1. OLD AT MATH
Information acquired: 2023
I got my Ph.D in Sydney, Australia. My achievements in the area of Higher Category Theory relate to four important aspects of it : I built higher operads for all weak higher transformations on the globular setting; I discovered a notion of fractal property which may be possessed by an ω-operad. Thanks to this notion, the difficult problem of the existence of the weak ω-category of weak ω-categories is replaced by a precise technical problem, which is to prove that a speciï¬c ω-operad of coendomorphism is contractible; I discovered the ï¬rst algebraic models of (∞,n)-categories (for each n ∈ N): My models of (∞,n)-categories are algebras for speciï¬c monads on the category of globular sets. Recently I described in the IHES the ï¬rst globular approach of Grothendieck ∞-topos. A paper related to this subject should be available soon. This approach is more natural than those proposed by Jacob Lurie using quasicategories for three reasons : First my ∞-toposes are very natural, in the sense that their constructions follow exactly the classical deï¬nition of Grothendieck toposes, just by using pure categorical methods. When truncated in the level 1, my ∞-toposes, provide exactly classical Grothendieck toposes. Secondly a notion of higher stacks emerged when we build it, which shows straight-away that our model of higher stacks do form an ∞-topos, where other didn’t prove such important fact. Also, our approach of Grothendieck ∞-topos is more general than those of Jacob Lurie, because he has built them with quasicategories which are well known models of (∞,1)-categories, whereas with our approach we use algebraic models of ∞-categories, where cells greater than 1 are not necessary invertibles. Very recently I found of a whole technical philosophy to build, probably, all kind of weak higher geometries by using natural models of higher moduli stacks under a suitable interaction between higher operads and some beautiful Quillen model structures. Sorry to say that, but my work must go beyond than those of Jacob Lurie, that I respect a lot, where applications of my approach much be much more easier and flexible for the future. At the moment I try to understand the simplicial approach of Lurie (unfortunately, even basics of simplicial sets is still mysterious for me at the moment; I will try to ask basics questions about it in this site) in order to be able to claim such thing. Last thing : I live alone in Paris (I am french). I have the Asperger Syndrom, which makes me difficult to apply anywhere, because I have some problems with applications, etc. My conditions (no moneys) makes me slow to interact, finish my work, etc. Also I do love other mathematics : Especially I had written a lot of manuscripts of others mathematics alone (Algebraic Geometry, Differential Geometry, Fiber Bundles, Measure Theory, Sets Theory, Forcing Methods, etc.) Please, feel free to contact me. ©
