Hadamard Lecturer at

Centre de Mathématiques Laurent Schwartz,

*École Polytechnique, France*

*Keywords:* Homotopical Algebra, Homological Algebra, Higher Category Theory, Derived Algebraic Geometry, Topological Quantum Field Theory.

Curriculum Vitae

*Operator-indexing categories and related subjects*(In collaboration with B. Le Grignou and Y.Harpaz). This is an outgrowth of my thesis work, that is an open-ended attempt to understand the homotopical algebraic structures from a variety of perspectives. The original goal was to understand the structure that is naturally formed by planar trees as appearing in the work of Kontsevich-Soibelman, and eventually this led to the question what sort of objects should one use in order to index algebraic operations. One proposed solution to that question is the notion of weak operad that I learned from Yonatan Harpaz. We now attempt to see if there is a certain procedure that explains why structures over and under the same weak operad*O*descend to structures over a certain product of*O*with the 1-disk operad. With Brice Le Grignou, our ongoing project is to better understand the origins of many of such operation-indexing categories, as well as the notion of a Segal category in an enriched setting.*Machine learning and finite algebraic structures*(In collaboration with B. Shminke and C. Simpson). The recent explosion of interest in machine learning and data analysis suggests many different research directions. One of them is to attempt to understand if existing industry-level machine learning methods can be applied to the combinatorics of finite algebraic objects, such as semigroups or n-categories. The number of such objects explodes quickly with the order of the set, so perhaps the question of complete classification is a wrong one to ask. Can a neural network, however, learn to fill multiplication tables with missing elements, especially if there are many ways to do so? What can it tell us about semigroups of order 11? See this page for what we currently have.

- Edouard Balzin,
*The formalism of Segal sections,*preprint, arXiv:1811.09601, 59 pages. - Edouard Balzin,
*Reedy model structures in families,*preprint (slightly updated), arXiv:1803.00681, 90+ pages. - Eduard Balzin,
*Grothendieck fibrations and homotopical algebra,*PhD thesis defended on 20 June 2016. Manuscript, Slides (French). - Edouard Balzin,
*Derived sections and categorical resolutions in homotopical context,*preprint arXiv:1410.3387, 50 pages, Applied Categorical Structures (2017) doi:10.1007/s10485-017-9483-1. - E. R. Balzin,
*Derived sections, Factorisation Algebras and Deligne Conjecture,*Mathematical Notes 100(1), 313-317. - E. R. Balzin,
*Resolutions of categories and derived sections,*Russian Mathematical Surveys**69**:5 (2014), pp. 918–920.

- Current Research Statement.
- Exposé Formal Theory of Monads (Following Street) at 2014 session of Kan Extension Seminar organised by Emily Riehl.
- Incomplete set of lecture notes for the model categories course given at the Independent University of Moscow in 2012.

- Homotopy Seminar at HSE. Drop a note to Nikita Markarian if you wish to give a talk.
- Algebraic Topology group seminar at University Paris XIII.