Special Session on Geometric Complexity Theory
During August 19 - 23, 2013 conference
Differential Geometry and its Applications, Brno, Cech Republic
Organized by J.M. Landsberg
A. Abdesselam (U. Virginia)
E. Briand (U. Sevilla)
P. Burgisser (U. Paderborn)
C. Ikenmeyer (TAMU)
S. Kumar (UNC Chapel Hill)*
J.M. Landsberg (TAMU)
*Kumar will also be a plenary speaker at the conference
Titles and abstracts:
Abdesselam: On the Foulkes-Howe conjecture and why it mattered to 19th century mathematicians
We will discuss the Foulkes-Howe conjecture and its straightforward "cabled" generalization
regarding the injectivity or surjectivity of a map from the p-th symmetric power of
a qr-th symmetric power into the analogous object with p and q exchanged.
We will explain this problem from the point of view of classical invariant and elimination theory,
in relation to the work of Hermite, Gordan, Hadamard and many others in their quest for a good
understanding of multidimensional resultants.
Briand: Recent progress on Kronecker coefficients.
I will present some recent results about Kronecker coefficients, in
particular about the sequences of Kronecker coefficients indexed by
three partitions with variable first part. These sequences are
eventually constant and their limit (the "stable Kronecker
coefficients") are interesting objects by themselves.
Besides I will examine closely the explicit formulas known for the
Kronecker coefficients indexed by three partitions of lengths 2, 2 and
4, and what this example may tell us about the general case.
Ikenmeyer: Explicit Lower Bounds via Geometric Complexity Theory
Kumar: Some geometric and representation theoretic aspects of the orbit
closures of determinant and permanent
We prove the lower bound R(M_m) \geq 3/2 m - 2 on the border rank of m 2
x m matrix multiplication by exhibiting explicit representation
theoretic (occurence) obstructions in the sense of the geometric
complexity theory (GCT) program. While this bound is weaker than the one
recently obtained by Landsberg and Ottaviani, these are the first
significant lower bounds obtained within the GCT program. Behind the
proof is the new combinatorial concept of obstruction designs, which
encode highest weight vectors in Sym^d\otimes(C^n)^* and provide new 3
insights into Kronecker coefficients.
Landsberg: New open questions in GCT
Recent advances in the study of shallow circuits
implies that the standard conjectures in algebraic complexity
theory (e.g. VP\neq VNP) can be phrased in terms of
secant varieties of Chow varieties and other natural
G-varieties. I will explain the advances and the new problems.