Number Theory

Properties of Elliptic Curves

• Investigating the group structure of rational points on specific elliptic curves \(E: y^2 = x^3 + ax + b\)
• Exploring the Nagell-Lutz theorem and computing torsion subgroups for families of curves
• Studying elliptic curves over finite fields \(E(\mathbb{F}_p)\) and their application in cryptography (ECC basics)
• Analyzing the distribution of points modulo \(p\) for a fixed elliptic curve (related to Sato-Tate conjecture)

Diophantine Equations and Approximation

• Solving Pell's equation \(x^2 - Dy^2 = 1\) for various \(D\) and exploring the connection to continued fractions of \(\sqrt{D}\)
• Investigating specific cases or generalizations of Waring's problem (e.g., sums of squares or cubes)
• Exploring properties of continued fractions for specific irrational numbers (e.g., quadratic irrationals, \(e\), \(\pi\))
• Studying Thue equations or other specific families of Diophantine equations

Distribution of Primes

• Exploring elementary proofs related to prime numbers (e.g., Chebyshev bounds, Bertrand's Postulate)
• Investigating patterns in prime gaps or twin primes computationally and heuristically
• Studying the distribution of primes in arithmetic progressions (Dirichlet's theorem) for specific moduli
• Analyzing properties of the Riemann zeta function \(\zeta(s)\) and its connection to prime numbers

Algebraic Number Theory

• Computing ring of integers, discriminants, and ideal factorization in specific quadratic or cyclotomic number fields
• Investigating unique factorization and the structure of ideal class groups for simple number fields \(\mathbb{Q}(\sqrt{d})\)
• Exploring Dirichlet's Unit Theorem by finding fundamental units in real quadratic fields

Computational Number Theory

• Implementing and comparing different primality testing algorithms (e.g., Fermat, Miller-Rabin)
• Implementing and analyzing integer factorization algorithms (e.g., Pollard's p−1, Pollard's rho, quadratic sieve basics)
• Exploring arithmetic dynamics, such as the Collatz conjecture or properties of aliquot sequences, through computation

Stochastic Processes

Markov Chains

• Analyzing convergence rates (mixing times) for specific Markov chains (e.g., card shuffling, random walks on finite graphs)
• Calculating hitting times, recurrence/transience properties, and stationary distributions for specific chains (e.g., Gambler's Ruin)
• Modeling simple real-world phenomena using Markov chains (e.g., basic queueing models, simple epidemic models like SIS/SIR on small populations)
• Exploring connections between random walks and electrical network theory

Random Walks

• Investigating properties (recurrence/transience) of simple symmetric random walks on different lattices (\(\mathbb{Z}\), \(\mathbb{Z}^2\), \(\mathbb{Z}^3\))
• Studying properties of loop-erased random walks or self-avoiding walks (often computational)
• Analyzing biased random walks and their limit theorems

Brownian Motion and Stochastic Calculus

• Simulating Brownian motion paths \(B_t\) and verifying properties like quadratic variation and non-differentiability
• Exploring the construction and properties of the Itô integral \(\int_0^T f(t)dB_t\) for simple integrands
• Studying solutions to basic Stochastic Differential Equations (SDEs) like the Ornstein-Uhlenbeck process \(dX_t = -\alpha X_t dt + \sigma dB_t\) or Geometric Brownian Motion \(dS_t = \mu S_t dt + \sigma S_t dB_t\) and their applications
• Investigating first passage times for Brownian motion

Point Processes

• Analyzing properties of the Poisson process \(N(t)\) (e.g., inter-arrival times, superposition, thinning)
• Modeling real-world events (e.g., arrival times, radioactive decay) using Poisson processes
• Exploring compound Poisson processes or simple spatial point processes

Other Topics

• Simulating and analyzing percolation thresholds \(p_c\) on different lattices (e.g., square, triangular)
• Investigating basic properties of random graph models (e.g., Erdős–Rényi model \(G(n,p)\))
• Exploring simple branching processes (e.g., Galton-Watson process) and conditions for extinction/survival

Functional Analysis

Operator Theory

• Studying the spectral theory of compact self-adjoint operators on Hilbert space, with examples (e.g., integral operators)
• Investigating properties of specific operators like shift operators, Volterra operator, or projection operators
• Exploring different operator topologies (norm, strong, weak) and their properties
• Characterizing Fredholm operators and index theory in simple cases

Banach Spaces

• Comparing geometric properties (e.g., uniform convexity, reflexivity) of different Banach spaces (\(L^p\), \(\ell^p\), \(C(K)\))
• Exploring applications of the fundamental theorems (Hahn-Banach, Uniform Boundedness, Open Mapping, Closed Graph) in specific contexts
• Investigating the structure of dual spaces for common Banach spaces
• Studying Schauder bases in specific separable Banach spaces

Applications to PDEs and Integral Equations

• Understanding the concept of weak solutions for simple elliptic PDEs (e.g., Poisson equation) using Sobolev space basics (\(H^1\))
• Applying the Lax-Milgram theorem to prove existence and uniqueness for certain PDEs
• Using functional analytic methods (e.g., spectral theory of compact operators) to solve Fredholm integral equations

Harmonic Analysis

• Investigating convergence properties of Fourier series (e.g., pointwise, uniform, \(L^2\))
• Studying properties of the Fourier transform on different function spaces (\(L^1\), \(L^2\), Schwartz space)
• Exploring basic wavelet theory, such as the Haar wavelet system and its properties/applications
• Studying Fourier analysis on finite abelian groups

C*-algebras and Operator Algebras

• Exploring the Gelfand representation for commutative C*-algebras
• Studying properties of specific C*-algebras, such as continuous functions \(C(X)\) or matrix algebras \(M_n(\mathbb{C})\)
• Investigating positive elements and states in C*-algebras