Modular arithmetic studies arithmetic “mod n.” Instead of ordinary equality, we consider that two integers are equivalent if their difference is a multiple of n. We write:
a ≡ b mod n means a − b is divisible by n.
This arithmetic is the backbone of cryptography, residue classes, cyclic groups, and more.
Addition mod n Simulator
Result: —
▼ Modular Arithmetic — Harmonic Engine
Langlands Program — A Grand Unifying Vision
What is the Langlands Program?
The Langlands Program proposes deep connections between number theory,
harmonic analysis, and geometry. Roughly speaking, it links:
Representations of Galois (or Weil) groups arising from number fields
Automorphic representations — analytic objects from harmonic analysis
Through these correspondences, problems about zeta and L-functions,
modular forms, and elliptic curves become facets of a single unifying
structure.
Core Structures & Concepts
Local and global fields: number fields, p-adics, and adèles
Reductive algebraic groups: such as GL(n) over various fields
Automorphic forms: analytic representations on adelic groups
Galois / Weil groups: encoding arithmetic symmetry
Functoriality: predicted transfers between representations
Why It Matters
The Langlands Program is often described as a grand unification of
mathematics, binding number theory, harmonic analysis, representation
theory, and geometry into a single conceptual framework.
Landmark results — such as the link between elliptic curves and modular
forms — can be understood as specific realizations of Langlands-type
correspondences.
