We will first introduce Summer UMS and then decide on the logistics for the next weeks. We will go over some potential textbooks we could cover and then pick one by vote. Every member will then have the opportunity to sign up to give a talk.

This summer, we will be studying graph theory and combinatorics: the study of discrete structures and counting. We will begin with an overview of graph theory, combinatorics, and the infinite analogues of these. We will then begin the study of graphs with basic definitions and examples, including the notions of a path, degree of a vertex, connectivity, complex and bipartite graphs, and more. Throughout the talk, I will also mention interesting applications to mathematics such as graph-theoretic proofs of the Cantor-Schroder-Bernstein theorem and Fermat's Little Theorem.

In this talk, we will begin by recalling the introductory notions of graph theory, briefly discussing connected graphs, graph isomorphisms, and metrics (as related to graph theory). We will then continue by defining the relevant terminology in the study of distances in graphs, invoking relevant intuition along the way. We will then delve into a relatively detailed discussion of adjacency matrices, and conclude with a recent (and interesting!) application of graph models using real-world data.

In tomorrow’s talk, we will begin by recalling the notions of complete graphs and subgraphs and introducing cliques and edge coloring. We will then use these notions to solve two problems, one of which once appeared in the Putnam Competition and the other is an algebra problem. After that, I will introduce Ramsey numbers and Ramsey’s Theorem. They will then help us prove Schur’s Theorem.

The notation of formalized counting and binomial properties can be confusing and jumping directly into identities and theorems can be overwhelming if they are imagined with only a loose connection to their real-world inspiration. We will go through each foundational concept slowly, recreating them through intuitive analogies. This will let us see the beauty of these structures unfold! We will go over basic binomial theorem principles, how they extend to multiple dimensions, and some interesting problems with generating functions.

The pigeonhole principle is a powerful tool that is found in all kinds of combinatorial problems. Using the principle we can easily construct contradiction proofs for almost any problem that requires some kind of counting. Moreover, the Pigeonhole principle is so useful that it has applications in combinatorial geometry, number theory, algebra, and even statistics. For example, thanks to it we know that in New York City, there are at least 2 people with the same number of hairs! We’ll go over Erdos-Szerekes’s Upper bound for Ramsey numbers, Kronecker Dirichet’s Approximation theorems, and some interesting Olympiad problems from the Moscow Math Olympiad, Ireland Math Olympiad, USAMO, and IMO.

How many ways are there to change a dollar into pennies, nickels, quarters, and dimes? How quickly can we figure it out? These questions and their immediate generalizations take us on a fascinating tour of combinatorics. In this talk, we analyze the change-making problem using generating functions, and we use our method to motivate the theory of counting partitions (where a partition is simply an arrangement of n objects into k distinct piles). In developing this theory, we arrive at powerful results such as Euler's Pentagonal Number Theorem and Hardy and Ramanujan's asymptotic formula for the partition function.

 

The Rado graph is the only graph on countably many vertices. There are alternative constructions of this graph that can all be shown to be equal. Furthermore, it has many remarkable properties such as universality, homogeneity, and robustness all of which are made possible by the extension property.