Blog

Carles Casacuberta (Universitat de Barcelona)

Homotopy reflectivity is equivalent to the weak Vopenka principle

Abstract: It is well known that the existence of homotopical localization with respect to every (possibly proper) class of maps between spaces or spectra is implied by suitable large-cardinal axioms. However, no concluding evidence had been given that the existence of such localizations could not be proved in ZFC. Using a recent result of Trevor Wilson, we prove that the existence of localizations with respect to classes of maps of spaces or spectra is equivalent to the weak Vopenka principle, stating that there is no full embedding of the opposite category of ordinals into any locally presentable category. In fact we prove that the weak Vopenka principle is equivalent to the claim that every colocalizing subcategory of the homotopy category of any stable locally presentable model category is reflective. This is joint work with Javier Gutiérrez.

Ferran Cedó (Universitat Autònoma de Barcelona)

Indecomposable solutions of the Yang-Baxter equation of square-free cardinality

Abstract: Let $p_1,\dots,p_n$ be distinct prime numbers. Let $m_1,\dots,m_n$ be positive integers such that $m_1+\cdots+m_n>n$ . In previous joint work with J. Okni\'{n}ski, we proved that there exist simple involutive non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation with $|X|=p_1^{m_1}\cdots p_n^{m_n}$. A natural question is asked: If $n>1$, is there a simple involutive non-degenerate set-theoretic solution $(X,r)$ of the Yang-Baxter equation with $|X|=p_1\cdots p_n$?

In this talk, I will answer this question.

This is joint work with J. Okni\'{n}ski

Wolfgang Pitsch (Universitat Autònoma de Barcelona)

Witt group and Maslov index

Abstract: The main subjects of this talk will be $W(k)$, the Witt group over a field $k$, and the Maslov index of three Lagrangians in a symplectic space, which is an invariant, originally introduced in topology, taking values in $W(k)$. I will show how the machinery of Sturm sequences and Sylvester matrices developed by Barge-Lannes can be used to prove that the equivalence class of Maslov’s 2-cocycle, associated to the homonymous index, is trivial modulo $I^2$, with $I$ being the fundamental ideal of $W(k)$.