🏛️ Hausdorff Trimester Program - Geometric Statistics 2026

Sat, 12 Dec 2026 00:00:00 +0000

I am honored to be invited to participate in the Dual Trimester Program on Geometric Statistics: theory, application, and computation at the Hausdorff Research Institute for Mathematics (HIM) in Bonn, Germany from September 14 to December 12, 2026.

This is an exceptional opportunity to engage with leading mathematicians and researchers working at the forefront of geometric and statistical methods in mathematics.

Program Overview

The trimester program spans three months and includes multiple schools, workshops, and conferences:

School on Geometric Statistics

September 14-18, 2026

Foundational lectures on geometric statistics
Theory and applications of statistics on manifolds
Computational methods for geometric data analysis

Workshop: Stochastic Analysis, Statistics, and Computation on Manifolds and Singular Spaces

October 12-16, 2026

Advanced topics in stochastic analysis
Statistical methods on curved and singular spaces
Computational algorithms and implementations

Workshop: Geometry, Topology, and Learning on Smooth and Singular Spaces

November 9-13, 2026

Learning theory perspectives on geometric problems
Topological data analysis methods
Applications to machine learning on manifolds

Conference: Interactions of Statistics and Geometry III (ISAG III)

December 7-11, 2026

Presentation of cutting-edge research
Dialogue between statisticians and geometers
Exploring new directions in geometric statistics

About Hausdorff Research Institute for Mathematics

The Hausdorff Research Institute for Mathematics (HIM) is an international center devoted to advancing research in mathematics and mathematical economics. HIM brings together:

A critical mass of expertise and scientists at one location
An inspiring mathematical atmosphere
Opportunity for mathematicians to work on challenging projects undisturbed
Direct discussion with leading experts in their field

This unique environment has consistently led to substantial output of research results of the highest quality, with many projects initiated, pursued, and sometimes completed during HIM programs.

Program Goals

By participating in this trimester program, I aim to:

Deepen knowledge in geometric statistics and their applications
Explore connections between geometry, topology, and machine learning
Engage with world-leading experts in the field
Collaborate on research projects addressing fundamental problems
Build connections with the international mathematics community

Research Interests Alignment

The program aligns perfectly with my research interests in:

Geometric and topological deep learning
Mathematical foundations of GNNs and topological methods
Applications of geometry to machine learning
Algebraic and differential geometric approaches

Learn More

For more information about HIM and the Geometric Statistics program:

Geometric Statistics Trimester Program

About HIM

Hausdorff Research Institute

🧭 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:

Mathematics | Raghuwansh Raj