# Josh Alman (MIT) - Limits on the Universal Method for Matrix Multiplication

## Limits on the Universal Method for Matrix Multiplication

We study the known techniques for designing Matrix Multiplication algorithms. The two main approaches are the Laser method of Strassen, and the Group theoretic approach of Cohn and Umans. We define a generalization based on restrictions of powers of tensors which subsumes these two approaches, which we call the Universal method. We then prove concrete lower bounds on the algorithms one can design by applying the Universal method to many different tensors. Our proofs use new tools for upper bounding the asymptotic slice rank of a wide range of tensors. Our main result is that the Universal method applied to any Coppersmith-Winograd tensor (the family of tensors used in all record-holding algorithms from the past 30+ years) cannot yield a bound on omega, the exponent of matrix multiplication, better than 2.16805.

No background knowledge about matrix multiplication algorithms is assumed. Some of the talk is based on joint work with Virginia Vassilevska Williams.

**Host:** *Aaron Elmore*

#### Josh Alman

I’m a Rabin Postdoc in Theoretical Computer Science at Harvard. I work on algorithm design and complexity theory, and I especially like using algebraic tools to solve problems throughout computer science.

I completed my PhD in Computer Science at MIT, advised by Ryan Williams and Virginia Vassilevska Williams. I spent the first half of grad school at Stanford until I moved to MIT with my advisors.

Before that, I was a Math major at MIT, where I lived on Floorpi in East Campus, and I helped organize the Undergraduate Math Association. I have also interned in the Theory Group at IBM Almaden, worked as a Software Engineer at Addepar, and participated in the REU in Combinatorics at the University of Minnesota, Twin Cities.