Resources Library

Foundations

Core Curriculum Refresh

• Pre-Algebra ➜ Fractions, ratios, modular thinking.
• Algebra I & II ➜ Polynomials, quadratics, systems.
• Geometry ➜ Euclidean proofs, coordinate geometry.
• Precalculus ➜ Sequences, complex numbers, vectors.

Competition Core

Intermediate Contest Prep

• AMC/AIME Bridge problems sets with video walkthroughs.
• Generating Functions primer for counting problems.
• Analytic Geometry challenges with transformations.
• Combinatorial identities & invariants practice.

Olympiad Labs

Advanced Exploration

• Functional equation clinics with substitution heuristics.
• Geometry labs on complex numbers & inversion.
• Number theory sprints: LTE, Hensel lifting, CRT variants.
• Combinatorics bootcamps on probabilistic method.

Local Methods

p-adic Valuations & Hensel Lifts

Illustrative prompt. Determine the largest integer k such that 3^k divides F_2n (the 2n-th Fibonacci number).

Approach cues. Track v₃ using lifting-the-exponent on the recursion, then apply Hensel to extend solutions of x² + 1 ≡ 0 (mod 3) into 3-adic roots.

Useful on TST/USAMO NT problems where congruence classes alone fail; p-adic lifts convert divisibility puzzles into local analysis statements.

Transformational Geometry

Projective & Möbius Toolkits

Illustrative prompt. Given a cyclic quadrilateral, prove the Miquel points of its three diagonal triangles coincide.

Approach cues. Switch to projective coordinates, use cross-ratio preservation, or deploy a Möbius map sending the circumcircle to the unit circle so spiral similarities become linear fractional transformations.

Central to IMO shortlist geometry where synthetic steps explode; mastering projective duals and Möbius conjugations simplifies concurrency/collinearity claims.

Polynomial Power

Combinatorial Nullstellensatz

Illustrative prompt. Show that any coloring of the cells of an n × n grid with fewer than n colors contains a monochromatic rook transversal.

Approach cues. Encode placements via a multivariate polynomial, track coefficient of x₁x₂…x_n, and invoke Nullstellensatz to guarantee a non-zero evaluation.

Shows up on EGMO/IMO combinatorics where counting arguments stall; polynomial interpretations turn extremal existence questions into algebraic coefficient hunts.

Analytic Combinatorics

Generating Functions & Saddle Points

Illustrative prompt. Estimate the growth of the number of ways to write n as a sum of distinct squares.

Approach cues. Build an exponential generating function, apply logarithmic differentiation, then use Cauchy’s integral with saddle-point asymptotics for bounds.

Relevant for APMO/IMO probability-combinatorics hybrids and computing precise error terms in recurrence solutions.

Graph Extremes

Spectral & Probabilistic Methods

Illustrative prompt. Prove the existence of a triangle-free graph on n vertices with more than n^3/2 edges.

Approach cues. Combine eigenvalue bounds (Hoffman) with Erdős’s probabilistic method to exceed deterministic constructions.

These tactics surface on RMM/IMO shortlist combinatorics when brute force double counting underestimates the extremal threshold.

Game & Recurrence Theory

Sprague-Grundy & Morphic Sequences

Illustrative prompt. Analyze a variant of Wythoff Nim where players may remove Fibonacci numbers of stones from either pile.

Approach cues. Encode legal moves as mex recurrences, exploit Beatty sequence structure, and use morphic-word representations to classify cold positions.

Appears in Titu Andreescu training sets and WOOT assignments; provides competitive edges on niche combinatorial game theory problems.

Algebraic Number Theory

Cyclotomic Fields & Algebraic Integers

Illustrative prompt. Classify the integer solutions to x² + xy + y² = 7^k.

Approach cues. Factor inside the Eisenstein integers, study unit groups of Ζ₃, and chase norms to force exponent parity.

Used on IMO/EGMO number theory where classical congruences stall; cyclotomic factorization turns Diophantine systems into ideal-factor arguments.

Arithmetic Geometry

Elliptic Curves & Descent

Illustrative prompt. Determine all rational points on y² = x³ - 2.

Approach cues. Compute torsion subgroups, run a 2-descent, and certify Mordell-type finiteness with height bounds.

Appears in Putnam-style algebra problems and high-tier TSTs; gives a blueprint for handling quartic Diophantine equations by embedding in elliptic curves.

Probabilistic Combinatorics

Entropy & Container Methods

Illustrative prompt. Show that the number of triangle-free graphs on n vertices is at most 2^O(n3/2).

Approach cues. Use entropy to encode adjacency lists, combine with graph container lemmas to bound independent-set counts.

Central to modern extremal problems (RMM, IMO SL C8 variants); trains contestants to translate randomness into rigorous counting arguments.

Analysis & Variational Methods

Euler–Lagrange Toolkits

Illustrative prompt. Among all smooth curves joining two points with fixed arc length, maximize the enclosed area.

Approach cues. Formulate a functional, compute Euler–Lagrange equations, and use transversality to characterize extremizers (circles).

Cultivates rigor for inequality proofs that cite calculus of variations; relevant to Balkan/AUSMO tasks on isoperimetric-type extremals.

Representation Theory

Characters & Burnside Applications

Illustrative prompt. Count non-equivalent colorings of the faces of a dodecahedron with three colors.

Approach cues. Apply Burnside’s lemma, upgrade to character tables for rotational symmetry, and tabulate cycle structures.

Shows up in WOOT and national olympiads where pure counting is messy; representation language streamlines symmetric enumeration problems.

Additive Combinatorics

Discrete Fourier & Energy Bounds

Illustrative prompt. Prove that a subset of {1,…,N} with no 4-term arithmetic progression has size O(N^2/3).

Approach cues. Use Fourier coefficients to measure additive energy, invoke the Balog–Szemerédi–Gowers theorem, and iterate density increment steps.

Equips contestants for cutting-edge number theory/combinatorics fusion problems (IMO SL N6/C6) where classical pigeonhole is insufficient.

Additive Combinatorics

Sum-Product & Energy Bounds

Illustrative scenario. Show that for a finite A ⊆ ℝ, either |A + A| or |A·A| is large.

Application cues. Apply Cauchy–Davenport, incidence geometry (Szemerédi–Trotter), and energy decompositions to force expansion.

Modernizes contest combinatorics: the same energy heuristics resolve hard AIME/IMO inequalities and algebraic identities.

Model Theory & Logic

O-minimal Geometries

Illustrative scenario. Prove that a definable subset of ℝⁿ has finitely many connected components.

Application cues. Use cell decomposition, dimension arguments, and quantifier elimination to control solution sets.

Bridges olympiad inequalities to structural results: o-minimal thinking clarifies when iterative substitutions exhaust all solutions.

