# Institute for Mathematical Sciences Programs & Activities

**Combinatorial and Toric Homotopy**

(1 - 31 August 2015)

**Combinatorial and Toric Homotopy**

(1 - 31 August 2015)

*on the occasion of Professor Frederick Cohen's 70th BirthdayJointly organized with Department of Mathematics, NUS*

Organizing Committee · Visitors and Participants · Overview · Activities · Venue

*Co-chairs*

- Jelena Grbic (University of Southampton)
- Zhi Lu (Fudan University)
- Jie Wu (National University of Singapore)

**Members**

- Alejandro Adem (University of British Columbia)
- Jon Berrick (National University of Singapore)
- Fengchun Lei (Dalian University of Technology)
- Ran Levi (The University of Aberdeen)
- Lek-Heng Lim (University of Chicago)
- Roman Mikhailov (St. Petersburg State University)
- Mikiya Masuda (Osaka City University)
- Dong Youp Suh (Korea Advanced Institute of Science and Technology)
- Dai Tamaki (Shinshu University)
- Vladimir Vershinin (University of Montpellier II)
- Yan Loi Wong (National University of Singapore)

This program aims to explore toric homotopy theory and combinatorial homotopy theory as well as their connections with other areas of mathematics. The toric varieties were investigated in algebraic geometry since 1970s. The notion of toric geometry was introduced after the work of Atiyah, Bott and Guillemin et al and other mathematicians. The topological explorations on toric objects given by Davis and Januszkiewicz as well as Goresky, Kottwitz and MacPherson led to the new area named as *toric topology* to be born during 2002-2006. Since then the subject has rapidly developed. The integral cohomology ring of the moment-angle complexes described by Buchstaber and Panov in terms of the Stanley-Reisner face ring led to a consequence recovering the Hochster theorem for the Betti numbers of the Staley-Reisner ring. The methods from homotopy theory enriched the development of toric topology. During the last few years, the decompositions of the generalized moment-angle complexes named as polyhedron products have been studied by Bahri, Bendersky, Cohen and Gitler; Grbic and Theriault as well as other mathematicians. On the other hand, combinatorial techniques arisen from toric topology were applied to determine the homotopy type of certain orbit configuration spaces.

The interplay between homotopy theory and toric topology pushes forward the development of *toric homotopy theory* as one of the new directions in homotopy theory, where the closed connections between polyhedron products, Whitehead products and Hopf invariants provide a fundamental clue for developing toric homotopy theory. The closely related subject is combinatorial homotopy theory. Dan Kan and John Moore gave the combinatorial definition of homotopy groups in 1950s. An explicit combinatorial description of homotopy groups of 2-spheres was found by Wu in 1990s. Recent progress made by Mikhailov and Wu led to an explicit combinatorial description of higher dimensional spheres using free product of pure braid groups with amalgamations. Some fundamental connections between homotopy groups of 2-spheres and braid groups were discovered by Berrick, Cohen, Wong and Wu. Further explorations led to fundamental connections between homotopy groups and link groups from the work of Lei, Li, and Wu. The simplicial methodology was used to determine the group of Brunnian braids over surfaces except the 2-sphere and the projective plane from the work of Badakov, Mikhailov, Vershinin and Wu as well as to explore the structure of mapping class groups in the work of Berrick, Hanbury and Wu with an application given in the work of Mikhailov, Passi and Wu. The interplay between homotopy groups and geometric group enriches the development of combinatorial homotopy theory. On the other hand, the combinatorial methods have been applied to homotopy theory with fruitful new results as well as connections to the modular representation theory from the work of Selick and Wu as well as other mathematicians.

The notion of configuration space had been introduced in physics in 1940s concerning the topology of configurations with various explorations on this important object since then. In mathematics, Fadell and Neuwirth introduced configuration spaces in 1962. Since then, various applications have been obtained. The connections between configuration spaces and the chromatic polynomials have been discovered recently by Eastwood and Huggett as well as Chen, Lu and Wu.

The notion of configurations was introduced in robotics community to study safe-control problems of robots since 1990s. Mathematically the topological robotics was born recently due to the work of Faber and Ghrist, where the topology of configuration spaces on graphs plays an important role in this newly created area.

Ideas, methods and constructions of Algebraic Topology are finding applications in various domains. A recent example would be the work of Chan, Carlsson and Rabadan (Proc. Natl. Acad. Sci. USA, 2013) on Topology of viral evolution. Drawing from Algebraic Topology (especially persistent homology) the authors give a method that effectively characterizes clonal evolution, reassortment, and recombination in RNA viruses, in avian influenza in particular. Applications of Algebraic Topology will be included in the activities of this program.

The program aims to achieve the following objectives:

- Some of the leading experts from different backgrounds will come together to:
- explore connections between these interdisciplinary topics;
- talking about the latest developments in algebraic topology with paying attention to the applications of algebraic topology to high technology and sciences;
- charting out new directions for research in toric and combinatorial homotopy theory;
- explore possible research collaborations.

- Introduce various aspects of the subject of topology to young researchers and graduate students by the leading experts, with the aim of future development of the area of topology.

**1. Young Topologist Seminar: 11 - 19 August 2015**

The seminar will consist of the following activities:

- Tutorials on advanced topics in algebraic topology.
- Seminar on toric topology.
- Seminar on combinatorial and applied topology.
- Seminar on geometric groups.

This seminar will focus on the training of young topologists. We will invite active researchers to give lectures on the topics. The seminar will also pay attention for the applications of topology to computer sciences and robotics.

**2. Workshop on Applied Topology: 20 - 21 August 2015**

This workshop will focus on applied topology as part of the structure of Young Topologist Seminar. The active researchers in applied topology will be invited to give talks.

**3. International Conference on Combinatorial and Toric Homotopy: 24 - 28 August 2015**

This conference will focus on toric topology, combinatorial groups, geometric groups and algebraic topology. The participants will explore geometric groups with their connections to homotopy theory, representation theory with its connections to topology, configuration spaces and toric topology. There will be a lot of discussions on these special topics.

Please note that our office will be closed on the following public holiday.

- 7 - 10 Aug 2015 (SG50 Public Holiday & National Day)

Students and researchers who are interested in attending these activities are requested to complete the online registration form.

The following do not need to register:

- Those invited to participate.

Organizing Committee · Visitors and Participants · Overview · Activities · Venue