题目: Contractads = Operads + Graphs

报告人: Denis Lyskov (Université Paris Cité)

时间：11月9日（周一），11月11日（周三），11月13日（周五），16：00-17：30

地点：东区第五教学楼5503教室

摘要：

This mini-course provides an overview of the theory of contractads, introduced and developed by the speaker and A. Khoroshkin. A contractad generalizes the notion of an operad: its operations are indexed by connected graphs, and its composition maps correspond to contractions of connected subgraphs. For example, restricting to complete graphs recovers the notion of an ordinary symmetric operad.

We show that many classical operads, including those governing commutative, Lie, associative, and pre-Lie algebras, as well as the little disks operad and the operad of moduli spaces of stable curves of genus zero, admit generalizations to contractads. We also discuss examples arising from graph theory that have not previously appeared in operad theory.

We explain how standard tools such as Koszul duality, Gröbner bases, and Hilbert series extend to contractads. We explore functional equations relating the Hilbert series of Koszul dual contractads and derive several applications, including computing Hilbert series of moduli spaces of stable curves, chromatic polynomials of graphs, and counting Hamiltonian paths and cycles.

Finally, we discuss applications of contractad theory to graphical configuration spaces. We explain how to compute their compactly supported cohomology using a graphical version of Chevalley-Eilenberg homology. A central question concerns the formality of little disks contractads, extending the results of Kontsevich and Tamarkin. In particular, we present a proof of the formality of the little 2-disks operad using logarithmic geometry and Hodge theory.