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Title:Faster linear algebra using treewidth

Abstract:

: We look at the complexity of solving sparse linear systems as a function of the treewidth of the instance. That is, the sparse matrix is associated with a sparse graph, and solutions can be found faster when that graph has low treewidth. We give a parameterized algorithm in system size and treewidth achieving the conjectured optimal performance.

This is joint work with Daniel Grier.

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Title:ÌýAlice, Bob, and colours: An algebraic approach to quantum advantage

Abstract:ÌýTwo-player one-round games exhibit quantum advantage if the availability of quantum resources results in non-classical correlations between the players' answers, which allow outperforming any classical strategy. The first part of the talk illustrates a link between (i) the occurrence of quantum advantage in a game, and (ii) the complexity of the corresponding classical computational problem. In particular, I will show that both (i) and (ii) are determined by the same algebraic invariants of the game, known as polymorphisms. This allows transferring methods from the universal-algebraic theory of constraint satisfaction to the setting of quantum advantage. In particular, this approach yields a complete characterisation of the occurrence of quantum advantage for games played on graphs. ÌýÌý

The second part of the talk outlines recent work on the quantum chromatic number introduced in [Cameron--Montanaro--Newman--Severini--Winter'07]. The gap between this parameter and its classical counterpart is a measure of quantum advantage for the graph colouring game. Using the polymorphic approach, I will show that the maximum gap is linear when the quantum chromatic number makes use of entangled strategies on a constant number of qubits. In contrast, in the case of unlimited entanglement resources, I will prove the existence of graphs whose quantum chromatic number is 3 and whose classical chromatic number is arbitrarily large, conditional to the quantum variants of Khot's d-to-1 Conjecture [Khot'02] and HÃ¥stad's inapproximability of linear equations [HÃ¥stad'01].

Title:ÌýLearning from Coarse Samples

Abstract:Coarsening occurs when the exact value of a sample is not observed.
Instead, only a subset of the sample space containing the exact value is known.
Coarse data naturally arises in diverse fields, including Economics, Engineering, Medical and Biological Sciences, and all areas of the Physical Sciences.
One of the simplest forms of coarsening is rounding, where data values are mapped to the nearest point on a specified lattice.

We survey applications of coarse learning to regression with self-selection bias, regression with second-price auction data, and present details of an SGD-based algorithm for coarse Gaussian mean estimation.

Based on joint work with Alkis Kalavasis and Anay Mehrotra ()

Title:Source characterization of the hypegraphic posets

Abstract: For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented $1$-skeleton of the hypergraphic polytope $\Delta_\mathbb{H}$, which is the Minkowski sum of the standard simplices $\Delta_H$ for each hyperedge $H \in \mathbb{H}$. In 2019, C. Benedetti, N. Bergeron, and J. Machacek established a remarkable correspondence between the transitive closure of the oriented $1$-skeleton of $\Delta_\mathbb{H}$ and the flip graph on acyclic orientations of $\mathbb{H}$. Viewing an orientation of $\mathbb{H}$ as a map $A$ from $\mathbb{H}$ to $[n]$, we define the sources of the acyclic orientations as the values $A(H)$ for each hyperedge $H \in \mathbb{H}$. In a recent paper, N. Bergeron and V.

Pilaud provided a characterization of $P_\mathbb{H}$ based on the sources of acyclic orientations for interval hypergraphs. Specifically, two distinct acyclic orientations $A$ and $B$ of $\mathbb{H}$ are comparable in $P_\mathbb{H}$ if and only if their sources satisfy $A(H) \leq B(H)$ for all hyperedges $H\in \HH$. The goal of this work is to extend this source characterization of $P_\mathbb{H}$ to arbitrary hypergraphs on $[n]$.

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

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.

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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.

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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,