Bucharest Topology Days

Schedule and abstracts

	Monday 21 July	Tuesday 22 July
09:30–10:30	Xiaolei Wu	Martin Palmer-Anghel
	Coffee break
11:00–12:00	Cristina Palmer-Anghel	George Altmann
	Lunch
14:00–15:00	Mihai-Cosmin Pavel	Awais Shaukat
	Coffee break
15:30–16:30	Anca Măcinic

Xiaolei Wu
Embedding groups into bounded acyclic groups
We first discuss various embedding results for groups in the literature. Then we talk about how could one embed a group of type F_n into a group of type F_n with no proper finite index subgroup quasi-isometrically. The embedding we have uses the so called labelled Thompson groups, and it is functorial. We also show that the labeled Thompson group is always bounded acyclic. As a corollary, one could embed any group of type F_n into a bounded acyclic group of type F_n quasi-isometrically. This is based on a joint work with Fan Wu, Mengfei Zhao and Zixiang Zhou. If time permitted, I will also talk about some ongoing work with Martin Palmer.

Cristina Palmer-Anghel
Universal link invariants via configuration spaces
Coloured Jones and Alexander polynomials are quantum invariants originating in representation theory. They are building blocks for 3-manifold quantum invariants, and their geometric information is an important open problem in quantum topology. We will describe them from a unified topological viewpoint. For a fixed level N , we define new link invariants: "Nth Unified Jones invariant" and "Nth Unified Alexander invariant". They globalise topologically all coloured Jones and all ADO link polynomials with (multi-)colours bounded by N. This shows that all coloured Jones and coloured Alexander polynomials at bounded (multi-)level are encoded by the same Lagrangian intersections in a fixed configuration space.

Then, asymptotically, we define geometrically a universal ADO link invariant and universal Jones link invariant. The question of providing a universal invariant recovering all ADO link polynomials was an open problem. A parallel question about semi-simple knot invariants is the subject of Habiro's famous universal invariant. Our universal Jones invariant recovers all coloured Jones polynomials, providing a new semi-simple universal link invariant. The first non semi-simple universal link invariant that we construct unifies geometrically all ADO link invariants.

Mihai-Cosmin Pavel
Moduli of higher-rank PT-stable pairs on threefolds
Stable pairs were introduced by Pandharipande and Thomas to define new curve-counting invariants on Calabi–Yau threefolds. It was soon observed that such objects can be understood via a generalized notion of stability on the derived category of coherent sheaves. This notion, known as PT-stability, extends the original construction and recovers the stable pairs of Pandharipande and Thomas as PT-stable objects of rank 1 and trivial determinant. One is naturally led to study the moduli theory of PT-stable objects on projective threefolds. However, unlike the original case, it is unknown whether the moduli space of higher-rank PT-stable objects is projective. In this talk, we present recent progress on this problem, based on joint work with Tuomas Tajakka.

Anca Măcinic
Geometric obstructions for free CL-arrangements
We discuss generalizations to conic-line arrangements of some geometric/numerical properties of free projective line arrangements.

Martin Palmer-Anghel
On the homology of asymptotic monopole moduli spaces
Magnetic monopoles were introduced by Dirac in 1931 to explain the quantisation of electric charges. In his model, they are singular solutions to an extension of Maxwell's equations allowing non-zero magnetic charges. An alternative model, developed by 't Hooft and Polyakov in the 1970s, is given, after a certain simplification, by smooth solutions to a different set of equations, the Bogomolny equations, whose moduli space of solutions has connected components M_k indexed by positive integers k (the "total magnetic charge"); these spaces M_k are non-compact manifolds. Partial compactifications have recently been constructed by Kottke and Singer, whose boundary strata formalise the notion of "ideal" or "asymptotic" monopole moduli spaces. I will describe joint work with Ulrike Tillmann in which we prove the existence of stability patterns in the homology of these spaces. I will also mention work in progress on calculating their stable homology.

George Altmann
Enriching Welded Knot Groups via Loop Braid Group Representations
Welded links are an extension of classical links introduced by Kauffman in 1999, which are closely related to the loop braid group, analogous to how classical links can be seen as closures of braids. The Wirtinger group of a welded knot is a well-known invariant, which, when restricted to classical knots, arises naturally as the fundamental group of the complement. Furthermore, we can enrich the Wirtinger group with some additional data known as the peripheral system, giving a complete invariant for classical links. For the class of welded knots, we can enhance the Wirtinger group W(D) further with a Z[W(D)]-module for any welded link D. This is motivated by a representation of the loop braid group, which acts non-trivially on the second homotopy of some topological space. Moreover, we can lift the peripheral system to this new invariant, which conjecturally can be realised via the topology of the ribbon torus image of a link under Satoh's tube map.

Awais Shaukat
Low Dimensional Topology in Cryptography
Based on the braid group and its problems, two cryptosystems were suggested about a decade ago: by Anshel, Anshel and Goldfeld in 1999 and by Ko, Lee, Cheon, Han, Kang and Park in 2000. These cryptosystems initiated a wide discussion about the possibilities of cryptography in the braid group especially. The Conjugacy problem in braid groups forms the basis for many proposed cryptosystems. Braids are of cryptographic interest because computations and data storage can be performed quite efficiently, but they are complex enough that at first glance it seems unlikely that they have any unexpected underlying structure. In this talk I will survey these systems with attacks done on these using braid representations and some suggestions for braid-based systems and cryptanalysis will be given.