Geometric Group Theory

Cayley Graph Quasi-Isometries

Illustrative scenario. Classify growth rates of a finitely generated group using its Cayley graph.

Application cues. Compare metrics via quasi-isometries, apply Gromov’s polynomial growth theorem, and convert to combinatorial bounds.

Sharpens contest instincts for counting walks/words and analyzing graph diameter bounds on RMM/IMO shortlist problems.

Variational Methods

Calculus of Variations & PDE Flows

Illustrative scenario. Optimize the shape of a membrane to minimize the first eigenvalue of the Laplacian.

Application cues. Derive Euler–Lagrange equations, test admissible perturbations, and leverage isoperimetric inequalities.

Feeds olympiad geometry/analysis crossovers — the same extremal principles justify inequality boundary cases.

Random Matrix Heuristics

Wigner Semicircle & Concentration

Illustrative scenario. Estimate the probability that the largest eigenvalue of a random symmetric matrix exceeds 2√n.

Application cues. Use moment methods, trace power expansions, and concentration inequalities (McDiarmid/Azuma).

Provides probabilistic intuition for matrix inequalities that show up as disguised olympiad linear algebra or graph spectral problems.

Sheaves & Topology

Cohomological Counting

Illustrative scenario. Use Lefschetz fixed-point formulas to count points on curves over finite fields.

Application cues. Translate combinatorial data into cohomology groups, evaluate traces of Frobenius, and reduce to reciprocity statements.

Connects olympiad-level generating functions to deeper algebraic topology frameworks, inspiring novel approaches to counting and symmetry.

Ergodic & Ramsey Methods

Multiple Recurrence Heuristics

Illustrative example. Show that any subset of the integers with density > 0.3 contains a 3-term arithmetic progression.

Typical problem. Translate density into ergodic averages, invoke van der Waerden/Szemerédi via multiple recurrence, and push back to a finite coloring argument.

Competition relevance: gives a conceptual playbook for coloring and invariance problems on IMO shortlist combinatorics where classic pigeonhole counts stall.

Modular Forms & q-Series

Theta Functions in Diophantine Problems

Illustrative example. Express the number of representations of an integer as a sum of four squares using theta series.

Typical problem. Build a modular form generating function, compare Fourier coefficients, and interpret congruence restrictions through transformation laws.

Competition relevance: upgrades number-theory bounding tricks for USAMO/IMO N6-type equations by connecting quadratic forms to analytic identities.

Geometric Measure Theory

Isoperimetric & Hausdorff Tools

Illustrative example. Prove that a plane curve enclosing area 1 must have length at least 2√π using measure arguments.

Typical problem. Approximate sets with smooth regions, integrate curvature bounds, and translate Hausdorff dimension estimates into inequality constraints.

Competition relevance: reinforces why equality cases in olympiad inequalities arise from circles/regular figures, informing construction of extremal witnesses.

Discrete Harmonic Analysis

Fourier Tiles & Restriction

Illustrative example. Bound the number of lattice points on a discrete parabola using Fourier restriction estimates.

Typical problem. Take the discrete Fourier transform, identify major/minor arcs, and apply restriction/uncertainty principles to limit solution counts.

Competition relevance: clarifies additive energy bounds and exponential sum cancellations that frequently appear in AIME/IMO analytic number theory.

Algebraic Topology

Homology for Counting Problems

Illustrative example. Determine whether a graph embedded on a torus can be edge-colored with two colors without creating a monochromatic cycle.

Typical problem. Lift the embedding to fundamental cycles, compute homology classes, and detect obstructions via boundary maps.

Competition relevance: equips contestants to reason about topology flavored combinatorics (e.g., US TST G5/C6) where planar intuition is insufficient.

Convex Optimization

Duality & Barrier Arguments

Illustrative example. Minimize a symmetric sum subject to linear constraints using convex duality.

Typical problem. Write the Lagrangian, derive KKT conditions, construct barrier functions, and interpret the dual solution as a weighted inequality.

Competition relevance: translates SOS/PSD intuition into a systematic approach for high-end inequalities on IMO/Putnam problems.

Arithmetic Dynamics

Rational Iterations & Heights

Illustrative prompt. Classify all rational periodic points of f(x) = x² − 2.

Approach cues. Reduce mod primes to prune cycles, apply the Northcott finiteness of canonical heights, and chase orbits through quadratic extensions.

Teaches contestants how to weaponize iteration/recurrence structure on US TST N-problems where naive bounding fails.

Tropical Geometry

Newton Polygons & Valuations

Illustrative prompt. Determine how many solutions the system x³ + y = 1, x + y³ = 1 has over the 3-adics.

Approach cues. Tropicalize the curves, inspect slopes of the Newton polygons, and refine the candidate valuations via balancing conditions.

Connects contest-level valuation tricks to polyhedral intuition; handy on IMO shortlist NT/Alg problems where p-adic lifting needs geometric insight.

Expander Paradigms

Spectral Gaps & Zig-Zag

Illustrative prompt. Construct a 3-regular graph on 2^m vertices with second eigenvalue at most 2√2.

Approach cues. Blend Alon–Boppana bounds with the zig-zag product, track adjacency spectra, and verify Cheeger inequalities.

Feeds extremal-combinatorics intuition for olympiad problems asking for dense graphs avoiding patterns or modeling rapid mixing in Markov chains.

Probabilistic Number Theory

Sieve Weights & Densities

Illustrative prompt. Show that the count of integers up to n with all prime factors ≤ n^1/3 is O(n^2/3).

Approach cues. Deploy Selberg weights, bound partial zeta products, and compare with Brun–Titchmarsh style integrals.

Sharpens contestants’ ability to quantify rarity (e.g., smooth numbers) when contest questions mix combinatorics with density estimates.

Algebraic Combinatorics

Young Tableaux & Symmetric Functions

Illustrative prompt. Count the number of standard Young tableaux of a 2 × n rectangle.

Approach cues. Apply the hook-length formula, interpret Schur polynomials as generating functions, and use bijective proofs to verify identities.

Empowers olympiad combinatorics when counting structured objects or proving inequalities between symmetric sums.

Optimization Theory

Sum-of-Squares & SDP Bounds

Illustrative prompt. Prove that the polynomial x⁴ + y⁴ + 1 − x²y² is nonnegative.

Approach cues. Set up a semidefinite program for Gram matrix coefficients, express the result as a sum of squares, and connect to weighted AM-GM.

Extends inequality playbooks with rigorous certificates, mirroring IMO solutions that disguise SOS or PSD-matrix arguments.

Nonlinear Recurrence Dynamics

Invariant Curves & Monodromy

Illustrative example. Classify eventual behavior of the Lyness recurrence a_n+2 = (a_n+1 + 1)/a_n for positive rationals.

