GAAG — IV

Geometry in Algebra and Algebra in Geometry IV, IME-USP, (05.11.2018 — 09.11.2018)

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Program

Titles and Abstracts

Mini-courses

• Luca Vitagliano (Università di Salerno). Calculus up to homotopy on leaf spaces.
  Lecture notes.
  Foliations are important objects in differential geometry. The main reason for their importance is that they are distinguished instances of more general, and ubiquitous, objects: differentiable stacks on one side, and solution spaces of PDEs on the other side. Given a foliation, its leaf space is of a particular interest, for instance, in reduction problems. However, leaf spaces are often highly non-regular, even in the case of a regular foliation, and it is natural to ask the question: how can we define a calculus on a leaf space? In this mini-course, I will propose an answer inspired by Homotopical/Derived Geometry. It turns out that functions, vector fields, differential forms, etc. on a leaf space can be understood as elements of suitable homotopy algebras. Accordingly, every leaf space supports a calculus up to homotopy.
• Peter Arndt (University of Düsseldorf). Abstract motivic homotopy theory
  Lecture notes.
  Motivic homotopy theory is a fusion of homotopy theory and algebraic geometry. In analogy to the homotopy category of topological spaces, obtained by making the unit interval contractible, one obtains a homotopy category of schemes by making the ane line contractible. This category has a rather toplogical avour; one can for example talk about suspensions, loop spaces and classifying spaces and represent cohomology theories by spectra. Standard results from algebraic topology can then be repeated, for example algebraic K-theory can be represented by Grassmannians and one can construct new invariants in the style of algebraic topology, most notably algebraic cobordism. After its rst spectacular successes, the resolution of the Milnor and Bloch-Kato conjectures, the subject has led to a multitude of results, for example on vector bundles over schemes, algebraic cycles, descent theorems in algebraic K-theory and computations of stable homotopy groups of spheres in topology.
  
  The basic construction steps of motivic homotopy theory out of the category of schemes have been repeated for complex and non-archimedian analytic geometry, and recently also derived algebraic geometry, and these new settings are being protably applied and explored. Further geometric settings suggest themselves as input for variants of motivic homotopy theory, in particular the many proposals for geometry over the eld with one element.
  
  In this talk I will present an abstract setup for motivic homotopy theory, encompassing most known variants and analogues of algebraic geometry. We start with a short introduction to motivic homotopy theory, paralleling the setup of unstable and stable homotopy theory for topological spaces. Then I will motivate what follows by presenting some notions of "schemes over deeper bases" and other alternative settings of algebraic geometry. In the main part of the talk I will start from a very general general input: a cartesian closed, presentable innity category, replacing the category of motivic spaces, and a commutative group object therein, replacing the multiplicative group scheme. For this starting data I will present constructions and results generalizing those of motivic homotopy theory. Among these are a representation theorem for line bundles, a Snaith type algebraic K-theory spectrum, Adams operations, rational splittings and a rational motivic Eilenberg-MacLane spectrum.
• Mark Colarusso (University of South Alabama). Integrable systems and Lie theory
  In this minicourse, we study the interaction between Lie theory and the theory of integrable systems. Integrable systems rst appeared in the 19th cen- tury as classical mechanical systems for which the equations of motion could be solved by a nite sequence of algebraic operations, integration, and function inversion. Such classical integrable systems include the motion of certain spin- ning tops, the free motion of a particle on an ellipsoid, and the motion of a rigid body in an ideal uid.
  
  In the last 40 years, Lie theory has played a large role in the study of in- tegrable systems. The symplectic geometry of Lie algebras is a natural place to understand certain integrable systems such as the Toda lattice. Aside from physics, integrable systems have proven themselves to be useful in Lie theory. In the 1980's Guillemin and Sternberg showed that a certain integrable system on conjugacy classes of nn Hermitian matrices can be thought of as a geometric analogue of the classical Gelfand-Zeitlin basis for irreducible representations of the unitary group. One of our main goals of the course will be to understand a complexifed version of the integrable system developed by Guillemin and Stern- berg and its interaction with the representation theory of innite dimensional Gelfand-Zeitlin modules introduced by Drozd, Futorny, and Ovseinko.
  
