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What it Means for a Machine to Learn, with Jihong Cai [notes] #
Abstract
This is part 5 in a series for MATRIX on modern theoretical mathematics. This talk is a summary of many of the topics covered in the MATH 490: Mathematics of Machine Learning course at UIUC taught by Prof. Kirkpatrick. We cover PAC-learnability, the no-free-lunch theorem, VC-dimension, and apply VC-dimension to threshold functions, halfspaces, and decision trees.All the Ways Linear Algebra Fails in Infinite Dimensions, with Jihong Cai [notes] #
Abstract
This is part 4 in a series for MATRIX on modern theoretical mathematics. The overarching purpose of this talk is to introduce basic functional analysis. To do so, we cover the concepts of undergraduate linear algebra and demonstrate that every concept fails in some way when applied to vector spaces with infinite dimensions. Functional analysis is the toolkit to handle vector spaces of infinite dimension, and we motivate this as well with an example of an infinite-dimensional vector space, then cover some of the tools functional analysis uses.Unique Factorization in Domains, with Jihong Cai [notes] #
Abstract
This is part 3 in a series for MATRIX on modern theoretical mathematics. We cover the idea of unique factorization, what domains are, and what the question of unique factorization means both in the integers and outside the integers.Twists, Turns, And Topology: A Journey From Bridges To Algebraic Wonders, with Jihong Cai [notes] #
Abstract
This is part 2 in a series for MATRIX on modern theoretical mathematics. We cover the development of modern algebraic topology, from Euler to Poincaré and beyond, and modern applications of algebraic topology. Along the way, we introduce homeomorphisms, fundamental groups, and homotopy and homology groups. Finally, we cover the math behind topological data analysis and exhibit its application in neuroscience.Straightedge and Compass Constructions, with Jihong Cai [notes] #
Abstract
This is part 1 in a series for MATRIX on modern theoretical mathematics. We start with straightedge and compass constructions and exhibit examples of constructible numbers and objects. We then discuss the constructability problem and introduce the tools of field extensions and the tower law to explain its resolution with the formalization of the constructible numbers. We end with some remarks on Fermat primes and impossible constructions. While the notes are rigorous and technical, the talk is aimed at all undergraduate levels. The notes are a guide and a reference for those who wish a rigorous explanation of the subjects.