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Prof. Dr. Alberto Cattaneo

**If you are interested, please contact the instructor: Ödül Tetik.**

Below is the current plan for the seminar.

Most of the words will probably not mean anything to anybody -- if this is the case for you, you should definitely take this seminar.

Weekly plan for talks, list of references (version 02.02.2021, details subject to change)

Here is a general guide to picking a talk according to your general interests:

- Part 1 (talks 2--6): This part is focused almost exclusively on geometric/topological aspects. It builds up to two (sketches of) proofs of the Recognition Principle, a classic result in topology and operad theory, which states that if (and only if) a ("group-like") space is an "algebra over the operad of little n-cubes", then it is, for all topological intents and purposes, an n-fold loop space. These little cubes operads are extremely important in all kinds of areas, from higher algebra/derived algebraic geometry to mathematical approaches to classical and quantum field theories. Roughly speaking, for n=1, this operad describes (homotopy-[see below])associative algebras, and as n goes up, it describes algebras that are closer and closer to being commutative.
- Part 2 (talks 7--10): This part is more algebraically-oriented, but the theory covered (that of quadratic and Koszul operads) contains important and well-known implications for many areas of mathematics and physics. Highlights are: 1) the formalisation and proof of the old idea of the "duality" between commutative and Lie algebras, and the self-duality of associative algebras, which dates back at least to Quillen 2) the cobar, D, and DD constructions, which are the starting points for the notion of Koszulity and the notion of "homotopy-algebraic structures", meaning algebraic structures where defining equalities, like the Jacobi identity, or the associativity equation, hold "only up to homotopy". Such structures are crucial in certain areas of modern algebra, geometry and physics. 3) Classic results like the Koszulity of the operads describing associative, commutative, and Lie algebras.
- Part 3 (talks 11--14): This part is a mixture of both, and the last talk also touches on an approach to physics that is mathematically very clean and satisfying. Highlights are: 1) a treatment of the Deligne conjecture (a theorem), a great illustration of some ideas of homotopy-algebra and Koszulity, important and interesting topologically, algebraically and physically in equal measure. 2) The homologies of (iterated) loop spaces carry ("shifted") Poisson structures (Poisson algebras generalise the structure of the Poisson bracket from classical mechanics)! A result which seems completely preposterous and opaque at first sight, but is a corollary to the recognition principle and the (beautiful) calculation of the homologies of configuration spaces, the baby case of which goes back to work of Arnold from the 60s. This is the beginning of a subject with deep ties to number theory, knot theory and deformation quantization, but we will not have time to go into this. 3) A physical interpretation/implementation of some these things.

**Modul:** MAT578 Seminar Essentials in Operads