Approach cues. Identify conserved quantities, study the birational map on elliptic curves, and use monodromy of level sets to detect periodicity.

Competition relevance: reframes tricky recurrences on IMO/Putnam problems as geometric flows, revealing hidden invariants beyond simple telescoping.

Infinitary Combinatorics

Ultrafilters & Compactness

Illustrative example. Prove a van der Waerden style theorem for colorings of the naturals using ultrafilter limits.

Approach cues. Build idempotent ultrafilters, apply Ellis semigroup compactness, and project to finite progressions.

Competition relevance: teaches how infinite arguments can be compressed into finite density proofs that often underlie hardest AIME/USAMO combinatorics.

Arithmetic Statistics

Distribution of Arithmetic Invariants

Illustrative example. Estimate the proportion of quadratic fields with class number 1 using Cohen–Lenstra heuristics.

Approach cues. Translate invariants into random matrix models, use moments of L-functions, and interpret probabilistic heuristics.

Competition relevance: equips contestants to justify density claims when olympiad NT asks “how often” a Diophantine pattern occurs.

Differential-Algebraic Synthesis

Ritt Decomposition & Functional Equations

Illustrative example. Solve f(f(x)) = x³ + 3x by decomposing rational functions.

Approach cues. Apply Ritt’s theory to split compositions, compare degrees via differential Galois groups, and enforce algebraic independence.

Competition relevance: formalizes functional equation tricks common on TST/IMO problems involving iterates or symmetry constraints.

Microlocal Geometry

Wavefront Sets & Stationary Phase

Illustrative example. Analyze asymptotics of I(t) = ∫₀¹ exp(2πi(t² + x³)) dx.

Approach cues. Track stationary points, control phases via Morse lemmas, and interpret decay using wavefront sets.

Competition relevance: sharpens oscillatory integral intuition so contestants can bound exponential sums or evaluate tricky limits with analytic flair.

Model-Theoretic Number Theory

Definable Sets & Pila–Wilkie

Illustrative example. Bound rational points of bounded height on the graph of exp(x) over [0,1].

Approach cues. Invoke o-minimality to stratify definable sets, apply Pila–Wilkie counting, and translate to Diophantine bounds.

Competition relevance: introduces a blueprint for taming functional equations and transcendence-style inequalities that appear on elite olympiad shortlists.

Higher-Order Fourier Analysis

Gowers Norm Diagnostics

Illustrative example. Estimate the number of 4-term arithmetic progressions inside a subset of {1,…,N} with density 0.3.

Approach cues. Measure bias via U³ norms, apply inverse theorems to detect quadratic phases, and decompose the set into structured + pseudorandom pieces.

Competition relevance: clarifies why combinatorial energy bounds work on USAMO/IMO problems about additive configurations, and suggests when to try exponential sum estimates or density increments.

Motivic Integration

Arc Spaces & Counting

Illustrative example. Compare the number of solutions of x² + y² = 1 modulo p^k for varying k.

Approach cues. Lift solutions through arc spaces, evaluate volumes via motivic measures, and interpret congruence counts as coefficients of generating series.

Competition relevance: reframes p-adic lifting/LTE ideas with a geometric lens, suggesting structured ways to push congruence identities on N-level olympiad questions.

Derived Category Maneuvers

Mutations & Exact Triangles

Illustrative example. Show that two projective plane curve complements are derived equivalent via a sequence of mutations.

Approach cues. Package complexes into exact triangles, mutate along exceptional collections, and read off cohomological invariants that remain unchanged.

Competition relevance: even if derived categories stay theoretical, the mutation mindset helps contestants reorganize algebraic identities or telescoping chains on olympiad problems.

Non-Archimedean Geometry

Berkovich Skeleta

Illustrative example. Analyze the reduction graph of an elliptic curve over Q_p and classify its component group.

Approach cues. Build the Berkovich skeleton, track lengths via valuations, and push intersection numbers through specialization maps.

Competition relevance: deepens intuition for Newton polygons and valuation trees so that AIME/USAMO solutions using Hensel or lifting arguments feel more geometric and less ad hoc.

Stochastic Calculus Bridges

Optional Stopping & Martingales

Illustrative example. Bound the probability that a biased random walk hits +10 before −4.

Approach cues. Model payoffs via martingales, apply optional stopping with carefully bounded increments, and convert expectations to inequality bounds.

Competition relevance: informs how to control random processes on APMO/Putnam problems and provides a rigorous path to the probabilistic method beyond simple linearity of expectation.

Topological Data Signatures

Persistent Homology Intuition

Illustrative example. Determine how many connected components persist in a Vietoris–Rips filtration of 12 lattice points.

Approach cues. Build filtration complexes, compute birth- death pairs via boundary matrices, and interpret barcodes as combinatorial invariants.

Competition relevance: offers a structured way to reason about evolving graphs or simplicial complexes, which appears in combinatorics and geometry shortlist problems disguised as dynamic connectivity tasks.

Cyclotomic Fields

Stickelberger & Splitting

Illustrative example. Determine all primes p ≡ 1 (mod 9) for which x³ + y³ + z³ = 0 has only trivial solutions modulo p.

Typical problem. Embed the congruence in ℚ(ζ₉), track principal ideals using Stickelberger elements, and read off splitting conditions from the class group.

Application note. Elevates primitive-root and CRT tactics for USAMO/IMO N-problems by showing when higher cyclotomic structure is the right lens.

Elliptic Curves

Two-Descent Playbook

Illustrative example. Classify rational solutions of y² = x³ − 2 via descent.

Typical problem. Compute the 2-Selmer group, analyze local solubility at small primes, and stitch the information into a rank bound that limits rational points.

Application note. Gives olympiad solvers a concrete method to push beyond Mordell curves built from infinite descent or factoring tricks.

Entropy Compression

Algorithmic Probabilistic Method

Illustrative example. Prove that every planar graph has a vertex-coloring with five colors that avoids a prescribed forbidden pattern.

Typical problem. Design a randomized recoloring algorithm, encode the run history, and compare the log of the trace count before/after compression to show termination with positive probability.

Application note. Sharpens the probabilistic-method instincts behind APMO/IMO combinatorics, especially when the Lovász Local Lemma feels too blunt.

Additive Energy

Fourier Decomposition Tactics

Illustrative example. Bound the number of solutions to a + b = c + d inside a dense subset of {1,…, N} lacking 4-term arithmetic progressions.

Typical problem. Express the indicator via the discrete Fourier transform, apply Parseval to quantify additive energy, and feed the bounds into Balog– Szemerédi–Gowers style density increments.

Application note. Explains the backbone of advanced AIME/USAMO additive-combinatorics solutions that juggle energies, sumsets, and Fourier bias.

Cluster Algebras

Mutation Recurrences

Illustrative example. Track the Markov number recurrence x² + y² + z² = 3xyz under mutations and prove positivity of all iterates.

