Summer School on

Operads and Universal Algebra

June 28 – July 3, 2010, Nankai University, Tianjin, China

The purpose of this summer school is to introduce graduate students and non-experts to the areas of operads and universal algebra. This school should prepare the audience through mini-courses for the International Conference on the same subjects in the following week when more advanced topics and current research will be discussed.

Program for Summer School

	Monday June 28	Tuesday June 29	Wednesday June 30	Thursday July 1	Friday July 2
8:30-9	Opening
9-10:15* 9-10**	Loday 1	Loday 2	Bai 1	Loday 3	Bai 2
10:15-10:45* 10:15-11:15**	Break	Break	Livernet 1	Break	Livernet 2
10:45-12* 11:30-12:30**	Curien 1	Curien 2	Chen 1	Curien 3	Chen 3
	Lunch	Lunch	Lunch	Lunch	Lunch
14:00-15:00	Guo 1	Guo 2	Questions and discussions	Chen 2	Questions and discussions
15:30-16:30	Vallette 1	Vallette 2		Vallette 3
16:30-17:30	Liu 1	Liu 2		Liu 3

* for Monday, Tuesday and Thursday.

** for Wednesday and Friday.

Summaries:

Chengming BAI: Pre-Lie algebras and Lie and Jordan analogues of Loday algebras.

Abstract: Introduction to the study of pre-Lie algebras, emphasizing the relationships with other topics in mathematics and mathematical physics. How pre-Lie algebras fit into a bigger framework of their relationships with dendriform dialgebras, which lead to the introduction of Lie and Jordan analogues of Loday algebras.

Pierre-Louis CURIEN: Higher category theory and programming languages

Abstract: Definitions of n-categories (operadic ones, Joyal's quasi-categories and associated model categories). Dendroidal sets (Moerdijk, Moerdijk-Cisinski) and associated model categories. Applications to programming languages, relevance of n-categories in the study of (some versions of ) type theory, where type equalities hold only up to homotopy and in the study of rewriting (Lafont-MŽtayer-Worytkiewicz).

Yuqun CHEN: Groebner-Shirshov bases for general algebras

Abstract: Groebner bases and Groebner-Shirshov bases theories were invented independently by A.I. Shirshov for (commutative,anti-commutative) non-associative algebras and for Lie algebras (explicitly) and associative algebras (implicitly), by H. Hironaka for infinite series algebras (both formal and convergent) and by B.Buchberger for polynomial algebras. Groebner bases and Groebner-Shirshov bases theories have been proved to be very useful in different branches of mathematics, including commutative algebra and combinatorial algebra.

For a general algebra, by using Groebner-Shirshov bases, we may solve the following problems: normal form; word problem; rewriting system; embedding of algebras into simple algebras and two-generated algebras; PBW theorem; extensions of groups and algebras, etc.

In this mini-course, I will introduce the basic concepts related Groebner-Shirshov bases, in particular, the Composition-Diamond Lemma for associative algebras, which plays a key role in such a theory. I will give a general method to establish Composition-Diamond Lemma for general algebras, for instance, Lie algebras, dialgebras, Omega-algebras, differential algebras, Rota-Baxter algebras, L-algebras and so on.

Li GUO: Dendriform type operads and Rota-Baxter type operators

Abstract: We consider a class of operads introduced by Loday as a natural generalization of his dendriform dialgebra. We also consider a class of linear operators related to the Rota-Baxter operator and study its relationship with dendriform type operads.

Muriel LIVERNET: Topological operads

Abstract: Loop spaces, Stasheff polytopes, Little cubes operads, double loop spaces. Delooping machines. Gerstenhaber operad, Deligne conjecture. e_{n}-operads.

Yun LIU: Universal algebra and some related topics in theoretical computer science

Abstract: Varieties (in universal algebra), Birkhoff theorem, modular commutator theory, tame congruence theory.

Jean-Louis LODAY: Algebraic operads

Abstract: Operads (definitions and variations). Generalized bialgebras. Operadic twising morphism and Koszul duality for operads.

Bruno VALLETTE: Koszul duality

Abstract: Twisting morphism and twisted tensor product. Koszul duality for associative algebras (theory, methods and examples). Homotopy algebras (definition and properties).