Related papers: Introduction to Shtukas and their moduli
This is a review article based on a mini-course comprised of four talks given by the author at Berkeley.
We survey recent work on moduli spaces of manifolds with an emphasis on the role played by (stable and unstable) homotopy theory. The theory is illustrated with several worked examples.
We introduce and discuss the concept of modular entanglement. This is the entanglement that is established between the end points of modular systems composed by sets of interacting moduli of arbitrarily fixed size. We show that end-to-end…
In this paper, we consider the versal deformations of three dimensional Lie algebras. We classify Lie algebras and study their deformations by using linear algebra techniques to study the cohomology. We will focus on how the deformations…
We discuss the relation between Liouville theory and the Hitchin integrable system, which can be seen in two ways as a two step process involving quantization and hyperkaehler rotation. The modular duality of Liouville theory and the…
We review some recent results concerning Landau levels and Tomita-Takesaki modular theory. We also extend the general framework behind this to quasi *-algebras, to take into account the possible appearance of unbounded observables.
We survey recent advances in the theory of moduli spaces of stable sheaves on hyperk\"ahler manifolds of dimension greater than $2$. We start by recalling the well-known theory in dimension $2$, i.e.~for $K3$ surfaces, emphasizing the…
In [1] we introduced the concept of structured space, which is a topological space that locally resembles some algebraic structures. In [2] we proceeded the study of these spaces, developing two cohomology theories. The aim of this paper is…
We present a general and comprehensive overview of recent developments in the theory of integral models of Shimura varieties of Hodge type. The paper covers the following topics: construction of integral models, their possible moduli…
The purpose of this paper is to describe a relationship between the moduli space of vortices and the moduli space of instantons. We study charge k vortices in U(N) Yang-Mills-Higgs theories and show that the moduli space is isomorphic to a…
In previous work, we defined certain virtual fundamental classes for special cycles on the moduli stack of Hermitian shtukas, and related them to the higher derivatives of non-singular Fourier coefficients of Siegel-Eisenstein series. In…
We study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and…
Relations between the string topology of Chas and Sullivan and the homotopy skein modules of Hoste and Przytycki are studied. This provides new insight into the structure of homotopy skein modules and their meaning in the framework of…
These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of poset topology. These notes include introductory material, as well as recent developments and open problems. Some…
Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the…
We introduce local invariants of algebraic spaces and stacks which measure how far they are from being a scheme. Using these invariants, we develop mostly topological criteria to determine when the moduli space of a stack is a scheme. As an…
In these expository notes we draw together and develop the ideas behind some recent progress in two directions: the treatment of finite type partial differential operators by prolongation, and a class of differential complexes known as…
This is a survey paper dealing with moduli aspects of curves over finite fields. It discusses counting points of moduli spaces, relations with modular forms and stratifications on moduli spaces.
We construct bases for the spaces of higher order modular forms of all orders and weights. We also provide a cohomological interpretation of these forms.
The moduli space of triangles is a two-dimensional space that records triangle shapes in the plane, considered up to similarity. We study the subset corresponding to \textit{lattice triangles}, which are triangles whose vertices have…