Zoran Skoda diplomski Spectra and reconstruction theorems

In many contexts, maps on a mathematical object with values in some standard gadget (e.g. ground field, category of sets) form a structure which is in some sense dual to the original object, hence capable to essentially reconstruct the original object. For example, the dual of a vector space consists of linear functionals; in finite dimensions the original vector space can be reconstructed as the dual of the dual: the original vectors are the corresponding evaluation functionals on the dual space. Finiteness conditions can often be relaxed by consideration of formal topologies and, more generally, pro-completions. The contravariant functors from a category to the category of sets form a presheaf category; by Yoneda lemma the original category embeds in it as the subcategory of representable presheaves, which are analogues of evaluation functionals on the original category; representability of functors is a fruitful property leading to important tools like classifying spaces or moduli spaces. In geometry, the algebra of functions (or in advanced analogies category of sheaves) is a sort of a dual representation of a space whose points can be reconstructed as (evaluation) functionals (or equivalently their kernels which are maximal ideals; or generalizations like prime ideals or distinguished classes of modules where certain orders on ideals/modules play a role). This is the idea of a spectrum of an algebra of functions, which is related to the original spectral theory of operators.

This diploma work will present several well known examples of reconstruction theorems, spectral constructions (along with duality phenomena) with an attempt to find common ideas of reconstruction and spectra in different contexts. Examples may include rudimentary versions of Tannaka duality, Gel’fand-Neimark theorem, Giraud’s theorem, Serre-Swan theorem or alike. Original research on new examples is possible but not required.

Literature will be around selected parts of

• $n$Lab http:ncatlab.org the wiki for categories, mathematics, physics and philosphy
• Pierre Cartier, A mad day’s work: From Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408
• Bodo Pareigis, Lectures on quantum groups and noncommutative geometry, Muenchen 2002
• Ross Street, Quantum groups, Australian Mathematical Society 2007
• Tom Leinster, Basic category theory, Cambridge University Press 2014
• André Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, in: LNM 1488 (Como 1990), Springer 1991
• Shahn Majid, Foundations of quantum group theory, chapter 9

Hrvatska skraćena verzija

U mnogim kontekstima, preslikavanja iz nekog matematičkog objekta u neku standardnu bazu kao što je osnovno polje ili kategorija skupova, sačinjavaju strukturu koja je u nekom smislu dualna početnom objektu čime sugeriramo da suštinski možemo rekonstruirati početni objekt. Npr. dual vektorskog prostora se sastoji od linearnih funkcionala i u konačno-dimenzionalnom slučaju može biti rekonstruiran kao dual duala, pri čemu su početni vektori pridruženi evaluacijskim funkcionalima na dualu. Uvjeti konačnosti mogu se oslabiti promatranjem formalnih topologija i, općenitije, kategorijskih pro-upotpunjenja. Kontravarijantni funktori iz zadane kategorije u kategoriju skupova čine kategoriju predsnopova; po Yonedinoj lemi početna kategorija se ulaže u kategoriju predsnopova kao potkategorija reprezentabilnih predsnopova, koji se mogu promatrati kao analogoni evaluacijskih funkcionalna na početnoj kategoriji. Reprezentabilnost funktora je korisno svojstvo koje vodi na plodne koncepte kao što su klasificiraju prostori odnosno prostori modula. U geometriji, algebra funkcija (a u naprednijim primjerima kategorija snopova) se može gledati kao dualna prezentacija prostora, koji može biti rekonstruiran kao prostor evaluacijskih funkcionala, ili ekvivalentno, njihovih jezgri, koje su dakle maksimalni ideali (važne su generalizacije u kojima tu ulogu preuzimaju prosti ideali, pa čak i posebni objekti u kategoriji modula, gdje uređaji na klasama modula igraju značajnu ulogu). To je ideja spektra algebre (ili operatorske algebre) funkcija, koja je iznikla iz spektralne teorije operatora.

Ovaj diplomski rad će izložiti rudimentarnu formu nekoliko poznatih rekonstrukcijskih teorema, odnosno spektralnih konstrukcija s naglaskom na usporedbu spektralnih ideja u raznim kontekstima.