Typical problem. Package variables into seeds, perform exchange relations, and monitor invariants (e.g., Laurent positivity) that survive every mutation.

Application note. Reframes olympiad recurrences and Diophantine dynamics as structured mutation games, reducing guesswork in long algebra chains.

Discrete Morse Theory

Gradient Matchings on Complexes

Illustrative example. Compute the Betti numbers of a simplicial complex formed from divisibility relations on {1,…,12}.

Typical problem. Construct a Morse matching on the face poset, count critical cells, and convert the data into homology groups.

Application note. Provides a principled language for topology-flavored combinatorics problems (e.g., IMO shortlist C/G tasks) that ask for counting cycles or proving connectivity statements.

Diophantine Approximation

Baker Bounds on S-Unit Equations

Illustrative example. Show that 2^x − 3^y = 1 has only the solutions (1,1), (2,1).

Typical problem. Encode the exponential Diophantine as a linear form in logarithms, invoke Baker’s lower bounds to trap the exponents, and finish with a short brute-force sweep.

Application note. Gives olympiad solvers a principled way to cap exponent sizes on LTE/Lifting problems instead of guessing when to stop an infinite descent.

Geometric Invariant Theory

Moment Maps & Stability Tests

Illustrative example. Determine when six weighted points on ℙ¹ are stable under the action of PGL₂.

Typical problem. Evaluate Hilbert–Mumford weights via one-parameter subgroups, read the answer off the moment polytope, and interpret the result in terms of cross-ratio inequalities.

Application note. Enhances contest geometry by reframing barycentric, projective, and inversion arguments as stability checks on weighted configurations.

p-adic Hodge Bridges

Newton Slopes vs. Hodge Filtrations

Illustrative example. Classify the valuations of solutions to x^p − ax = b over ℚ_p.

Typical problem. Build the Newton polygon of the Frobenius polynomial, compare slopes with the Hodge polygon, and use the resulting inequalities to pin down possible valuations.

Application note. Translates Hensel and lifting arguments into visual slope comparisons, sharpening number-theory instincts for AIME/USAMO congruence hunts.

Extremal SDP Frameworks

Flag Algebra Certificates

Illustrative example. Prove that every K₄-free graph has triangle density at most 2/9.

Typical problem. Encode subgraph densities as algebraic variables, set up semidefinite constraints using flag algebras, and interpret the certificate as a weighted inequality.

Application note. Suggests systematic ways to craft extremal inequalities on olympiad combinatorics problems beyond ad hoc double counting.

Automorphic Bridges

Theta Correspondences & Counts

Illustrative example. Express the number of representations of n as x² + 2y² via a modular theta series and deduce congruence restrictions on n.

Typical problem. Construct a theta lift, match Fourier coefficients with representation numbers, and use modular transformations to read off density statements.

Application note. Links generating functions and congruences in a disciplined way, clarifying when to invoke modular-form heuristics on USAMO counting problems.

Matroid Theory

Tutte Polynomials & Flow Duality

Illustrative example. Compute the Tutte polynomial of a cographic matroid arising from a 3-regular graph and interpret evaluations at (1,1) and (2,0).

Typical problem. Run deletion–contraction recurrences, translate the polynomial’s specializations into counts of spanning trees or nowhere-zero flows, and dualize the data for planar complements.

Application note. Reinforces how invariants survive graph transformations, a skill that unlocks creative manipulations on olympiad graph and network flow tasks.

Transcendence Theory

Schneider–Lang Barriers

Illustrative example. Prove that e^1/3 and e^2/3 are algebraically independent over ℚ.

Typical problem. Translate relations into linear forms in logarithms, invoke Schneider–Lang or Nesterenko bounds, and squeeze the possible degrees until only the trivial relation survives.

Application note. Sharpens when to escalate from LTE to analytic number theory in olympiad exponentials, replacing guesswork with explicit height bounds.

Motivic Cohomology

Higher Chow Intuitions

Illustrative example. Compute the regulator map for a cycle in CH²(P² ∖ D) and relate it to dilogarithm values.

Typical problem. Build a cubical complex model, chase the boundary maps, and evaluate the regulator integral to recover special zeta values.

Application note. Gives a principled language for contest problems that secretly ask for evaluating polylogarithmic sums or relating areas to zeta constants.

Geometric Representation Theory

Crystal Graph Playbooks

Illustrative example. Use Littelmann paths to compute the multiplicities of weights in a tensor product of sl₃-modules.

Typical problem. Model highest-weight modules via crystal graphs, apply Kashiwara operators to enumerate tableaux, and extract multiplicities from saturation-type inequalities.

Application note. Illuminates why certain combinatorial identities (Littlewood–Richardson, hook formulas) solve olympiad counting problems with representation-theoretic flair.

Hypergraph Extremes

Container & Absorber Methods

Illustrative example. Bound the number of 3-uniform hypergraphs on n vertices that avoid a Fano plane.

Typical problem. Build hypergraph containers, iterate the pruning to limit independent sets, then deploy absorption to construct near-extremal configurations.

Application note. Turns contest combinatorics into a disciplined density-control exercise, clarifying when to move past simple Turán-style bounds.

Topological Dynamics

Multiple Recurrence Engines

Illustrative example. Deduce van der Waerden-type colorings from Furstenberg’s multiple recurrence theorem.

Typical problem. Convert combinatorial statements into measure-preserving systems, apply ergodic decomposition, and translate recurrence back into arithmetic progressions.

Application note. Adds a dynamical viewpoint to additive-combinatorics olympiad problems, showing how ergodic heuristics justify density increment moves.

Information Geometry

Entropy & Divergence Flows

Illustrative example. Optimize an inequality of the form ∑ a_i log(a_i/b_i) subject to linear constraints using convex duality.

Typical problem. Model the objective with Kullback– Leibler divergence, apply Lagrange multipliers in the exponential family, and interpret the minimizer via mirror descent.

Application note. Bridges olympiad inequalities and coding-theory problems with the entropy method, elevating Jensen/AM-GM tricks into structured variational arguments.

Higher-Order Fourier

Gowers Norm Diagnostics

Illustrative example. Bound the number of 4-term arithmetic progressions inside a subset of {1,…, N} with density 0.28.

Typical problem. Measure the subset’s U³ norm, decompose it into structured nilsequence plus uniform error, and transfer the estimate into progression counts.

Application note. Explains why certain additive combinatorics olympiad problems require splitting into structure/randomness pieces instead of iterating Cauchy–Schwarz blindly.

Polynomial Optimization

Moment & SOS Hierarchies

Illustrative example. Certify that x⁴ + y⁴ + z⁴ ≥ xyz for all reals with x + y + z = 0.