  We will begin discussing some ideas from Hamiltonian mechanics and sym- plectic geometry and then use these ideas to analyze some classical integrable systems. In the following talks, we will see how Lie theory and Poisson geome- try can be used to describe some modern integrable systems including the Toda lattice and Gelfand-Zeitlin integrable systems. Finally, we will discuss how the geometry of Gelfand-Zeitlin integrable systems.
• Simone Marchesi (UNICAMP). An introduction to instanton bundles
  Lecture notes.

Talks

• Abdelmoubine Amar Henni (UFSC). The fixed rank 2 Instanton sheaves on P3
  We give some properties of the fixed rank 2 Instanton sheaves on P3 under the natural action of the 3-dimensional torus. This allows us to relate then to Pandharepande-Thomas stable pairs. Moreover, we classify all the supports and give a lower bound on the number of irreducible components of the fixed locus.
• Alejandro Cabrera (UFRJ). Formal symplectic realizations and beyond
  We will review a construction of a formal symplectic realization for any Poisson structure on R^d. We will show how weighted sums over graphs naturally appear in this problem, and how they coincide with the semi-classical part of the Kontsevich star product formula. This is based on joint work with B. Dherin. We speculate about possible extensions of such interpretation to the full star product formula.
• Alessia Mandini (PUC-Rio). Hyperpolygons and parabolic Higgs bundles
  TBA
• David Martinez Torres (PUC-Rio). Coadjoint orbits and standard symplectic structures
  We will survey on some results on describing the symplectic structure of some (non-compact) coadjoint orbits as a standard one, providing an alternative approach to some known constructions.
• Eugenia Martin (UFPR). Irreducible components of varieties of Jordan algebras
  In 1968, F. Flanigan proved that every irreducible component of a variety of structure constants must carry an open subset of nonsingular points which is either the orbit of a single rigid algebra or an infinite union of orbits of algebras which differ only in their radicals.
  
  In the context of the variety Jor_n of Jordan algebras, it is known that, up to dimension four, every component is dominated by a rigid algebra. In this work, we show that the second alternative of Flanigan's theorem does in fact occur by exhibiting a component of JorN5 which consists of the Zariski closure of an infinite union of orbits of five-dimensional nilpotent Jordan algebras, none of them being rigid.
  
  This is a joint work with I. Kashuba.
• Jean Valles (UPPA-LMA).Free curves in a pencil
  A plane curve is called free when its associated logarithmic sheaf is a sum of two line bundles. In this talk, I will explain how to produce free curves from a pencil of curves with same degree and smooth base locus.
• Matias del Hoyo (UFF). Morita equivalences of vector bundles
  When working with Lie groupoids, representations up to homotopy arise naturally, and they are useful, for instance, to make sense of the adjoint and coadjoint representations. The idea behind them is to use graded vector bundles and allow non-associativity. This brings up our interest in understanding the symmetries of a graded vector bundle, and in particular, of a 2-term graded vector space. In a joint work with D. Stefani we show that they can be encoded in a general linear 2-groupoid, a higher analogue of the classic general linear group. I will present our contributions and discuss several lines for further research.
• Oliver Lorscheid (IMPA). Representation type via quiver Grassmannians
  The representation type of a quiver Q can be characterized by the geometric properties of the associated quiver Grassmannians: (a) Q is representation finite if all quiver Grassmannians are smooth and have cell decompositions into affine spaces; (b) Q is tame if all quiver Grassmannians have cell decompositions into affine spaces and if there exist singular quiver Grassmannians; (c) Q is wild if every projective variety occurs as a quiver Grassmannian.
  
  With the exception of (extended) Dynkin type E, this result is proven in joint work with Thorsten Weist, based on previous results by Haupt, Reineke, Hille and Ringel. The result for type E was recently completed by Cerulli Irelli-Esposito-Franzen-Reineke. In this talk, we will give an introduction to quiver Grassmannians, explain this result and outline its proof.