[Topology.beta] DDT&G seminar @Leiden on 7 March: Guy Boyde on Temperley-Lieb algebras. AG by Lies

[Rot, T.O. (Thomas)]

Dear all,

Next week we have the next edition of the DDT&G. Please see the announcement below. Please let interested master students know.

Of course we will also have an AG on Tuesday. Lies will talk about Extremal Betti Numbers. Abstract below.

Best,
Thomas

Dear colleagues,

You are cordially invited to attend the next Dutch Differential Topology and Geometry seminar, which will take place next week in Leiden.

Speaker: Guy Boyde<https://guyboyde.wordpress.com/>
Title: "The homology of Temperley-Lieb algebras" (see below for the abstract)
Date: Friday 7 March, 14:30-17:00
Location: Leiden, BW.0.32 (Gorleaus)

The schedule will be as follows:

2:30-3:30 p.m.: first part of the seminar (little to no prior knowledge assumed)
4:00-5:00 p.m.: second part of the seminar (slightly more advanced level)

Please notice that the first part of the seminar (2:30-3:30 p.m.) is intended for general mathematical audience, and master students with an interest in geometry and topology are especially encouraged to participate.

Please visit the seminar's webpage for additional information and an overview of past and upcoming talks:

https://www.few.vu.nl/~trt800/<https://www.few.vu.nl/~trt800/ddtg.html>ddt<https://www.few.vu.nl/~trt800/ddtg.html>g.html<https://www.few.vu.nl/~trt800/ddtg.html>

We hope to see many of you next week!

Alvaro, Federica and Thomas

Abstract

  Temperley-Lieb algebras are certain (very pictorial) finite-dimensional associative algebras coming originally from statistical physics and knot theory. There are several ways to define homology groups for an algebra: we will focus on just one. The punchline of the talk is that the homology of Temperley-Lieb algebras has a rich algebraic structure coming from the "geometry" of the original algebras.
  In the first half of the talk, I'll give a friendly introduction to these algebras: we'll see why they're important in knot theory, and prove that their dimension is given by the Catalan numbers. We'll also discuss the homological algebra that we will need in the rest of the talk (which will be very elementary and hands on). In the second talk I'll explain what we know about their homology, ending with joint work in progress with Rachael Boyd, Oscar Randal-Williams, and Robin Sroka.

AG: 4-3-2025 11:00 Lies Beers: Extremal Betti Numbers

[Joint work with Magnus Botnan] Let T(n,k+1) be the Turan graph with n vertices and k+1 partition classes. We study extremal Betti numbers and persistence in edgewise filtrations of flag complexes. For a graph G on n vertices, the kth Betti number of its flag complex is maximized when G = T(n,k+1). Extending this, we construct an edgewise filtration in which each graph attains the maximal kth Betti number among all graphs with the same number of edges. Moreover, the persistence barcode achieves the maximal number of intervals and total persistence among all edgewise filtrations with |E(T(n,k+1))| edges.
For k=1, we analyze edgewise filtrations of the complete graph. The maximal number of barcode intervals occurs precisely when T(n,2) appears in the filtration. Among these, we characterize those achieving maximal total persistence. We also prove that no filtration optimizes the first Betti number for all graphs in the filtration and conjecture that our constructions maximize total persistence over all edgewise filtrations of the complete graph.


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