PhD Course - Polynomials and algebraic complexity theory

October 4 - December 20

Organizers:
Radu Curticapean, Nutan Limaye

Lecturers:
In each session, one participant is responsible for presenting the covered topic and leading the discussion. Each participant must present at least once during the course to earn credits.

Dates of the course:
Joint sessions: October 4 2022 to December 20 2022.
Project deadline: February 1 2023.

Time:
Tuesday from 10:00 to 12:00.

Room:
Hybrid – Zoom and any meeting room with occupancy of more than 13 at ITU

Course description:
The course consists of 11 joint sessions and a project phase. During each 2-hour session, a participant will present part of the listed course material. The presentation should last 60-70 minutes. The remaining time is used for joint discussion of the covered material. Every participant should read the covered material before the session.
The final two sessions are reserved for discussing interesting research papers/ideas. The exact topics will be determined during the course.
The goal of this course is to give the participants an overview of algebraic complexity theory as well as some of the connections to other topics in theoretical computer science.

The projects will be based on a research paper not covered during the joint sessions. Participants will read the selected paper, summarize its main ideas along with the necessary mathematical background required to understand them, provide additional examples, and possibly simplify arguments from the original paper.

Reading list:
- Amir Shpilka and Amir Yehudayoff. (SY) Arithmetic Circuits: a survey of recent results and open questions
- Markus Bläser and Christian Ikenmeyer. (BI) Introduction to geometric complexity theory (Lecture notes)
- Relevant research papers.

Note: Being written in mathematical style, these resources are difficult to read. They will often require slow reading, analysing mathematical concepts, and consulting several external resources for background knowledge.

Programme:
04/10/2022 – Introduction to the algebraic complexity theory
11/10/2022 – Universal circuits, homogenisation, partial derivatives. (Section 2.1, 2.2, 2.3 in SY)
18/10/2022 – Fall break
25/10/2022 – Chapter 2, Depth reduction (Sections 2.4, 2.5 in SY).
01/11/2022 – Algebraic complexity classes and comparisons (Sections 5-7 in BI)
08/11/2022 – Lower bounds for monotone and non-commutative computation. (Section 3.3, 3.4 in SY).
15/11/2022 – Lower bounds for constant depth and multilinear circuits (Section 3.5, 3.6 in SY).
22/11/2022 – Approaches for proving general lower bounds (Section 3.8 in SY)
29/11/2022 – Math for Algebraic Complexity and Waring rank (Appendix A, Section 2 in BI)
06/12/2022 – Tensors for computer scientists (Section 13 in BI)
13/12/2022 – Complexity of bilinear maps (Section 14 in BI)
20/12/2022 – Border rank (Section 15 in BI)
01/02/2023 – Project submission deadline

Prerequisites:
Knowledge of basic probability theory, discrete mathematics, and linear algebra. The first session is an introduction to the topic.

Exam:
Each student must present at least once during the course. At the end of the course, the student must hand in a project report. Radu Curticapean and Nutan Limaye will evaluate the report.

Credits:
5 ECTS (~137,5 hours)

Amount of hours the student is expected to use on the course:
Participation: 22 hours
Preparation (reading/prepare presentation): 54 hours
Project work: 55 hours
Total: 131 hours

How to sign up:

Contact Nutan Limaye at nuli@itu.dk