An LMS online lecture course in algebraic geometry.
I will first describe the notion of Bridgeland stability conditions on triangulated categories. Then I will focus on stability conditions on the bounded derived category of coherent sheaves on curves, surfaces and threefolds. In the end, some recent applications of Bridgeland stability conditions in classical algebraic geometry and Donaldson-Thomas Theory will be explained.
An LMS online lecture course in mirror symmetry.
Mirror symmetry conjecturally associates to a Fano orbifold a (very special type of) Laurent polynomial. Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial. We apply this to construct new Fano 3-folds with terminal Gorenstein quotient singularities.
In this series of talks, I will go through some of the basics of toric geometry to showcase how one can use combinatorial data to systematically build geometric objects. We will restrict our attention to the well-studied Fano case, for which there is concrete evidence that the mirror theorem holds in many cases.
Many prominent dualities in mathematics are instances of a common construction centered on the notion of spectral functor. Roughly stated, one starts with a locally finitely presentable category, equipped with a subcategory of distinguished local objects encoding point-like data and a factorization system (Étale maps, Local maps) where the etale maps behave as duals of distinguished continuous maps. Several manners of axiomatizing the correct relation between those ingredients have been proposed, either through topos theoretic methods by "localizing" local objects with a Grothendieck topology generated by étale maps, or in an alternative (though tightly related) way based on the notion of local right ajoint (or equivalently stable functor). Then the spectrum of a given object is constructed as a topos classifying etale maps under this given object toward local objects, equipped with a structural sheaf playing the role of the "free local object" under it. This defines a spectral functor from the ambient locally finitely presentable category to a category of locally structured toposes, forming an adjunction with a corresponding global section functor. This construction provide a convenient template for several prominent 1-categorical examples, as dualities for rings in algebraic geometry, or also Stone-like dualities for different classes of propositional algebras. The strong analogy between those dualities and their corresponding first order syntax-semantics dualities suggests the later could be understood as instances of a convenient 2-dimensional spectral construction. In this talk we will expose the ongoing work devoted to concretize this intuition.
After recalling the 1-dimensional version of the construction and the details of some prominent Stone-like examples, we introduces a notion of stable 2-functor and provide a method to construct an associated notion of spectral 2-sites, defining the spectrum as the associated Grothendieck 2-topos equipped with a distinguished structural stack. In particular we give a special interest in determining the local objects and the factorization system associated to doctrines corresponding to fragments of first order logics, as Lex, Reg, or Coh; in those situations, the construction simplifies as the spectral site happens to be 1-truncated so that one recover the corresponding 1-dimensional notion of classifying topos of a theory as the spectrum, and the geometry of the spectrum actually arises from the geometric properties of local toposes and étale geometric morphisms.
In this talk we discuss a notion of birational equivalence suitable for Poisson affine varieties: namely, that their function fields are isomorphic as Poisson fields. Some very interesting questions on non-commutative birational geometry, such as the Gelfand-Kirillov Conjecture, make perfect sense in the quasi-classical limit, and naturally leads one to consider the Poisson birational class of the algebras they quantize. In this setting, we study the behaviour of Poisson birational equivalence on the quasi-classical limit of rings of differential operators. With this idea we solve a Poisson analogue of Noether's Problem, introduced by Julie Baudry and François Dumas, in a constructive fashion, for essentially all finite symplectic reflection groups. As applications of our method, we show the Poisson rationality of the Generalized Calogero-Moser spaces, introduced by Etingof and Ginzburg in 2002, and surprisngly for this author, all Coloumb branches of 3d, N=4 SUSY gauge theories - an important object in mathematical physics recently given a rigorous formulation by Nakajima in 2015, and later Nakajima, Braverman, Finkelberg in 2016.
In this talk we will delve into the background details of the previous talk by introducing syntactic proof systems and their categorical semantics, including the construction of syntactic categories and κ-classifying toposes, as well as the role of certain properties of Grothendieck topologies and
Kripke-Joyal semantics. We will then study some topos-theoretic completeness theorems for certain infinitary logics that generalize results of Deligne and Joyal.
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