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Title:Constraining moduli space cohomology by counting graphs

Abstract: In 1992, Kontsevich defined complexes spanned by graphs. TheseÌý
complexes are increasingly prominent in algebraic topology, geometricÌý
group theory and mathematical physics. For instance, a 2021 theorem byÌý
Chan-Galatius and Payne implies that the top-weight cohomology of theÌý
moduli space of curves of genus g is equal to the homology of a specificÌý
graph complex. I will present a new theorem on the asymptotic growthÌý
rate of the Euler characteristic of this graph complex and explain itsÌý
implication on the cohomology of the moduli space of curves. The proofÌý
involves solving a specific graph counting problem.

Ìý

Title:Splitting algorithms for monotone inclusions with minimalÌýdimension

Abstract: Many situations in convex optimization can be modeled as the problem ofÌýfinding a zero of a monotone operator, which can be regarded as aÌýgeneralization of the gradient of a differentiable convex function. InÌýorder to numerically address this monotone inclusion problem, it isÌývital to be able to exploit the inherent structure of the operatorÌýdefining it. The algorithms in the family of the splitting methodsÌýachieve this by iteratively solving simpler subtasks that are defined byÌýseparately using some parts of the original problem.

In the first part of this talk, we will introduce some of the mostÌýrelevant monotone inclusion problems and present their applications toÌýoptimization. Subsequently, we will draw our attention to a commonÌýanomaly that has persisted in the design of methods in this family: theÌýdimension of the underlying space —which we denote as lifting— of theÌýalgorithms abnormally increases as the problem size grows. This hasÌýdirect implications on the computational performance of the method as aÌýresult of the increase of memory requirements. In this framework, weÌýcharacterize the minimal lifting that can be obtained by splittingÌýalgorithms adept at solving certain general monotone inclusions.ÌýMoreover, we present splitting methods matching these lifting bounds, and thusÌýhaving minimal lifting.

Ìý

Title:The external activity complex of a pair of matroids

Abstract: In 2016, Ardila and Boocher were investigating the variety obtained by taking the closure of a linear space within A^n in its compactification (P^1)^n; later work named this the "matroid Schubert variety". Its Gröbner degenerations led them to define, and study the commutative algebra of, the _external activity complex_ of a matroid. If the matroid is on n elements, this is a complex on 2n vertices whose facets encode the external activity of bases.

In recent work with Andy Berget on Speyer's g-invariant, we required a generalisation of the definition of external activity where the input was a pair of matroids on the same ground set. We generalise many of the results of Ardila--Boocher to this setting. Time permitting, I'll also present the tropical intersection theory machinery we use to understand the external activity complex of a pair.

For those who attended my talk at this year's CAAC on this paper, the content of the present talk is meant to be complementary.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm,