📐 London Geometry and Machine Learning (LOGML) Summer School 2026

Fri, 17 Jul 2026 00:00:00 +0000

I had the privilege to attend the London Geometry and Machine Learning (LOGML) Summer School 2026 held at Imperial College London from July 13-17, 2026.

This was my first summer school during my PhD journey, and it was an incredible opportunity to engage with the vibrant community of doctoral students, researchers, and experts working at the intersection of mathematics, geometry, and machine learning.

Program Overview

The summer school featured intensive lectures and tutorials covering:

Graph Representation Learning - Understanding and learning from graph-structured data
Learning Graphs from Data - Discovering graph structures from raw data
Equivariant Machine Learning - Building models that respect symmetries and invariances
Algebraic Geometry in Deep Learning - Mathematical foundations for neural network design
Discrete Structures for 3D Geometric Learning - Geometric approaches to 3D data
Graph Foundation Models - Pre-trained models for graph-based tasks
Physics-Informed Graph Neural Networks - Incorporating physical priors into GNNs

Key Learnings

I was particularly inspired by lectures from world-leading experts including:

Xiaowen Dong - Graph signal processing and learning
Dorina Thanou - Geometric methods in machine learning
Stefanie Jegelka - Graph neural networks and expressiveness
Olga Fink - Physics-informed machine learning

Their discussions on mathematical and physical priors in building graphical models, and the integration of AI in physical and health sciences, were enlightening.

Collaborative Project

During the summer school, I had the opportunity to work on a group project exploring:

Canonicalization of 3D Molecular Point Clouds - Collaborating with mentor Snir Hordan, we investigated how symmetry-aware preprocessing can help conventional neural architectures learn from molecular geometry more efficiently.

This work bridges the gap between geometric deep learning theory and practical molecular applications.

Key Takeaways

Deepened understanding of geometric and topological approaches in deep learning
Networked with talented researchers from institutions worldwide
Gained hands-on experience with cutting-edge techniques in equivariant learning
Explored applications in molecular geometry and 3D learning
Built connections with mentors and peers in the GDL community

Gratitude

I’m grateful to the LOGML Organizers, especially Dr. Lennart Bastian, the speakers, mentors, and fellow participants for creating such an intellectually stimulating and collaborative environment.

Learn More

For more information about LOGML:

Visit LOGML Official Website

Imperial College London

🧭 Spherical Harmonics and the Spherical Coordinate System

Mon, 16 Mar 2026 00:00:00 +0000

Spherical harmonics are one of the most important mathematical tools for analyzing functions defined on the surface of a sphere. They appear in many areas including quantum mechanics, geophysics, computer graphics, and modern geometric deep learning.

This post introduces the spherical coordinate system and explains how spherical harmonics arise naturally when studying functions on the sphere.

The Spherical Coordinate System

In three-dimensional space, we often represent points using Cartesian coordinates

\[ (x, y, z) \]

However, when working with spherical objects or rotationally symmetric problems, it is more natural to use spherical coordinates:

\[ (r, \theta, \phi) \]

where:

\(r\) is the distance from the origin
\(\theta\) is the polar angle (measured from the z-axis)
\(\phi\) is the azimuthal angle (measured in the xy-plane)

The relationship between Cartesian and spherical coordinates is:

\[ x = r \sin\theta \cos\phi \]\[ y = r \sin\theta \sin\phi \]\[ z = r \cos\theta \]

This coordinate system is particularly useful when studying functions defined on the surface of a sphere, where \(r\) is constant.

What Are Spherical Harmonics?

Spherical harmonics are a set of special functions defined on the sphere that form an orthogonal basis for functions on the sphere.

They are analogous to Fourier series, but instead of decomposing functions on a circle, they decompose functions on a sphere.

A spherical harmonic is usually written as

\[ Y_l^m(\theta, \phi) \]

where:

\(l\) is the degree
\(m\) is the order
\(-l \le m \le l\)

These functions arise as solutions to Laplace’s equation in spherical coordinates.

Why Spherical Harmonics Matter

Spherical harmonics appear in many scientific fields:

Physics

They are used to solve:

Schrödinger equation
gravitational potentials
electromagnetic fields

Computer Graphics

They are used for:

lighting models
environment maps
rendering reflections

Geometric Deep Learning

Modern AI models that operate on spheres, rotations, or 3D structures often rely on spherical harmonics for:

equivariant neural networks
molecular modeling
3D vision

Visualizing the Harmonics

Each spherical harmonic corresponds to a particular frequency pattern on the sphere.

Low-degree harmonics represent smooth variations, while higher-degree harmonics represent finer oscillations.

Conceptually:

Geometric Deep Learning | Raghuwansh Raj