Typical problem. Build the moment matrix, search for a sum of squares decomposition subject to linear constraints, and interpret the resulting certificate as a sharpened inequality.

Application note. Encourages olympiad solvers to translate inequality proofs into structured algebra (SOS / PSD matrices) whenever AM-GM/Jensen runs out of steam.

Arithmetic Statistics

Selmer Averages & Heuristics

Illustrative example. Predict the average size of the 2-Selmer group for elliptic curves of the form y² = x³ + ax + b with |a|, |b| ≤ H.

Typical problem. Parametrize local conditions, evaluate mass formulas over adelic orbits, and average the resulting densities to guess a global count.

Application note. Provides intuition for when to expect many or few rational solutions, guiding olympiad attempts that mix descent with counting arguments.

Geometric Flows

Curvature-Driven Inequalities

Illustrative example. Show via mean-curvature flow that any convex surface enclosing volume 1 has surface area at least that of the unit sphere.

Typical problem. Run a normalized flow, monitor monotone quantities (entropy, area, isoperimetric deficit), and freeze the evolution when equality emerges.

Application note. Reframes olympiad inequality puzzles as evolution arguments, clarifying why symmetric extremals (balls, regular polygons) appear in contest solutions.

Discrepancy Methods

Randomized Rounding & Beck-Fiala

Illustrative example. Two-color the columns of a 0/1 matrix with row sums ≤ 5 so that each row’s signed sum is bounded by 4.

Typical problem. Apply partial coloring or entropy-based random walks, round fractional assignments, and iterate until all constraints are satisfied.

Application note. Offers a structured approach to balancing arguments that show up in AIME/USAMO combinatorics, replacing ad hoc parity checks with principled discrepancy bounds.

Model-Theoretic Combinatorics

VC-Dimension & Stability

Illustrative example. Show that a definable family of subsets of {1,…, n} with VC-dimension 2 has at most O(n²) distinct traces.

Typical problem. Apply Sauer–Shelah, stabilize the configuration via indiscernible sequences, and deduce structural theorems that bound extremal counts.

Application note. Equips contestants to detect when combinatorial explosions are impossible, which streamlines solutions to olympiad problems about set systems, incidence graphs, or iterative colorings.

Perfectoid Techniques

Tilted Lifting Games

Illustrative example. Classify solutions to y^p − y = x^p − x over ℤ_p by passing to a perfectoid cover.

Typical problem. Tilt the tower to characteristic p, analyze the Frobenius-fixed points, and descend the structure back to mixed characteristic to recover integral solutions.

Application note. Sharpens instincts for Hensel/LTE puzzles by showing when repeated lifting hides inside a structured perfectoid or Witt-vector viewpoint.

Microlocal Analysis

Sheaf-Theoretic Stationary Phase

Illustrative example. Estimate exponential sums of the form ∑ exp(2πi f(n)/p) by studying the microsupport of the associated sheaf.

Typical problem. Encode the sum as the trace of Frobenius on an ℓ-adic sheaf, microlocalize near critical points, and apply stationary phase to extract power savings.

Application note. Reinforces exponential-sum heuristics that appear in olympiad inequalities, showing how stationary-phase reasoning controls oscillatory behavior.

Sato–Tate Heuristics

Angle Distributions & Moments

Illustrative example. Predict the bias of quadratic residues mod primes p where an elliptic curve has trace of Frobenius 0.

Typical problem. Model the Frobenius angles via the Sato–Tate measure, compute expected moments, and compare to explicit character sums to bound deviations.

Application note. Guides contest number theory when deciding whether to expect cancellation or constructive density in sequences defined by elliptic curves or character sums.

Geometric Representation Theory

Perverse Filtrations

Illustrative example. Compute the intersection cohomology of a Schubert variety by leveraging the decomposition theorem.

Typical problem. Resolve the variety, track perverse sheaf filtrations, and use weight arguments to derive character formulas.

Application note. Links symmetry arguments in olympiad algebra to deeper representation-theoretic structures, clarifying when Young-tableaux counting or hook-length strategies appear.

Optimal Transport

Synthetic Curvature via Wasserstein

Illustrative example. Prove Brunn–Minkowski type inequalities on a discrete graph using displacement convexity.

Typical problem. Construct geodesics in the Wasserstein space of probability measures, analyze entropy convexity, and translate the curvature bounds into combinatorial inequalities.

Application note. Illuminates why convexity arguments dominate olympiad inequality proofs and offers a transport-based perspective for discrete isoperimetric questions.

Probabilistic Spectral Graphs

Local Laws & Expanders

Illustrative example. Estimate the second eigenvalue of a random d-regular graph to certify expander properties with high probability.

Typical problem. Apply matrix concentration plus trace methods to bound spectral norms, then convert the bounds into mixing or diameter guarantees.

Application note. Strengthens contest combinatorics intuition for graph eigenvalues, helping solvers justify expansion-based arguments beyond purely deterministic constructions.

Parahoric Geometric Langlands Labs

Hecke Eigensheaf Gluing

Illustrative example. Construct an eigensheaf for a wildly ramified local system on a nodal curve by gluing parahoric level structures across the normalization.

Typical problem. Track how modifications at marked points alter the Hecke correspondence, compute the resulting vanishing cycles, and verify the eigenvalue character on global sections.

Application note. Mirrors olympiad habits of reconciling local constraints at multiple vertices or moduli to produce a consistent global configuration.

Cylindrical Contact Homology Charts

Reeb Dynamics Spectral Sequences

Illustrative example. Compute the contact homology of a prequantization bundle by resolving Morse–Bott families of Reeb orbits and organizing the resulting differentials into a spectral sequence.

Typical problem. Index simple and multiply covered orbits, evaluate counts of punctured holomorphic cylinders, and chase how filtration levels collapse to recover symplectic invariants.

Application note. Encourages competition solvers to catalogue periodic behaviors before summing contributions—exactly how generating functions or recursion trees are tamed on AIME/USAMO problems.

Hodge–Newton Polytope Diagnostics

Slope Profile Comparisons

Illustrative example. Given an F-isocrystal arising from a unit-root L-function, plot its Hodge and Newton polygons and isolate the break points that control admissibility.

Typical problem. Compute Hodge numbers from a filtered isocrystal, determine Newton slopes via Frobenius eigenvalues, and test Mazur’s inequality by comparing the convex hulls.

Application note. Rehearses the contest skill of bounding one diagram by another (e.g., area under a curve, cumulative sums) to constrain the feasible numerical profiles in inequality or combinatorics tasks.

Spectral Network Wall-Crossing

WKB Triangulation Maps

Illustrative example. Track how a spectral network on a quadratic differential mutates when a phase crosses a critical value, and record the induced cluster transformation.

Typical problem. Integrate WKB paths between branch points, enumerate finite webs, and compute how framed BPS invariants jump when new trajectories emerge.

