Mathematics and Statistics

An introduction to point set topology, a foundational topic for much of modern mathematics. We will cover topological spaces, separation axioms, quotient spaces, compactness, connectedness, path connectedness, and homotopy. In the last part of the course we will cover the fundamental group, the most basic algebraic topological invariant.

An introduction to set-theoretic probability with applications to mathematical statistics. Topics include probability spaces, discrete and continuous random variables, single and multivariate distributions, and limit theorems, leading into the mathematical theory of estimators, sampling distributions, confidence intervals, and hypothesis testing.

This course covers statistical methods in machine learning such as decision trees, random forests and support vector machines The course will use a project-based approach to give students hands-on experience using these techniques by analyzing large and complex real-world datasets. More importantly, they will learn the statistical principles behind these procedures, such as loss functions, maximum likelihood estimation and bias-variance trade-off as well as why these principles matter in real world settings.

Number theory is the study of the properties of the positive integers. Topics include divisibility, congruences, quadratic reciprocity, numerical functions, Diophantine equations, continued fractions, distribution of primes. Applications will include cryptography, the practice of encrypting and decrypting messages, and cryptanalysis, the practice of developing secure encryption and decryption protocols and probing them for possible flaws. Speaking Intensive.

Our world is built of networks: the internet, social networks, transportation networks, communication networks, biological networks. Natural and useful question include "What makes a network robust?" "Can we predict where failures might occur?" "What can we do to slow propagation of viruses along a network?" This courses will cover abstract mathematical properties of networks that can help us answer these questions. These will be examined in the context of both theoretical and real world networks. Further, student groups will analyze and report on a real world network of their choice.