimal border rank", play an important role in proving upper
bounds for the complexity of matrix multiplication. For that reason
Problem 15.2 of "Algebraic Complexity Theory" by Bürgisser...
Chapter 14 of the classic text "Computational Complexity" by
Arora and Barak is titled "Circuit lower bounds: complexity
theory's Waterloo". I will discuss the lower bound problem in the
context of algebraic complexity where there are barriers...
The second lecture features the nuts and bolts of the invariants
from first lecture, which we call foundations. We explain the
structure theorem for foundations of ternary matroids, which is
rooted in Tutte's homotopy theorem. We show how this...
