Abstract for the main lecture series by Prof. Nitin Nitsure
Differential 1-forms and tangent vector fields have been of vital importance in differential geometry for a very long time. To extend these gadgets to algebraic geometry -- where one is commonly confronted with spaces and morphisms that have singularities, and with the need to work in a relative set-up -- radically new ideas were needed. Alexander Grothendieck addressed this with his invention of cotangent complexes in 1961. Their theory was further developed by his student Luc Illusie in his doctoral thesis of 1971.
The cotangent complex is today a fundamental tool in algebraic and arithmetic geometry. This series of lectures aims to introduce young mathematicians, who are already somewhat familiar with the basics of schemes and derived categories, to the theory of cotangent complexes.
After setting up cotangent complexes, we will illustrate the theory with examples of how they can work as powerful tools. This will include applications to the local properties of morphisms, intersection theory and to lifting problems in deformation theory.