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
contact: jml@math.tamu.edu

Confirmed speakers:

A. Abdesselam (U. Virginia)
E. Briand (U. Sevilla)
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 TheoryWe prove the lower bound R(M_m) \geq 3/2 m^2 - 2 on the border rank of mx m matrix multiplication by exhibiting explicit representationtheoretic (occurence) obstructions in the sense of the geometriccomplexity theory (GCT) program. While this bound is weaker than the onerecently obtained by Landsberg and Ottaviani, these are the firstsignificant lower bounds obtained within the GCT program. Behind theproof is the new combinatorial concept of obstruction designs, whichencode highest weight vectors in Sym^d\otimes^3(C^n)^* and provide newinsights into Kronecker coefficients.
Kumar: Some geometric and representation theoretic aspects of the orbit closures of determinant and permanent

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.