Do you like puzzles, symmetry, prime numbers, or the bridges between mathematics and physics, coding, or biology? Join me for an adventure in modern algebra. We start from concrete, visual, and computable questions and follow them into deep ideas used across mathematics and science. You do not need to know everything in advance. Bring curiosity, grit, and a willingness to learn. Projects are tailored for advanced undergraduates or master’s students.
• Pick a theme that excites you (see the menu below).
• Begin with a core problem (proof, computation, build).
• Follow enrichment paths (generalizations, deeper theorems).
• Add applications (physics, coding, geometry, biology) where it fits.
• Count and classify the symmetries of the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Prove, for instance, that the rotational symmetry group of the icosahedron is isomorphic to A5.
• Build or 3D-print models; use pictures to track orbits and stabilizers.
• Technical names: polyhedral groups; A4, S4, A5; point groups; orbit–stabilizer; centralizers; conjugacy classes.
• Compute character tables for small groups (S3, D8, Q8); decompose representations; verify orthogonality relations numerically.
• Use characters to predict spectral lines or vibrational modes in simple molecules.
• Technical names: character theory, irreducible representations, induction/restriction, Fourier analysis on finite groups.
• Start with the simplest Lie algebra sl2(C): build highest-weight modules, draw weight diagrams, and work through tensor products (spin addition).
• See how the same math explains angular momentum in quantum mechanics.
• Technical names: Lie algebras; highest-weight representations; SU(2); Clebsch–Gordan rules.
• Decode Dynkin diagrams (A, B, C, D and exceptional E6, E7, E8); compute small root systems like A2, B2, G2.
• Understand how these diagrams classify all simple Lie algebras.
• Technical names: root systems, Weyl groups, Cartan subalgebras, classification of simple Lie algebras.
• Deform symmetry by a parameter q to meet quantum groups and produce knot invariants (e.g. the Jones polynomial).
• Work small examples and see braids become matrices.
• Technical names: Uq(sl2), R-matrices, braid group representations, ribbon/monoidal categories.
• Mix “even” and “odd” symmetries in Lie superalgebras such as osp(1|2) or sl(m|n); explore typical vs. atypical modules.
• Connect to algebraic models of supersymmetry in physics.
• Technical names: Lie superalgebras, parity, simple/typical/atypical modules, representation theory of superalgebras.
• A guided taste of VOAs and Zhu algebras, with examples tied to moonshine and conformal field theory.
• Technical names: vertex operator algebras, modules, Zhu algebras, modularity phenomena.
• Explore cyclotomic polynomials Φn(x): prove irreducibility for primes (Eisenstein), compute examples like Φpq, and examine coefficient patterns.
• Technical names: Gauss’s lemma, Eisenstein’s criterion, Dedekind’s proof, Galois groups (Z/nZ)
• Explain Gauss’s 17-gon and classify which regular polygons are constructible.
• Draw field towers that make straightedge-and-compass constructions possible.
• Technical names: constructibility, Fermat primes, degree 2 extensions.
• Watch primes split or ramify in cyclotomic fields like Q(ζn); compute discriminants and sample class groups.
• Technical names: Frobenius elements, splitting/ramification, discriminants, class groups, cyclotomic units.
• Build small BCH or Reed–Solomon codes from cyclotomic ideas; decode toy examples by hand or with Sage/Python.
• Technical names: minimal polynomials, generator/locator polynomials, distance/weight enumerators.
• Explore regular vs. irregular primes, connections to Fermat’s Last Theorem, and modern directions such as Vaughan pairs and Dirichlet characters.
• Technical names: class number problems, Stickelberger (ideas), Dirichlet characters, Vaughan pairs.
• Study the cube, tetrahedron, octahedron, dodecahedron, and icosahedron. Prove exactly
which symmetries they have (including the A5 magic for the icosahedron).
• Relate rotations in SO(3) to unit quaternions (double cover by SU(2)).
• Technical names: rotational/reflection groups, SO(3), A5, quaternions and the SU(2) → SO(3) covering.
• Classify frieze groups and the 17 wallpaper groups; spot glides, rotations, and reflections in real images or your own tessellations.
• Technical names: planar isometry groups, orbifolds (intro), crystallographic restrictions.
• Link constructible n-gons to cyclotomic fields; give compass-and-straightedge “recipes” for small n; explain why some n are impossible.
• Technical names: minimal polynomials of cos(2π/n), field towers, Galois groups.
• Draw quivers (nodes + arrows), form path algebras, and study quiver representations as linear algebra along the arrows.
• Track indecomposables in tiny examples and visualize the Auslander–Reiten quiver.
• Technical names: Gabriel’s theorem (types A, D, E), tame vs. wild representation type, Auslander–Reiten quiver.
• Think in categories: objects, morphisms, functors, natural transformations; re-see standard algebra via universal properties.
• Technical names: limits/colimits, adjoint functors, Yoneda lemma (conceptual level).
• Study how tensor products “fuse” (begin with sl2 and Clebsch–Gordan); peek at fusion categories and modular tensor categories.
• Technical names: monoidal categories, fusion rings, modular tensor categories, Tannaka–Krein duality (idea).
• Explore quiver gauge theories (string-inspired) or use quivers to model workflows and automata.
• Technical names: Nakajima quiver varieties, path algebras in algorithms, transition monoids.
• Use braid groups and quantum groups to produce knot invariants; compute small examples.
• Technical names: braid groups Bn, Burau representation (low-dim), Jones polynomial, ribbon categories.
• Explore the strange link between the Monster group and modular functions, with a peek at vertex operator algebras.
• Technical names: Conway–Norton moonshine, modular forms, VOAs, Borcherds’ theorem (story level).
• Model codons with modular arithmetic and use group actions to study mutations; find symmetries in biological or social networks via automorphism groups.
• Technical names: Z_4^3 codon models, group actions, graph automorphisms, equitable partitions.
• Analyze chords and progressions via transformations; experiment with simple tools to explore musical symmetry.
• Technical names: T /I group, P LR transformations, Tonnetz, DFT on Z12.
Getting Started
1. Email a short note about the theme(s) that interest you and your background.
2. We’ll meet to scope a project: a core problem, 1–2 enrichment paths, and any applications you want to try.
3. You’ll write in LATEX, keep a living bibliography, and present a short talk near the end.
4. Ambitious? We can aim for a poster, preprint, or software demo.
Tools you might use SageMath/Python, lightweight CAS experiments, and simple visualization (drawing root systems, quivers, or polyhedra).
Friendly Acronyms You May See
BCH (coding), CAS (computer algebra system), CSS (quantum codes), CRT (Chinese Remainder Theorem), DFA (automata), DFT (discrete Fourier transform), ECC (elliptic-curve cryptography), FFT (fast Fourier transform), LDPC (codes), PLR/TI (music theory operations), PQC (postquantum cryptography), PSL/GL/SU (matrix groups), TQFT (topological quantum field theory), UFD (unique factorization domain), VOA (vertex operator algebra).