Short Intensive course

**Course Title:**

Topologies and Completions

**Resource Person: **Tiberiu Dumitrescu

**Abstract:**

Completion is an important tool in Algebraic Geometry, Number Theory, Functional Analysis, Commutative Algebra and other math areas. We give a quick introduction to the subject with examples and relevant proofs.

**Schedule:**

Starting from November 16, 2018.

Friday at 10:30-12:30

**Syllabus:**

Recap on inverse limits.

The completion with respect to a filtration.

Examples: power series ring and p-adic numbers.

Hensel’s Lemma (HL) – without proof.

Applications of HL in number theory, geometry and algebra.

Applying HL for lifting idempotents.

Finite algebras over a local complete ring.

Basic properties of completion.

Relating completion with the associated graded ring.

Applications.

Completion as an exact functor.

Maps from power series rings.

Proof of Hensel’s Lemma.

**Books:**

Atiyah and Macdonald, Introduction to commutative algebra

Eisenbud, Commutative Algebra With A View Toward Algebraic Geometry

Matsumura, Commutative Algebra

Matsumura, Commutative Ring Theory

Zariski and Samuel, Commutative algebra, Vols. I and II.

**Course Title: **

Clifford algebras and spin groups Advanced Algebraic Topology

**Resource Person: **Johann Davidov

**Abstract: **

The purpose of the course is to present basic facts about Clifford algebras, spin representation and Dirac spinors. The nation of tensor and exterior products of vector spaces will briefly be recalled.

Auxiliary material about topological groups and covering space will also be given. In order to cover the whole programme, the couse will be continued in the spring term of the acadmic year 2018/2019.**Prerequisites: **Standard facts about vector spaces, linear maps and multilinear forms.

**Schedule:**

Starting from: November 26, 2018

Monday (14:00-15:45)

Friday (14:30-16:15)

For further detail and registration in the course, please contact events@sms.edu.pk