Application note. Teaches competitors to follow phase changes meticulously—akin to watching how sign flips or inequalities change during substitution-heavy olympiad manipulations.

Randomized Schubert Calculus

Stochastic Littlewood–Richardson Sampling

Illustrative example. Approximate intersection numbers in a flag variety by running a random walk on puzzles or hives that encode Littlewood–Richardson coefficients.

Typical problem. Implement jeu-de-taquin shuffles, measure convergence of sampling, and validate the probabilistic counts against determinantal formulas.

Application note. Emphasizes the contest tactic of simulating small cases or randomized invariants to conjecture patterns before committing to a rigorous proof.

Nonlinear Brunn–Minkowski Flows

Convex Heat Blueprinting

Illustrative example. Evolve a convex body under Gauss curvature flow and verify that mixed volumes satisfy a strengthened Brunn–Minkowski inequality along the trajectory.

Typical problem. Differentiate support functions, control anisotropic curvature terms, and derive entropy-like quantities that remain monotone during the flow.

Application note. Reinforces olympiad approaches where smoothing or averaging arguments (e.g., mixing sequences, balancing vectors) convert a rigid inequality into a manageable evolution.

Prismatic Hodge Correspondence

Windows Between Hodge–Tate & de Rham

Illustrative example. Compare the prismatic cohomology of a supersingular K3 surface with its Hodge–Tate and de Rham realizations, tracing how the Nygaard filtration captures slope information.

Typical problem. Construct prismatic crystals, evaluate their Frobenius actions, and verify compatibility with filtered φ -modules to transfer integral structures between cohomology theories.

Application note. Reinforces contest instincts about translating a configuration into multiple equivalent encodings (angles vs. vectors vs. complex numbers) to exploit whichever perspective yields the cleanest invariant.

Noncommutative Crepant Resolutions

Derived Equivalence Blueprints

Illustrative example. Resolve a threefold toric singularity by building the endomorphism algebra of a tilting bundle and proving it yields a noncommutative crepant resolution equivalent to a small crepant blowup.

Typical problem. Choose a reflexive module collection, compute Ext quivers, verify Calabi–Yau properties of the derived category, and compare mutation sequences that connect different NCCRs.

Application note. Models olympiad solutions that swap between dual constructions (e.g., combinatorial vs. geometric) until one viewpoint linearizes the constraints.

Tropical Riemann–Hilbert Bridges

Irregular Connection Skeletons

Illustrative example. Tropicalize the Stokes data of an irregular connection on ℓ ² and recover the corresponding piecewise-linear wall structure that determines the monodromy.

Typical problem. Extract exponential factors, construct the associated spectral network, and match the resulting broken lines with a Riemann–Hilbert factorization.

Application note. Echoes the contest trick of turning angular or trigonometric data into linearized coordinates (projective or barycentric) before solving a functional equation.

Polyhedral Expander Lifts

Ramanujan 2-Lift Toolkits

Illustrative example. Produce a new bipartite Ramanujan graph via a 2-lift guided by a signed matching on the base polyhedron, then check the resulting characteristic polynomial factors.

Typical problem. Optimize signings, estimate eigenvalue interlacing, and certify expansion by bounding the nontrivial spectrum through interlacing family arguments.

Application note. Mirrors competition strategies that construct auxiliary graphs or double coverings to impose parity or symmetry conditions on an otherwise messy combinatorial count.

Stochastic Frobenius Trace Models

Random Matrix Companions

Illustrative example. Approximate the distribution of Frobenius traces for a family of hyperelliptic curves by matching them with eigenvalues of the compact symplectic ensemble and validating the resulting moment predictions.

Typical problem. Define monodromy groups, run equidistribution heuristics, compute low-degree moments, and compare to Katz–Sarnak predictions for point-count statistics.

Application note. Encourages olympiad contestants to benchmark complicated arithmetic behavior against a tractable random model, much like testing a conjectured recurrence or inequality with small randomized inputs before formal proof.

Configuration Stack Stability

Factorization Homology Ranges

Illustrative example. Show that configuration spaces of a high-genus surface exhibit homological stability by expressing them as mapping stacks into the Ran space and applying factorization homology.

Typical problem. Set up spectral sequences from stratifications, track how E_n-algebra structures act on homology, and prove stability bounds that extend classical configuration-space theorems.

Application note. Reminds competitors that reorganizing a counting problem by gradually adding elements (induction on size, gluing vertices) often reveals monotonic behavior that can be formalized as stability.

Categorical Donaldson–Thomas Frames

Wall-Crossing Stability Scripts

Illustrative example. Track DT invariants of a Calabi–Yau threefold across a Bridgeland wall by analyzing how the heart of a t-structure tilts and how the Hall algebra product changes.

Typical problem. Compute invariants for a quiver with potential, determine the scattering diagram that records wall-crossing factors, and verify consistency relations among chambers.

Application note. Mirrors contest habits of partitioning configuration space into regimes where the invariant is stable, then matching boundary behavior to stitch a global answer.

Geometric Quantization of Higgs Landscapes

Bohr–Sommerfeld Lattice Maps

Illustrative example. Quantize the Hitchin system for a genus-2 curve by selecting a polarization, computing Bohr–Sommerfeld fibers, and relating the resulting Hilbert space to automorphic representations.

Typical problem. Evaluate the symplectic form on the moduli of Higgs bundles, classify integral affine structures on the base, and derive how the quantization changes under twisting by local systems.

Application note. Reinforces olympiad-ready reasoning about dual coordinate systems (action-angle vs. geometric) and encourages translating between them to expose conserved quantities.

Nonlinear Fourier Decoupling

Multiscale Restriction Plans

Illustrative example. Apply Bourgain–Demeter decoupling to bound exponential sums over polynomial phases by partitioning frequency space into curved caps and estimating their L^p contributions.

Typical problem. Iteratively rescale a surface, establish induction-on-scales inequalities, and combine wave packet decompositions with number-theoretic input to prove Vinogradov-type bounds.

Application note. Encourages contest solvers to break a hard sum/product into scale-based chunks, estimate each, and then recombine—exactly the logic behind bounding tricky inequalities or generating function coefficients.

Arithmetic Topology Correspondences

Knot–Prime Analogies

Illustrative example. Compare the Iwasawa theory of a cyclotomic field with the Alexander polynomial of a fibered knot, highlighting how linking numbers mirror ideal class pairings.

Typical problem. Translate arithmetic duality statements into topological exact sequences, compute torsion growth in towers, and test conjectured parallels such as Greenberg ↔ Thurston norm.

Application note. Nurtures the contest reflex of mapping one domain onto another to reuse tools—similar to switching between combinatorial and algebraic interpretations of the same counting problem.

