Ardila’s matroid course, rewritten by topic: independence, duality, flats, minors, and representability.
Vectors, graphs, and matchings as three pictures of independence.
The independent-set axioms, and linear, graphic, and transversal matroids.
Basis axioms and the basis-exchange property.
The greedy algorithm characterises matroids.
Circuit axioms and elimination.
Dual matroids, and duality for linear, graphic, and transversal matroids.
Rank functions and closure operators.
Flats, geometric lattices, and simple matroids.
Deletion, contraction, and minors in the three models.
Forbidden minors, the Fano matroid, and representability.
Algebraic dependence and algebraic matroids.
Notes from Chen Ting’s Fourier analysis lectures: Lp, the maximal function, the transform, and tempered distributions.
Lp, weak Lp, convolution, approximate identities, and Marcinkiewicz interpolation.
Lp spaces, weak Lp, and Chebyshev.
Distribution functions and the layer-cake representation.
Young's inequality and the basic properties of convolution.
Approximate identities and convergence in Lp.
Riesz–Thorin and Marcinkiewicz interpolation.
The weak (1,1) bound, Lebesgue differentiation, and the HLS inequality.
The weak (1,1) bound and the dyadic maximal function.
Pointwise convergence of operators and Lebesgue points.
Pointwise convergence, Poisson integrals, and the radial maximal function.
Riesz potentials and the Hardy–Littlewood–Sobolev inequality.
The L1 theory, inversion, Plancherel, and Hausdorff–Young.
The Fourier transform on L1.
The inversion formula and uniqueness on L1.
Plancherel and the L2 theory.
Fourier transforms of complex measures.
Convolution of complex measures.
The Schwartz space and the Fourier transform on tempered distributions.
The Schwartz space and its Fourier transform.
Distributions and tempered distributions.
The Fourier transform on tempered distributions.
Topological and smooth manifolds, examples, paracompactness, products, and quotients.
Second countability, Hausdorff, atlases, and orientability.
Stereographic charts on the sphere and affine charts on real projective space.
Manifolds are locally compact, paracompact, and metrizable.
Products, connected sums, and free properly discontinuous quotients.
Series index: polynomial avoidance, the rank formula, tree coordinates, and positive realization.
The only forced homogeneous relation on the core is the zero boundary sum.
The differential rank equals the number of vertices minus the components of variable non-flat edges.
Weights on a non-flat spanning tree are local analytic coordinates with a rational inverse.
Local coordinates do not prescribe which extensions positive weights realize globally.
Series index: the isometric obstruction, the highway metric, and the tube theorem.
A vertex of degree at least three blocks an isometric embedding into a smooth Riemannian manifold.
A conformal factor that is cheap on the graph and expensive off it.
On the tripod, a hard tube loses a fixed multiple of delta; a soft penalty loses delta times lambda to a negative power.
Ribbon surfaces avoid planar crossings, and shortcuts are confined to vertex disks.
The highway metric is a variational model, not a replacement for Dijkstra.
A 3-dimensional thickening usually creates shortcuts that the graph does not have.
Series index: boundary injectivity, vertex Morse, extrema, and nodal domains.
Weak unique continuation fails, but a generic nonconstant eigenfunction is still injective on the boundary.
On the core the extension is vertex Morse and the index sum equals the Euler characteristic.
A path has j+1 extrema, a cycle has 2 ceil(j/2), and lambda_1 on a tree is extremal at the leaves.
When interior components meet the boundary in at most two vertices, nodal counts on V and on the support graph agree.
Homework boxes, notes side-bars, and a textbook class for lecture notes.
Notes on Szemerédi’s graph regularity lemma: density, energy, and the tower bound.
Notes on sequence problems in additive combinatorics.
Personal notes and improvements on Csaba’s Ramsey–Turán paper.
Course report on the Ramsey–Turán problem for K₄, split by section.
Final stretch of modern graph theory assignments.
Final homework for financial engineering.
Homework on extremal and structural graph theory.
Advanced financial engineering homework.
Szemerédi-type arguments in graph theory.
Financial engineering problem set.
Regular pairs and bipartite subgraphs.
Portfolio hedging and stochastic models.
Tableau method for the transportation problem.
Problem set on graph parameters.
Advanced topics in modern graph theory.
Financial engineering exercises on derivatives.