When defining a monoid structure on an arbitrary type in HoTT, one should require a multiplication that is not only homotopy-associative, but also has an infinite tower of higher homotopies. For example in dimension two one should have a condition similar to Mac Lane’s pentagon for monoidal categories. We call such a monoid a monoid up to coherent homotopy. The goal of my internship in Stockholm was to formalize them in Agda. It is well-known that infinite towers of homotopies are hard to handle in plain HoTT, so we postulate a variant of two-level type theory, with a strict equality and an interval type. Then we adapt the set-theoretical treatment of monoids up to coherent homotopy using operads as presented by Clemens Berger and Ieke Moerdijk [1,2].

Our main results are: (a) Monoids up to coherent homotopy are invariant under homotopy equivalence (b) Loop spaces are monoids up to coherent homotopy.

In this talk I will present the classical theory of monoids up to coherent homotopy, and indicates how two-level type theory can be used to formalize it.

References

1. Axiomatic homotopy theory for operads (arxiv.org/abs/math/0206094)

2. The Boardman-Vogt resolution of operads in monoidal model categories (arxiv.org/abs/math/0502155)

When defining a monoid structure on an arbitrary type in HoTT, one should require a multiplication that is not only homotopy-associative, but also has an infinite tower of higher homotopies. For example in dimension two one should have a condition similar to Mac Lane’s pentagon for monoidal categories. We call such a monoid a monoid up to coherent homotopy. The goal of my internship in Stockholm was to formalize them in Agda. It is well-known that infinite towers of homotopies are hard to handle in plain HoTT, so we postulate a variant of two-level type theory, with a strict equality and an interval type. Then we adapt the set-theoretical treatment of monoids up to coherent homotopy using operads as presented by Clemens Berger and Ieke Moerdijk [1,2].

Our main results are: (a) Monoids up to coherent homotopy are invariant under homotopy equivalence (b) Loop spaces are monoids up to coherent homotopy.

In this talk I will present the classical theory of monoids up to coherent homotopy, and indicates how two-level type theory can be used to formalize it.

References

1. Axiomatic homotopy theory for operads (arxiv.org/abs/math/0206094)

2. The Boardman-Vogt resolution of operads in monoidal model categories (arxiv.org/abs/math/0502155)

I will introduce a new tool for quantum algorithms called quantum fast-forwarding (QFF). The tool uses quantum walks as a means to quadratically fast-forward a reversible Markov chain. In a number of quantum walk steps that scales as the square root of the classical runtime, and inversely proportional to the 2-norm of the classical distribution, it retrieves a quantum superposition that encodes the classical distribution. This shows that quantum walks can accelerate the transient dynamics of Markov chains, thereby complementing the line of results that proves the acceleration of their limit behavior.

This tool leads to speedups on random walk algorithms in a very natural way. Specifically I will consider random walk algorithms for testing the graph expansion and clusterability, and show that QFF allows to quadratically improve the dependency of the classical property testers on the random walk runtime. Moreover, this new quantum algorithm exponentially improves the space complexity of the classical tester to logarithmic. As a subroutine of independent interest, I use QFF to determine whether a given pair of nodes lies in the same cluster or in separate clusters. This solves a robust version of s-t connectivity, relevant in a learning context for classifying objects among a set of examples. The different algorithms crucially rely on the quantum speedup of the transient behavior of random walks.

In algorithmic enumeration, one is typically asked to output the set of all objects of a certain type related to an input (e.g. all the cycles of a graph, all its maximal cliques, etc.), rather than finding just one. The problem of enumerating all the minimal dominating sets of a graph received an important attention, partly because of its strong links with a fundamental hypergraph problem. A long standing open question is whether the above problem can be solved in (output-) polynomial time. In this talk, I will introduce the problem and present an output-polynomial algorithm for Kt-free graphs. This is joint work with Marthe Bonamy, Oscar Defrain, and Michał Pilipczuk.

Link to the article: https://arxiv.org/abs/1810.00789