An Introduction to Spectral Graph Theory
Christopher Storm
Spectral Graph Theory is a relatively new field of mathematics which seeks to use ideas from continuous geometry to learn more about graphs. In this talk, I will introduce graphs and the adjacency operator on graphs. I will briefly outline some of the applications of graphs and some classical problems to give an idea of this rich field.
Then we will turn to the spectrum of the adjacency operator. Surprisingly, these eigenvalues have great impact on very tangible properties of the graph. I will show how they provide a bound on the diameter.
Time permitting, I will motivate the definition of Ramanujan Graphs, probably the most well-known class of graphs. The talk will be pitched with a heavy linear algebra component so that anyone who has had linear algebra should be able to follow the vast majority.
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Lee's Antepenultimate Talk at Dartmouth
Lee Stemkoski
This will be the traditional anecdotal advisorial fifth-year talk.
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To Be Or Knot To Be
Allison Henrich
Knot theory is a relatively young and decidedly exciting field of mathematics. It is one of the few areas of advanced math that can be made accessible to non-experts, yet very few mathematicians have had occasion to study knots. By way of introduction to this beautiful subject, I will introduce knots and discuss two of the most celebrated knot invariants--the Jones polynomial and Kauffman bracket.
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Why We Love the Isomorphism Theorems
Alison Setyadi
Have you ever wondered why we care about the Second Isomorphism Theorem (besides wanting to pass the algebra qual)? We give two applications. The First Isomorphism Theorem and the Correspondence Theorem also come into play.
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The Great Theorems of Mathematics: 100 Theorems in 50 Minutes
Dominic Klyve and Lee Stemkoski
Abstract: See title.
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An Introduction to Fast Fourier Transforms
Marty Malandro
I'll give a brief recap of the classical theory of Fourier transforms, and then I'll show how these ideas are useful in modern computer-based signal analysis. Along the way, we will encounter the Discrete Fourier Transform and see a fast method for computing it. If time permits, I'll show how these ideas generalize to Fourier transforms on other objects such as groups. This talk should be accessible to everyone.
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Some Results on Primality Testing
Jonathan Bayless
In 1975, V. R. Pratt showed that PRIMES is in NP by showing that M(p), the number of multiplications necessary to certify the primality of p, is bounded above by a constant times (log p)2. We discuss recent work of mine on a lower bound for M(p).
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Cohen Forcing and the Independence of the Continuum Hypothesis
Brooke Andersen
In 1963 Paul Cohen introduced forcing to show that the Continuum Hypothesis and the Axiom of Choice are independent of the axioms of set theory (ZF). I will introduce the forcing machinery, give examples of how it is used, and sketch Cohen's proof that there is a model in which the Continuum Hypothesis fails.
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Introduction to Coordinate Percolation
Elizabeth Moseman
If each vertex in the square lattice is declared to be open or closed with a given probability, will the graph induced by the open vertices have infinite components? This is the problem of percolation, and I will talk about what is known about this model. Another model of percolation, called coordinate percolation, introduces some dependency in which vertices are open. This second model has many of the same questions asked about it, but the answers are more detailed. These answers will be discussed, leading to a surprising connection to worms.
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