** Principal faculty advisor:**
Haynes Miller, MIT

** Palestine faculty advisors: **

Professor Khalid Adarbeh, An-Najah National University

Professor Reema Sbeih, Birzeit University

**Overview:**

This project will explore one of several possible new directions in
homological algebra, depending upon the interests of the students.
The starting point is the study of cohomology of groups (going back
to the 1930s). What if the word "group" is replaced by other algebraic
structures (such as ring, or commutative algebra, or monoid). What
is the analogous theory, and how do we compute it? There are actually
two competing definitions of cohomology in these cases, and in this
project we will compare them and try to find out when they agree.

**Participants:**

Hadeel AbuTabeekh, An-Najah National University

Mohamed Damaj, Birzeit University

Nasief Khlaif, Birzeit University

Daishi Kiyohara, MIT

Kenta J Suzuki, MIT

Ali Tahbub, Birzeit University

top row: Kenta Suzuki, Haynes Miller, Nasief Khlaif

middle row: Hadeel AbuTabeekh, Ali Tahboub, Mohammad Damaj

bottom row: Daishi Kiyohara

This project led on to three separate research projects over the spring
of 2022:

Quillen homology of Lie algebras

Hochschild systems and alternative algebras

Harrison homology and the Quillen homology
of commutative monoids