Zoran Skoda hom11connections

LECTURES Wednesdays at 16 00 (exactly! sometimes announced extra lecs on Mon. 18 00) Math, room 002/105/109: lec 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26?,27?,28?,29?,30?,31?,zadaci

In 2011-2012 I teach the graduate course in mathematics at the University of Zagreb with title Geometry of connections and integrability, once a week starting Wed 26. 11. 2011., 2011. The official proposal is here: koneksije.pdf (in Croatian), connection.pdf (in English).

Prerequisites: basic notions from topology, on chain complexes, and differentiable manifolds (tangent bundle, vector fields, differential forms, frame bundle, Riemannian structure); notions of a Lie group and Lie algebra; we will also use the basic language of categories and functors which will develop much further during the course; from algebra, modules and bimodules over associative rings/algebras and basics on tensor products of vector spaces and (bi)modules.

Description: In differential geometry, a connection is an additional structure which allows comparing infinitesimal quantities at different points in space, e.g. to allow differentiation. In physics, the connections may be dynamical quantities, “gauge fields”. A connection has an invariant, its curvature. When the curvature is non-zero its value leads to cohomological invariants called characteristic classes. Curvature is zero iff so-called Maurer-Cartan equation is satisfied. This condition, called flatness or integrability appears widely in modern mathematics. In geometry, by integrability of an infinitesimal structure we mean that it appears as a local expression of a globally defined geometric object, e.g.

• integral curves of a vector field
• Lie group integrating a Lie algebra
• solution of a system of differential equations.

Similarly, the existence of the deformation of a structure in algebra and geometry, requires a solution of a Maurer-Cartan equation in a different setup of a dg-Lie algebra.

We will start with the connections on vector bundles (and on Lie algebroids) in differential geometry, which have several distinct descriptions. We will show several examples of geometric integrability problems which can be stated in terms of flatness of a connection. Our true aim is, however to present how this classical setup vastly extends. The first extension is to extend connections from vector bundles to sheaves. In the case of flat connections, this essentially amounts to a deep subject of D-modules, which is a subject of another course this year. Sheaves allow more singular situations, hence it is beneficial to extend the very notion of differential calculus, from intrinsic geometric point of view. This has been done by Grothendieck. Following his approach we will introduce structured spaces and quasicoherent modules on them. We then define the filtration of infinitesimal neighborhoods and dual differential filtrations on Hom-bimodules, leading to regular differential operators. D-modules appear in the form of the related descent data, so-called crystals; such a descent datum can be expressed as a flat connection (Grothendieck connection). Main local properties of spaces and morphisms are defined in similar terms. For example, smoothness is defined in terms of the lifting of maps to infinitesimal neighborhoods; this same picture also inspired a similar approach in synthetic differential geometry. Allowing all orders of infinitesimals leads to formal power series and to formal schemes. If time permits, at this place we shall insert some basic notions of deformation theory.

After the geometric study of infinitesimals, we complete the study of the two principal examples, Chern-Weil theory of characteristic classes and the integration of Lie algebras/algebroids to Lie groups/groupoids, and show the importance of characteristic zero and finite dimensionality. Lie algebras under these conditions have Koszul duals which are (Chevalley-Eilenberg) dg-algebras, which can be, using ideas of Sullivan in rational homotopy theory, turned into simplicial sets, hence into topological spaces. The Koszul duality itself has a very clean form in terms of dg-(co)algebras and a solution of a generalized Maurer-Cartan equation.

Texts: lectures will be gradually posted online and linked from http://ncatlab.org/zoranskoda/hom11connections. Some sources include

BASIC REFERENCES

• Dale Husemöller, Fibre bundles, 3rd edition, Springer Graduate Texts in Mathematics 1994, gBooks

• J. L. Dupont, Fibre bundles and Chern-Weil theory, Univ. of Aarhus 2003, 115 pp. pdf

• M. Crainic, R. L. Fernandes, Lectures on integrability of Lie brackets, math.DG/0611259

• J. Lurie, Crystals and D-modules, pdf

• M M Postnikov, Lectures on geometry, semester IV (in Russian), differential geometry; semester VI, Riemannian geometry (semester VI exists in Springer English version, Enc. of Math. 91)

• Shigeyuki Morita, Geometry of characteristic classes, Transl. Math. Mon. 199, AMS 2001

• R. Bott, L. W. Tu, Differential forms in algebraic topology, Springer

• Dennis Sullivan, Infinitesimal computations in topology, Publ. IHES 47 (1977), p. 269-331, numdam, MR58:31119

• Ezra Getzler, Lie theory for nilpotent $L_\infty$ algebras, Annals of Mathematics 170 (2009), 271–301, math.AT/0404003, MR2010g:17026, doi

Some other, general differential geometry references can be found in entry differential geometry.

Notice that Prof. Željka Milin teaches another course in differential geometry this year “Pseudo-Riemannian geometry” (see announcement, pdf) which has some overlap at the beginning. She will cover some prerequisites on differentiable manifolds and the elementary theory of affine and pseudo-Riemannian connections. My course will be at the beginning in the setup of differentiable manifolds, but some other similar or generalized frameworks will be introduced in the later part of the course. Prof. H. Kraljević teaches course on D-modules (see announcement, MS word) which are a sheaf theoretic analogue of a bundle with flat connection, and we will touch on this notion and the algebraic approach of crystals briefly, to elucidate deeper geometric origin of the theory. Kraljević will instead emphasise on the point of view of representation theory.

Last revised on March 1, 2013 at 00:45:01. See the history of this page for a list of all contributions to it.