This is the second edition of a twice-yearly workshop that will alternate between the Southampton Centre for Geometry, Topology and Applications (CGTA) and the Beijing Institute of Mathematical Sciences and Applications (BIMSA). The aims are to promote exciting new work in homotopy theory, with an emphasis on that by younger mathematicians, and to showcase the wide relevance of the subject to other areas of mathematics and science.

Speakers

Anthony Bahri (Rider)

Matthew Burfitt (Aberdeen)

Sebastian Chenery (Southampton)

Xin Fu (Ajou University)

Alexander Grigor'yan (Bielefeld)

Fedor Pavutnitskiy (HSE)

Tseleung So (Regina)

Vladimir Vershinin (Montpelier)

Juxin Yang (Hebei)

Schedule of talks

**London Time Beijing Time Speaker**

*Monday, May 2*

12:00-12:50 19:00-19:50 Alexander Grigor'yan

13:00-13:50 20:00-20:50 Matthew Burfitt

14:00-14:50 21:00-21:50 Fedor Pavutnitskiy

*Tuesday, May 3*

12:00-12:50 19:00-19:50 Xin Fu

13:00-13:50 20:00-20:50 Sebastian Chenery

14:00-14:50 21:00-21:50 Anthony Bahri

*Wednesday, May 4*

12:00-12:50 19:00-19:50 Vladimir Vershinin

13:00-13:50 20:00-20:50 Juxin Yang

14:00-14:50 21:00-21:50 Tseleung So

Titles and Abstracts

**Title: Symmetric products and a realization of generators in the cohomology of a polyhedral product**

**Speaker: Anthony Bahri**

**Abstract:**

Polyhedral products, which are determined by a simplicial complex and a family of CW pairs, behave sufficiently well with respect to symmetric products as to allow for a description of the cohomology in terms of the link structure of the simplicial complex and the cohomology of the CW pairs. The process works particularly well under certain freeness conditions which include the use of field coefficients. Moreover, generators obtained via this process are robust enough to compute products. This talk however will focus on the way in which symmetric products are used to obtain the additive structure. Applications include the computation of Poincar series. The results are the culmination of a project which had its genesis in 2014 and is joint work with Martin Bendersky, Fred Cohen and Sam Gitler.

**Slides**

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**Title: Topological data analysis of Fast Field-Cycling MRI images**

**Speaker: Matthew Burfitt**

**Abstract: **

Fast Field-Cycling MRI (FFC MRI) has the potential to recover new biomarkers from a range of diseases by scanning a number of low magnetic eld strengths simultaneously. The images produced by an FFC scanner can be interpreted in the form of a sequence time series of 2-dimensional grey scale images with each times series corresponding to each of the magnetic eld strengths. I will investigate the applications of topological data analysis and machine learning to brain stroke images obtained by the FFC MRI scanner.

A main obstacle to achieving good results lies in multiplicative brightness errors occurring in the data. A simple solution might be to consider pixelwise image feature vectors initially only up to multiplication by a constant. This can be thought of as splitting the data point cloud within a product by rst embedding into standard n-simplices. We observe that this point cloud embedding can provide good information on tissue types which when modelled against the other component of the data point cloud can be used to highlight stroke damaged tissue.

A drawback of the rst method is that it discarded the pixel spatial locations information of the image. However, this can be captured by persistent homology in a parameter choice free process. A direct comparison between pixel intensity histograms and the Betti curves reveals that homology captures and emphasises tissue signals. Moreover, additional geometric and topological information about the images is captured with persistent homology. From here the ultimately aim is to extract persistent homology features useful for machine leaning and develop new visual diagnostics for the FFC data.

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**Title: The rational homotopy type of homotopy fibrations over connected sums**

**Speaker: Sebastian Chenery**

**Abstract: **

We provide a simple condition on rational cohomology for the total space of a pullback fibration over a connected sum to have the rational homotopy type of a connected sum, after looping. This takes inspiration from recent work of Jeffrey and Selick, in which they study pullback fibrations of this type, but under stronger hypotheses compared to our result.

**Slides**

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**Title: The homotopy classification of four-dimensional toric orbifolds**

**Speaker: Xin Fu**

**Abstract: **

Quasitoric manifolds are compact, smooth 2n-manifolds with a locally standard $T^n$-action whose orbit space is a simple polytope. The cohomological rigidity problem in toric topology asks whether quasitoric manifolds are distinguished by their cohomology rings. A toric orbifold is a generalized notion of a quasitoric manifold, and there are examples of toric orbifolds that do not satisfy cohomological rigidity. In this talk, we see that certain toric orbifolds in four dimensions, though not cohomologically rigid, are homotopy equivalent if their integral cohomology rings are isomorphic. We achieve this goal by decomposing those spaces up to homotopy. This is joint work with Tseleung So (University of Regina) and Jongbaek Song (KIAS).

**Slides**

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**Title: Path homology and join of digraphs**

**Speaker: Alexander Grigor'yan**

**Abstract: **

We introduce the path homology theory on digraphs (=directed graphs) and present Kunneth like formulas for path homology of various joins of digraphs.

**Slides**

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**Title: Homology of Lie rings**

**Speaker: Fedor Pavutnitskiy**

**Abstract: **

Homology of Lie algebras over fields is usually defined in terms of the Chevalley-Eilenberg chain complex. Similarly to group homology there are other equivalent definitions in terms of simplicial resolutions and Tor functors. Turns out that these definitions in general are no longer equivalent for Lie algebras over commutative rings. In the talk we will discuss these different approaches to homology of Lie rings and some theorems relating them. This is joint work with Sergei O. Ivanov, Vladislav Romanovskii and Anatoliy Zaikovskii.

**Slides**

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**Title: Suspension splittings of manifolds and their applications**

**Speaker: Tseleung So**

**Abstract: **

In order to study the topology of a space, it is useful to decompose the space into smaller pieces, analyse the pieces and reassemble into a whole. We say that a space has a suspension splitting if its suspension decomposes into a wedge of smaller spaces. In this talk I will talk about suspension splittings of 4-dimensional and 6-dimensional smooth manifolds, and their applications to computing generalized cohomology theories and gauge groups of 4- and 6-dimensional smooth manifolds.

**Slides**

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**Title: On homotopy braids**

**Speaker: Vladimir Vershinin**

**Abstract: **

The homotopy braid group b Bn is the subject of the work. First, linearity of b Bn over the integers is proved. Then we prove that the group b B3 is torsion free. Also we conjecture that the homotopy braid groups are torsion free for all n.

The talk is based on the joint work with Valerii Bardakov ans Wu Jie, Forum Math. 34 (2022), no. 2, 447-454.

**Slides**

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**Title: On the homotopy groups of the suspended Quaternionic projective plane**

**Speaker: Juxin Yang**

**Abstract: **

In this talk, I'll report on my computation of the homotopy groups $\pi_{r+k}(\sum^k\mathbb{H}P^2)$ (for r ≤ 15 and k ≥ 0) localized at 2 or 3, especially the unstable ones. Then I'll give some applications of them, including two classification theorems of a kind of 3-local CW complexes, and some decompositions of the self smash products

**Slides**

The link of the first Advances in Homotopy Theory Workshop: https://www.southampton.ac.uk/cgta/pages/soton-bimsa-biannual-2021-09.page