#### Andrew Brooke-Taylor

University of Leeds, Leeds

About Borel Reducibility of Equivalence Relations

Classification has always been a central theme in mathematics. The study of **Borel Reducibility of Equivalence Relations** deals with the classification of points of standard Borel spaces up to equivalence relations by explicit, or Borel, mappings between such spaces.

This idea gives rise to a notion of complexity of equivalence relations, and tools from **Descriptive Set Theory** are used to compare such relations and measure their complexities.

University of Leeds, Leeds

Kurt Gödel Research Centre, Vienna

Université Claude Bernard Lyon 1, Lyon

Università di Torino, Turin

10:30-11:30 | Amphipôle, Room 340 | Julien Melleray | TBA | TBA | |

TBA | Lunch Break | ||||

13:00-14:00 | Amphipôle, Room 340 | Andrew Brooke-Taylor | Borel completeness of a knot invariant | The algebraic structures called "quandles" constitute a complete invariant for knots: there is a uniform way to assign a quandle to any tame knot, in such a way that different knots give rise to different quandles. However, knot theorists are generally unimpressed with quandles, complaining that they seem to be too hard to tell apart. I will talk about joint work with Sheila Miller using the concepts of Borel reducibility to show just how difficult this problem is. If time permits, I will also discuss ongoing work with Miller and Filippo Calderoni extending these results by considering the embeddability relation of quandles. | |

Coffee Break | |||||

14:30-15:30 | Amphipôle, Room 340 | Luca Motto Ros | TBA | TBA | |

Coffee Break | |||||

16:00-17:00 | Amphipôle, Room 340 | Raphaël Carroy | Topological embeddability between functions | A function f embeds topologically in a function g if there are two topological embeddings b and c such that bf=gc. This notion was considered by Sławomir Solecki to find finite bases for non sigma-continuous functions between Polish spaces. Since then, finite bases have been found for several other classes of functions, including a finite basis for non Baire class one functions (joint work with Benjamin Miller). As a quasi-order, topological embeddability is however far from being always "simple". As a matter of fact, even on spaces of continuous functions it is very often analytic complete. I will pinpoint exactly when topological embeddability is an analytic complete quasi-order on continuous functions between Polish 0-dimensional spaces. This gives that on these spaces of functions, topological embeddability is either analytic complete or well-founded and without infinite antichains (joint work with Yann Pequignot and Zoltan Vidnyanszky). |

Julien Melleray

TBA

Andrew Brooke-Taylor

The algebraic structures called "quandles" constitute a complete invariant for knots: there is a uniform way to assign a quandle to any tame knot, in such a way that different knots give rise to different quandles. However, knot theorists are generally unimpressed with quandles, complaining that they seem to be too hard to tell apart. I will talk about joint work with Sheila Miller using the concepts of Borel reducibility to show just how difficult this problem is. If time permits, I will also discuss ongoing work with Miller and Filippo Calderoni extending these results by considering the embeddability relation of quandles.

Luca Motto Ros

TBA

Raphaël Carroy

A function f embeds topologically in a function g if there are two topological embeddings b and c such that bf=gc. This notion was considered by Sławomir Solecki to find finite bases for non sigma-continuous functions between Polish spaces. Since then, finite bases have been found for several other classes of functions, including a finite basis for non Baire class one functions (joint work with Benjamin Miller). As a quasi-order, topological embeddability is however far from being always "simple". As a matter of fact, even on spaces of continuous functions it is very often analytic complete. I will pinpoint exactly when topological embeddability is an analytic complete quasi-order on continuous functions between Polish 0-dimensional spaces. This gives that on these spaces of functions, topological embeddability is either analytic complete or well-founded and without infinite antichains (joint work with Yann Pequignot and Zoltan Vidnyanszky).

**UniL, Amphipôle**, 1015 Lausanne, Room **340**

University of Leeds, Leeds

Andrew Brooke-Taylor has travelled the world pursuing his mathematics career. After growing up in northern Australia, he studied at the Australian National University, MIT, and the University of Vienna. He held postdoctoral positions in Bristol and Kobe, and then was awarded his EPSRC fellowship, allowing him to return to the UK. Read more on his personal website.

Kurt Gödel Research Centre, Vienna

Since June 2014, Raphaël Carroy is postdoctoral fellow at the Kurt Gödel Research Center of the University of Vienna, in Austria. His general field is logic and descriptive set theory, and he is more specifically interested in Borel functions, better-quasi-orders (bqo), and infinite games. For more information, visit his personal website.

Université Claude Bernard Lyon 1, Lyon

Julien Melleray is Maître de Conférences at Université Lyon 1 since 2007. He obtained his PhD at Paris 6 under the supervision of Prof. J. Saint Raymond and later was Research Assistant Professor at the University of Illinois at Urbana-Champaign. He enjoys playing tennis. More information is available on his personal website.

Università di Torino, Turin

Luca Motto Ros is a tenure-track researcher (ricercatore TD di tipo B) at the Mathematics Department of the University of Turin. Previously, he was a post-doc for three years (Oct 2007-Sep 2010) at the Kurt Gödel Research Center for Mathematical Logic (chair: Sy D. Friedman) at the University of Vienna, and then an assistant professor at the Logic Department of the University of Freiburg (Oct 2010-Sep 2014). He got his PhD from the Polytechnic of Turin in 2007 under the direction of Riccardo Camerlo, professor at the Polytechnic of Turin, and Alessandro Andretta, professor at the University of Turin. More information is available on his personal website.

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