Theory on Yahel Uffenheimer

Algebraic Techniques for Fast Integer Multiplication

Sun, 18 Jan 2026 00:00:00 +0000

Prerequisites

Ring Theory: Homomorphisms, Ideals, Quotient Rings, Chinese Remainder Theorem (see Appendix for definitions).
Polynomials: Polynomial multiplication, division, convolutions.
Algorithm Analysis: Basic recursive complexity and $O$-notation.

Given two integers $a,b\in \mathbb{N}$, how fast can we multiply them?

Fast Fourier Transform over Finite Fields

Sun, 18 Jan 2026 00:00:00 +0000

Prerequisites

Ring Theory: Basic definitions (Rings, Fields, Polynomials).
Group Theory: Cyclic groups and generators.
Basic FFT: Familiarity with the standard divide-and-conquer strategy on $\mathbb{C}$ is helpful but not required.

The Fast Fourier Transform (FFT) is one of the most important algorithms in history. Usually, it is introduced over the field of complex numbers $\mathbb{C}$, relying on geometric intuition about roots of unity lying on the unit circle.

Fast Matrix Multiplication - Part 4

Fri, 16 Jan 2026 00:00:00 +0000

Prerequisites

Part 3: Familiarity with Tensor Rank, Direct Sums, and Tensor Products.
Combinatorics: Basic understanding of the Multinomial Theorem.
Asymptotic Analysis: Limits and roots of polynomials.

In this post, we will formalize the tools needed to compare different tensor algorithms and prove the powerful $\tau$-Theorem (also known as the Asymptotic Sum Inequality). This theorem is the engine behind many modern improvements in the exponent of matrix multiplication, allowing us to derive bounds on $\omega$ from sums of disparate tensors.

Fast Matrix Multiplication - Part 3

Thu, 15 Jan 2026 00:00:00 +0000

Prerequisites

Part 2 of this series: Familiarity with the tensor product of vector spaces and bilinear algorithms.
Linear Algebra: Vector spaces, Bases, Dual spaces, and Tensor Products (basic definition).

In this post, we continue building the foundation for fast matrix multiplication algorithms. We will discuss essential tensor operations—product, sum, and restriction—and establish the Triple Product Condition, setting the stage for the group-theoretic approach.

Fast Matrix Multiplication - Part 2

Wed, 14 Jan 2026 00:00:00 +0000

Prerequisites

Part 1 of this series: Familiarity with standard Matrix Multiplication (MM) and $\omega$.
Linear Algebra: Vector spaces, Bases, Dual spaces, and Tensor Products (basic definition).
Abstract Algebra: Fields and Polynomial rings (helpful for the tensor intuition).

In the previous post, we discussed Strassen’s algorithm and the definition of the exponent $\omega$. We demonstrated that the complexity of matrix multiplication is dominated by the number of multiplications in the base algorithm, assuming a recursive approach.

Fast Matrix Multiplication - Part 1

Tue, 13 Jan 2026 00:00:00 +0000

Prerequisites

Linear Algebra: Basic definitions (matrix multiplication, inner products, block matrices).
Asymptotic Notation: Big-O notation and the Master Theorem.
No prior knowledge of Group Theory or Representation Theory is required for this post.

In this series of posts, we will discuss a fascinating line of work that aims to use group theory and representation theory to design fast algorithms for matrix multiplication. We will start with the basics of matrix multiplication algorithms and build up to the group theoretic approach. This post is intended for readers without any prior knowledge of groups or representation theory, or of matrix multiplication algorithms. My goal is to build an intuitive understanding of the problem space before we dive deeper in future posts.

Basic Matroid Theory

Sat, 10 Jan 2026 00:00:00 +0000

Basic Matroid Theory and Infinite Extensions

Prerequisites:

Linear Algebra: Vector spaces, basis, dimension, and linear independence.
Graph Theory: Graphs, connected components, cycles, and spanning trees.
Set Theory: Basic cardinality and Zorn’s Lemma (for the infinite section).

In this post, we will discuss the basics of Matroid Theory. While often taught merely as a theoretical framework for Greedy Algorithms, matroids are a rich combinatorial structure in their own right. We will start with the finite foundations and then explore how these definitions behave (and break) when extended to the infinite case.