Higher-Rank RSK Crystals

Cluster Tableau Evolutions

Illustrative example. Construct the crystal graph for a GL_n representation using generalized RSK insertion, then relate promotion operators to cluster mutations.

Typical problem. Enumerate tableaux with specified charge, compute energy functions, and interpret the data via solvable lattice models or Littlewood–Richardson coefficients.

Application note. Strengthens olympiad combinatorics by practicing bijections, recording invariants at each move, and ensuring the evolution stays within allowable states—key habits for constructive proofs.

Stochastic Geometric PDE Bootstraps

Randomized Regularity Cascades

Illustrative example. Study the Navier–Stokes equation on a torus with random initial data, proving almost-sure regularity by iterating probabilistic energy estimates.

Typical problem. Combine Bernstein inequalities, frequency envelopes, and concentration bounds to push local solutions globally, quantifying how randomness dampens nonlinear resonances.

Application note. Echoes contest strategies where introducing randomness or averaging (probabilistic method) simplifies otherwise brittle extremal arguments.

Motivic Period Pairings

Regulator Polylog Bridges

Illustrative example. Express the period matrix of a mixed Tate motive over ℚ through polylogarithm values and compare it with the Beilinson regulator of a K₃ class.

Typical problem. Build a cycle representative, evaluate its iterated integral, match the result with motivic cohomology generators, and track how the regulator changes under specialization.

Application note. Echoes olympiad habits of converting geometry into integrals or sums, then pairing dual viewpoints to isolate the key invariant (area vs. barycentric moments, sum vs. telescoping pairings).

Hybrid Subconvexity Amplifiers

Shifted Convolution Ladders

Illustrative example. Bound an L-function on the critical line by combining conductor-lowering, delta-symbol decompositions, and a bilinear form over shifted convolution sums.

Typical problem. Choose amplification weights, separate diagonal/off-diagonal terms, apply Voronoi or Poisson summation, and optimize parameters to beat convexity.

Application note. Reinforces contest strategies where a clever weighting or substitution suppresses the hardest terms—mirroring how one tilts sums or introduces auxiliary variables on AIME/USAMO sums and recurrences.

Microlocal Hodge Filtrations

Nearby Cycle Portraits

Illustrative example. Compute the perverse filtration on the cohomology of a surface fibration by analyzing vanishing cycles and their microlocal support on the cotangent bundle.

Typical problem. Resolve singularities, describe the characteristic cycle, and match the resulting pieces with the Hodge filtration to pinpoint which weights survive degeneration.

Application note. Models olympiad scenarios where understanding how a configuration degenerates (points collide, lines become parallel) reveals which invariants persist and guides the final counting argument.

Polylogarithmic Deformation Stacks

Bridgeland Wall Blueprints

Illustrative example. Track how periods of a Calabi–Yau threefold vary across the complex moduli space and express the resulting normal functions as polylogarithmic solutions to Picard–Fuchs equations.

Typical problem. Construct the deformation stack, determine its variation of mixed Hodge structure, and solve the associated differential system to identify wall-crossing behavior.

Application note. Mirrors contest techniques where monitoring how a parameterized family changes (sweeping a point, varying an angle) exposes the critical transition that simplifies the final computation.

Adaptive Container Schemes

Entropy-Split Extremals

Illustrative example. Classify triangle-free subgraphs of a sparse pseudorandom host by iteratively applying container lemmas with entropy-sensitive parameters.

Typical problem. Choose an initial scoring function, construct containers with bounded edge density, and iterate with adaptive thresholds to refine extremal counts.

Application note. Extends olympiad combinatorics where you progressively prune a search space by tracking invariants (parity, degree bounds) to isolate the feasible configurations.

Randomized Hodge–Laplacian Dynamics

Spectral Flow Calibration

Illustrative example. Study the long-term behavior of heat flow on a random simplicial complex by decomposing cochains via the Hodge Laplacian and tracking how stochastic perturbations move mass between harmonic, exact, and coexact pieces.

Typical problem. Compute spectral gaps, couple the dynamics with concentration inequalities, and estimate hitting times for cohomology classes under random updates.

Application note. Resonates with contest tactics that analyze eigenvalues or invariants of a transition process (Markov chains, chip-firing) to conclude convergence, parity, or invariance properties in combinatorial games.

Rapoport–Zink Tower Atlases

Local Shimura Flowcharts

Illustrative example. Describe deformations of a basic p-divisible group with additional structure by charting the associated Rapoport–Zink space and explaining how its period morphism lands in a flag variety.

Typical problem. Match isogeny classes with strata, compute the Newton and Ekedahl–Oort polygons, and verify how Hecke correspondences travel up the tower.

Application note. Mirrors contest workflows where you stratify a hard configuration, study each layer’s constraints, then stitch them into a global classification.

Multiplicative Hitchin Stacks

Endoscopic Gluing Schemes

Illustrative example. Analyze the multiplicative Hitchin fibration for GL_n, determine the cameral cover, and outline how endoscopic data control the fiber components.

Typical problem. Compute characteristic polynomials of monodromy, construct spectral data in the Picard stack, and match it with trace identities in the relative character variety.

Application note. Reinforces competition habits of reducing a global question to spectral pieces before recombining them—just like splitting polynomials into factors or decomposing graphs into color classes.

Noncommutative Hodge Correspondence

NC Hodge–Deligne Ladders

Illustrative example. For a smooth proper dg-category, compute periodic cyclic homology and compare its filtration with the Betti/de Rham comparison map predicted by noncommutative Hodge theory.

Typical problem. Present the category via a quiver with potential, evaluate Hochschild complexes, and identify the pieces corresponding to weights in the Hodge diamond.

Application note. Encourages olympiad thinkers to keep parallel invariants in sync (degree sequences vs. weight sums), ensuring transformations preserve the quantities that solve the problem.

Motivic Character Sheaves

Categorical Trace Amplifiers

Illustrative example. Build a character sheaf on a reductive group, compute its trace function over finite fields, and compare it with automorphic coefficients predicted by the categorical Langlands program.

Typical problem. Describe the relevant Hecke correspondences, evaluate the sheaf-function dictionary, and analyze stability under Fourier–Deligne transforms.

Application note. Echoes contest strategies where you translate between combinatorial and analytic descriptions of the same quantity to leverage whichever side yields the cleanest estimate.

Probabilistic Floer Homologies

Random Morse Tunneling

Illustrative example. Introduce randomness into Hamiltonian perturbations, study the resulting Floer chain complexes, and track how stochastic gradients alter continuation maps.

Typical problem. Estimate expected counts of pseudo-holomorphic strips, bound energy via probabilistic compactness, and show convergence to a limiting homology.

Application note. Reminds contest solvers that randomization can smooth rugged landscapes, a tactic equally useful for bounding recurrences or averaging over residues.

