Queen Mary University London

Questions in algebra, while deep and interesting, can be incredibly difficult. Thankfully, when studying the representation theory of the symmetric groups, one can often take algebraic properties and results and write them in the language of combinatorics; where one has a wide variety of tools and techniques to use. In this talk, we will look at the specific example of the submodule structure of 2-part Specht modules in characteristic 2, and answer which hook Specht modules are uniserial in characteristic 2. We will not need to assume the Riemann hypothesis for this talk.

In 1972 Robert May argued that (generic) complex systems become unstable to small displacements from equilibria as the system complexity increases. His analytical model and outlook was linear. I will talk about a “minimal” non-linear extension of May’s model – a nonlinear autonomous system of N ≫ 1 degrees of freedom randomly coupled by both relaxational (’gradient’) and non-relaxational (’solenoidal’) random interactions. With the increasing interaction strength such systems undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically non-trivial regime where equilibria are on average exponentially abundant, but typically all of them are unstable, unless the dynamics is purely gradient. When the interaction strength increases even further the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. One can investigate these transitions with the help of the Kac-Rice formula for counting zeros of random functions and theory of random matrices applied to the real elliptic ensemble with some of the mathematical problems remaining open. This talk is based on collaborative work with Gerard Ben Arous and Yan Fyodorov.

Reducing a chain complex (whilst preserving its homotopy-type) using algebraic Morse theory ([1, 2, 3]) gives the same end-result as Gaussian elimination, but AMT does it only on certain rows/columns and with several pivots (in all matrices simultaneously). Crucially, instead of doing costly row/column operations on a sparse matrix, it computes traversals of a bipartite digraph. This significantly reduces the running time and memory load (smaller fill-in and coefficient growth of the matrices). However, computing with AMT requires the construction of a valid set of pivots (called a Morse matching).

In [4], we discover a family of Morse matchings on any chain complex of free modules of finite rank. We show that every acyclic matching is a subset of some member of our family, so all maximal Morse matchings are of this type.

Both the input and output of AMT are chain complexes, so the procedure can be used iteratively. When working over a field or a local PID, this process ends in a chain complex with zero matrices, which produces homology. However, even over more general rings, the process often reveals homology, or at least reduces the complex so much that other algorithms can finish the job. Moreover, it also returns homotopy equivalences to the reduced complexes, which reveal the generators of homology and the induced maps $H_{*}(\varphi)$. 

We design a new algorithm for reducing a chain complex and implement it in MATHEMATICA. We test that it outperforms other CASs. As a special case, given a sparse matrix over any field, the algorithm offers a new way of computing the rank and a sparse basis of the kernel (or null space), cokernel (or quotient space, or complementary subspace), image, preimage, sum and intersection subspace. It outperforms built-in algorithms in other CASs.

References

[1]  M. Jöllenbeck, Algebraic Discrete Morse Theory and Applications to Commutative Algebra, Thesis, (2005).

[2]  D.N. Kozlov, Discrete Morse theory for free chain complexes, C. R. Math. 340 (2005), no. 12, 867–872.

[3]  E. Sköldberg, Morse theory from an algebraic viewpoint, Trans. Amer. Math. Soc. 358 (2006), no. 1, 115–129.

[4]  L. Lampret, Chain complex reduction via fast digraph traversal, arXiv:1903.00783.