Summer Graduate School

The school will consist of two courses: Homological Mirror Symmetry and Algebraic Models for Spaces. These courses will be planned and taught by organisers with the help of teaching assistants for the problem sessions. The school will be aimed at a wide range of graduate students, from students with a Bachelor degree to beginning PhD students. The lectures and problem sessions will be complemented by a poster session in week one and a total of four introductory research talks on Friday afternoons. 

School Structure

Most days will consist of two lectures and two problem sessions.  A poster session and reception will be held on the first Tuesday.  On Friday afternoons, there will be research seminars with two one-hour talks.

Prerequisites

Prerequisites are divided into “basic” and “advanced”. Familiarity with basic prerequisites is expected for everyone taking part in the course. Prerequisites involve category theory (CT), algebraic topology (AT) and algebraic geometry (AG). Students will be asked in advance to rate their familiarity with each of these three domains on a scale from 1 to 5, with 1 corresponding to the basic list, 3 corresponding to the advanced list, and 5 corresponding to research level, typically for PhD students working in this field. The items in the following list will be considered as basic prerequisites:

(CT) Category theory language, eg. sections 1.2 - 1.4 (12 pages) of Categories for the working mathematician - S. Mac Lane

(AG) Tensor products, eg Sections 'Tensor product of modules', 'Restriction and extension of scalars', 'Exactness properties of the tensor product', and 'Algebras' (7 pages) of Introduction to Commutative Algebra - M. F. Atiyah and I. G. MacDonald

(AT) Chain complexes, eg Sections 1.1-1.4 (18 pages) of An Introduction to Homological Algebra - Charles A. Weibel

(AT) Homotopy of spaces, eg Chapter 0: until 'Cell Complexes', and Chapter 1: Section 1.1: until 'Paths and Homotopy' (15 pages) of Algebraic Topology - Allen Hatcher

(AG) Algebraic varieties, eg Chapter I: Sections 1 and 2 (14 pages) Algebraic Geometry - Robin Hartshorne

The items in the following list constitute optional, more advanced study material, which will be recalled during the courses when used. The interactive problem sessions will be organised taking differences in background knowledge into account.

(CT) Yoneda’s lemma, limits and colimits, adjunctions and equivalences: sections 3.1 - 3.4 and 4.1 - 4.4 of Categories for the working mathematician - S. Mac Lane

(AT) Simplicial complexes and simplicial sets: Sections 2 and 3 of An elementary illustrated introduction to simplicial sets - Greg Friedman: https://arxiv.org/pdf/0809.4221.pdf

(AT) Homotopy theory: Section 1.1 until 'Induced Homomorphisms', and Section 2.1 until 'Homotopy Invariance' of Algebraic Topology - Allen Hatcher

(AG) Schemes and sheaf cohomology: Chapter II: Sections 1,2 and 5, and Chapter III: Sections 1-5 Algebraic Geometry - Robin Hartshorne

Application Procedure

SLMath is only able to support a limited number of students to attend this school.  Therefore, it is likely that only one student per institution will be funded by SLMath.

For eligibility and how to apply, see the Summer Graduate Schools homepage.

Venue

Lectures will take place at the Middelheim Campus of the University of Antwerp.