Higher-Sparsity Sum–Product Duality

Fractal Expander Cascades

Illustrative example. Show that a sparse subset of a local field cannot be simultaneously sum- and product-stable by constructing multi-scale energy decompositions that force growth.

Typical problem. Blend incidence geometry bounds with entropy increments, track additive/multiplicative doubling constants, and deduce expansion in a prescribed range.

Application note. Strengthens olympiad instincts about enforcing growth: whenever a structure stays too rigid under multiple operations, you search for the contradiction via scaling arguments or invariants.

Iwasawa Eigenvariety Patchworks

Slope-Lattice Navigation

Illustrative example. Track how the slopes of a family of overconvergent eigenforms vary as the weight moves through an affinoid, then detect the jumps by studying the patched Hecke algebra.

Typical problem. Build the patched module, compute its characteristic power series, and identify companion points where the eigencurve intersects boundary components.

Application note. Echoes olympiad tactics of parameter sweeping—you monitor how roots or invariants shift when a variable moves to catch discontinuities that unlock a clean classification.

Holomorphic Anomaly Recursions

Topological String Ladders

Illustrative example. Use the BCOV holomorphic anomaly equation to compute higher-genus Gromov–Witten invariants on a Calabi–Yau threefold from boundary data on the moduli space.

Typical problem. Determine propagators, integrate modular derivative constraints, and match constant-map contributions with enumerative predictions to fix holomorphic ambiguities.

Application note. Reflects contest routines where recursion with precise boundary conditions (e.g., telescoping, induction with base checks) drives a complex counting argument to completion.

Nonlinear Kakeya Renormalization

Multiscale Needle Cascades

Illustrative example. Establish a decoupling inequality for curved Kakeya sets by iteratively rescaling wave packets and bounding overlap through induction-on-scales.

Typical problem. Quantify transversality of direction tubes, set up a Bourgain–Guth style iteration, and combine multilinear restriction with geometric combinatorics to drive the exponent down.

Application note. Reinforces olympiad instincts about dissecting a geometry problem into scale-sensitive cases (small/large angles, local vs. global counts) to reveal the tightest bound.

Derived Tropical Correspondence

Stable Pair Scattering

Illustrative example. Relate curve counts on a toric threefold to weighted tropical curves by building a derived logarithmic enhancement and translating obstruction theories across the correspondence.

Typical problem. Assemble the degeneration formula, compute balancing conditions for tropical types, and compare virtual multiplicities to verify the enumerative match.

Application note. Mirrors contest habits of switching between combinatorial and geometric encodings (lattice points vs. area, graph vs. path) to leverage whichever model simplifies the count.

Stochastic Quantum Walk Expanders

Interference Mixing Labs

Illustrative example. Analyze a coined quantum walk on an expander graph with random phase noise and show how mixing times remain near-logarithmic by bounding interference decay.

Typical problem. Diagonalize the noiseless walk, insert random perturbations, track eigenvalue spread via matrix martingales, and extract hitting-time guarantees.

Application note. Encourages contest solvers to blend deterministic structure with controlled randomness—akin to random walks on graphs or probabilistic invariants in combinatorial games.

Sheafified Discrepancy Potentials

Perverse Balance Flows

Illustrative example. Control the discrepancy of point sets on a surface by encoding incidence relations inside a constructible sheaf and applying perverse filtration bounds to the associated energy functional.

Typical problem. Build the incidence correspondence, pushforward to a parameter space, compute Euler characteristics fiberwise, and extract cancellation from the resulting spectral sequence.

Application note. Resonates with olympiad methods where you rephrase a counting problem as a flow or potential argument to expose the imbalance that forces a sharp estimate.

Relative Fargues–Fontaine Geometries

Prismatic Slope Atlases

Illustrative example. Classify vector bundles on the relative Fargues–Fontaine curve by exhibiting their Harder–Narasimhan polygons and comparing them with prismatic Hodge filtrations along a fixed untilt.

Typical problem. Track how modifications at characteristic points alter slopes, compute the associated G-bundle, and verify compatibility with local shtuka parameters.

Application note. Mirrors contest moves where you monitor how invariants evolve under controlled transformations—just like following a polygon of side lengths as you apply triangle moves.

Polylogarithmic Chabauty–Kim Loci

Selmer Facet Tracking

Illustrative example. Use non-abelian Chabauty to bound rational points on a genus-two curve by computing polylogarithmic iterated integrals and matching them with Selmer conditions.

Typical problem. Assemble the unipotent fundamental group, evaluate Coleman integrals along chosen tangential base points, and isolate the finite set of points satisfying all height constraints.

Application note. Encourages olympiad solvers to mix algebraic and analytic views of the same object—akin to cross-checking a Diophantine answer via modular arithmetic and bounding inequalities.

Microlocal Stokes Sheaf Stacks

Irregular Riemann–Hilbert Bridges

Illustrative example. Describe the Stokes data of a meromorphic connection by constructing its enhanced ind-sheaf, then read off the asymptotic sectors from the microlocal skeleton.

Typical problem. Compute exponential factors near each singular direction, glue the corresponding perverse sheaves across sectors, and verify the compatibility with the wild Riemann–Hilbert correspondence.

Application note. Reinforces the contest habit of breaking a tough functional equation into directional regimes before stitching together a global formula.

Derived Skein Quantization

Quantum Character Seeds

Illustrative example. Categorify the Kauffman skein algebra of a surface by constructing its derived representation stack and tracing how quantum traces act on the Fukaya-type category.

Typical problem. Present the skein relations as a derived intersection, compute deformation complexes, and extract quantum traces that coincide with character varieties at q = 1.

Application note. Echoes olympiad strategies of lifting a combinatorial relation into an algebraic structure to expose conservation laws or invariants that make the counting tractable.

Entropy-Preserving Hypergraph Limits

Flag-Container Harmonies

Illustrative example. Model a convergent sequence of dense hypergraphs via limit objects that record entropy deficits and explain how stability forces near-extremal structure.

Typical problem. Combine hypergraph containers with flag algebras, quantify the entropy contribution of each template, and deduce a Turán-type bound with exponentially many near-extremal models.

Application note. Reminds contest competitors to organize counting arguments by templates and error terms—the same mindset that separates main terms from small perturbations in combinatorial estimates.

Motivic Simpson Flowlines

Nonabelian Period Maps

Illustrative example. Track how a family of Higgs bundles evolves under a nonabelian Hodge flow, identify the resulting Betti moduli point, and express the comparison in motivic periods.

Typical problem. Solve the gradient flow for the Hitchin functional, match the limiting local system with its de Rham incarnation, and analyze the period map’s singularities.

Application note. Encourages olympiad training to keep track of dual viewpoints of the same structure—analogous to toggling between vector and complex forms of a geometry problem to reveal hidden simplifications.