The 1st GTSS
GEOMETRY-TOPOLOGY SUMMER SCHOOL
Nesin Mathematics Village, Şirince, İzmir
July 23-29, 2018
Lie groups, Lie algebras and representation theory
This mini-course is an introduction to Lie groups and Lie algebras. The topics covered include the structure theory and classification of complex semisimple Lie algebras using root systems, some elements of representation theory, with special emphasis on the complex semisimple case, weight theory. Towards the end, a more advanced topic might be discussed, time permitting, such as universal algebras, the Poincare-Birkhoff-Witt theorem, or another topic.
Textbook or/and course webpage:
1. Anthony W. Knapp. Lie groups beyond an introduction,
volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, second edition, 2002.
Linear Algebra, Algebra, Differentiable Topology (not a must)
Lectures on G2 Geometry
In this lecture series we present a combinatorial approach to the exceptional Lie group G2. We give a survey of various results about the algebraic structure. For the sake of completeness we decided to present them in a self-contained way to be easily accessible for future usage. We also present some applications to geometry. Manifolds with G2 structures, Decomposition of the exterior algebra into irreducible G2 representations, Metric of a G2 structure, 16 classes of G2 structures, Deformations of G2 structures. Lie algebra of G2, roots and their spaces, order, Killing form.
Language: TR, EN
We will be following the Reference:
Karigiannis, Spiro - Deformations of G2 and Spin(7) structures.
Canad. J. Math. 57 (2005), no. 5, 1012–1055.
Author's Ph.D. Thesis also available on the Arxiv.org.
Bryant, Robert L. Some remarks on G2-structures.
Proceedings of Gökova Geometry-Topology Conference 2005, 75–109, Gökova Geometry/Topology Conference (GGT), Gökova, 2006.
Other useful references are:
Anthony W. Knapp. Lie groups beyond an introduction, second edition.
volume 140 of Progress in Mathematics. Birkhauser Boston, Inc., Boston, MA, 2002.
G2 Geometry 1: 4 categories of vector cross product structures.
G2 Geometry 2: Decomposition of A*(M) into irreducducible
G2 Geometry 3: Metric of a G2 structure.
G2 Geometry 4: More into cross product identities.
G2 Geometry 5: 16 classes of G2 structures.
G2 Geometry 6: Deformations of G2 structures.
Introduction to Manifolds with Special Holonomy
Manifolds with special holonomy attract significant interest in both mathematics and mathematical physics. They appear in many contexts in Riemannian geometry, particularly Ricci-flat and Einstein geometry, minimal submanifold theory and the theory of calibrations, and string theory. We will start with basics of Riemannian geometry. The emphasis will be on discussing the Ricci-flat geometries that occur, then the holonomy classifications will be studied.
Textbook or/and course webpage (not necessary):
D. Joyce, “Riemannian Holonomy groups and Calibrated Geometry”
Prerequisites: Basic Differential Geometry (not a must but preferable)
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Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society