Zoran Skoda Seminar on structured spaces

There is a discussion going on with my students and a couple of colleagues from the mathematics faculty in Zagreb to possibly introduce a new seminar in graduate school of the departement of mathematics focusing on sheaf theory and modern geometry. If we are going to have it accepted maybe the emphasis on the dominant algebraic geometry component will be useful as it is most blatantly non/represented subfield of mathematics at our university.

Thus Seminar on algebraic geometry and structured spaces is a good title. It is customary at Uni Zagreb to say seminar in …field rather than seminar on …field.

Scientific outline

There is an idea, often quoted in noncommutative geometry, that a space is determined by all the beings which can live (physically: be observed, be measured) on the space. An example of that philosophy is the Gel’fand-Neimark duality between a category of commutative $C^\ast$-algebras and Hausdorff topological spaces where the algebra in question appears as the Banach algebra of continuous functions vanishing at infinity; restricting to compacts one obtains the unital commutative $C^\ast$-algebras. But more complicated objects than algebras are living on spaces and they typically organize into the categorifications of an algebraic structure, for example one can look at a category of sheaves of sets on a topological space. This category retains all the essential features of the space, for example its cohomologies.

Henri Cartan, J-P Serre and A Grothendieck introduced the modern approach to geometry based on the notion of a topological space equipped with a structure sheaf, which was in their situations typically a ring or a topological ring; see ringed space. Cartan was working with coherent analytic sheaves, what was later followed not only by French school but also soon by Germans, most notably Grauert and Remmert. Serre extended the method to the study of varieties, and Grothendieck to the very definition of scheme and then many generalizations followed, like ringed site, ringed topos and so on. Lawvere studied more general picture of the duality between a space and “quantity” where the latter is (by my opinion) somewhat unapt name for the collection of structures living on the space and characterizing it (word quantity would more correspond to a single such object than the collection of them, like a function and not an algebra of all functions).

More recently generalizations of smooth manifolds like smooth spaces, homotopic versions of structured spaces (Simpson, Toen, Vezzosi, Lurie…), derived categories of sheaves and their dg- and $A_\infty$-variants appeared.

Clearly, algebraic, operator-algebraic, topological, homotopical and categorical methods are central in the study of structured spaces. Hence it looks that no seminar currently offered is appropriate in its width and modern geometric emphasis to handle more systematically this area.

Much material can be found in nLab on these topics. Besides, below are some of the surveys and key references from our point of view:

General surveys of reasonable size

Kaledin Tokyo lectures on nc geometry

Cartier (mad day)

Toen Vezzosi (local algebra paper)

Schreiber-Škoda Categorified symmetries

Škoda (equivariant paper)

Some monographs

Lurie

Some fundamental papers of close interest

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