Algebra

A graduate-level exploration of expected value applications in finance, covering utility theory, portfolio optimization, risk-neutral pricing, and tail risk metrics. Students transition from theoretical foundations to computational implementation using Monte Carlo methods.

A graduate-level sequence on constrained optimization, covering Lagrange Multipliers, KKT conditions, and sensitivity analysis for economics and engineering applications.

A comprehensive graduate-level exploration of numerical optimization algorithms, moving from first-order gradient descent to second-order Newton methods and computationally efficient Quasi-Newton approaches. Students analyze convergence rates, stability, and strategies for navigating complex, non-convex landscapes.

This sequence establishes the rigorous mathematical underpinnings necessary for advanced optimization work, moving beyond procedural calculus to analysis-based proofs. Students explore the intersection of topology, set theory, and multivariate calculus to determine the existence and uniqueness of optimal solutions.

A graduate-level exploration of dynamical systems, focusing on the qualitative analysis of stability, phase portraits, and topological changes in nonlinear differential equations. Students move from linear classification to advanced stability proofs using Lyapunov functions and bifurcation theory.

A graduate-level sequence exploring systems of equations through the lens of multivariate modeling, economic interdependence, and linear optimization. Students progress from network flow conservation to the conceptual foundations of the Simplex method.

This advanced sequence explores systems of linear equations through the lens of computational linear algebra. Graduate students will move from theoretical existence and uniqueness in vector spaces to matrix factorizations, algorithmic complexity, numerical stability, and iterative methods for large-scale systems.

A comprehensive calculus sequence for undergraduate students focused on the rigorous application of derivatives to industrial, geometric, and economic optimization problems. Students progress from basic modeling to multi-constraint capstone analysis.

This graduate-level sequence focuses on the rigorous deconstruction of arguments through formal logic. Students move beyond informal fallacies to analyze structural validity, propositional logic, syllogistic forms, and modal scope, culminating in symbolic proofs of their own research arguments.

A graduate-level sequence exploring the gradient vector as the foundational tool for modern optimization. Students move from the geometric interpretation of multivariate derivatives to the implementation of stochastic algorithms used in machine learning.

A comprehensive introduction to predicate logic for undergraduate students, moving from basic atomic structures to complex relational propositions with multiple quantifiers. Students learn to map the internal architecture of language to rigorous logical symbols.

This graduate-level sequence bridges univariate statistics and multivariate geometry, exploring how variability manifests in high-dimensional spaces through covariance matrices, generalized variance, and principal component analysis.

This sequence for undergraduate students focuses on complex algebraic structures involving exponentials, including quadratic forms, distinct bases, inequalities, systems, and transcendental limitations. It prepares students for higher-level calculus and engineering mathematics through rigorous analytical techniques.

A graduate-level exploration of the mathematical foundations of discrete-time Markov chains, focusing on state classification, limiting behavior, and time reversibility. This sequence emphasizes formal derivation, proofs, and the application of linear algebra to stochastic systems.

An advanced graduate-level sequence exploring the mathematical foundations of model selection, including bias-variance decomposition, information criteria (AIC/BIC), resampling methods, and high-dimensional diagnostic strategies.