Why is it that graphs play such a key role in a essentially all applications of mathematics to real-life problems? The main reason is that there are so many situations in natural sciences/engineering/social sciences and more where we are trying to understand a complex system that consists of interacting pairs. These can be molecules in some biological system or two companies engaged in a business transaction etc. But what do you do when the underlying interactions may involve more than two parties? We are currently much less prepared to deal with such problems. It is natural to introduce into the game higher-dimensional simplicial complexes (a graph is a one-dimensional simplicial complex). This leads us into a fascinating set of problems and challenges, some of which already resolved and many still widely open. As it turns out graphs are just one example where a basic combinatorial object is inherently one-dimensional and where higher-dimensional counterparts suggest themselves. I will introduce recent work on high-dimensional permutations and tournaments.

These lectures are based on joint papers with several coauthors, e.g.

• http://arxiv.org/abs/1010.1400
• http://arxiv.org/abs/0903.1359
• http://arxiv.org/abs/1106.0649
• http://arxiv.org/abs/1108.5042
• http://arxiv.org/abs/1203.3312
• http://arxiv.org/abs/1208.4218
• http://arxiv.org/abs/1302.1684
• http://arxiv.org/abs/1307.2684

​The first session of this talk will take place on Friday, September 6 from 11:00 am – 12:30 pm.