Stanford Kiddie Colloquium

Title: Reciprocity Laws
Abstract: Gauss gave eight proofs of the law of quadratic reciprocity, and many more have been found subsequently. Quadratic reciprocity has also been generalized to laws of cubic reciprocity, Eisenstein reciprocity, and Artin reciprocity. These generalizations have been an important part of the development of algebraic number theory and class field theory. I'll explain what some of these are and how seemingly abstract statements relate to quadratic reciprocity, and how Gauss' proof using Gauss sums relates to a modern proof using cyclotomic fields.
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Title: The Combinatorics of the Beta Sieve
Abstract: We will be mainly concerned with the following problem: how long can a sequence of consecutive integers be before it is forced to contain a number with no prime factors less than some bound z? The Chinese Remainder Theorem and the Principle of Inclusion and Exclusion both provide upper bounds for the length of such a sequence, but both upper bounds grow exponentially in z. Combinatorial sieves are natural generalizations of the principle of inclusion and exclusion that provide better upper bounds on the lengths of such sequences. In this talk we will discuss optimal combinatorial sieves, and give evidence that the beta sieve is optimal.
Food: Jeremy

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Title: Inter-universal Teichmüller theory: Fact or Fiction?
Abstract: Just kidding. I'll actually talk about spanning multiples of curves with discs. Not as catchy of a title, but way more fun, I promise! This will involve minimal surfaces, the isoperimetric inequality, the monotonicity formula, among other things, so good times will be had by all.
Food: Paul

Title: Oriented bordism: calculation and application
Abstract: Classifying manifolds is hard, but it becomes easier if one only wants to classify them up to some coarse equivalence relation. Thom came up with a class of equivalence relations that sits exactly on the sweet spot between impossible to calculate and trivial and I'll talk about one particular instance of this equivalence relation: oriented bordism. In the talk I'll do a subset of the following things: (0) define oriented bordism, (i) by geometric tricks classify manifolds up to oriented bordism of dimensions 0 to 3 (dimensions 0 to 2 shouldn't surprise you, 3 maybe should), (ii) sketch how to do the complete classification, (iii) prove as an application the beautiful Hirzebruch signature theorem (involving complex analysis and maybe even some number theory), (iv) ramble about cool relations with other fields.
Food: Evita

Title: Ultraproducts and what to do with them
Abstract: Ultraproducts are a model-theoretic construction with surprisingly many applications to non-foundational mathematics, besides having a kick-ass name. I'll discuss how to construct them, a few reasons why one might care, and then use an ultraproduct to help connect ergodic theory and additive combinatorics. I will not assume any familiarity with ultrafilters or ultraproducts.
Food: Jeremy

Title: A quantum error correcting code
Abstract: I'll introduce the notion of a quantum error correcting code, and present an interesting example, Kitaev's toric code. This code uses systems arranged in the shape of a torus, and is constructed so that localised errors are protected against, with errors needing to wrap around the torus to corrupt the encoded information. I will then discuss the efficiency of codes of this form, leading to a nice graph theory problem. For sufficiently credulous viewers, no prior knowledge of quantum mechanics is required.

Title: Representations of the Symmetric Group and Chocolate Tableaux
Abstract: Knowing the irreducible representations of a group gives a lot of information about the group itself! In this talk, I will construct the irreducible representations of the symmetric group using Young tableaux. I will start talking about Young Tableaux, but I will mainly focus on the construction of Specht modules and the fact that they form a complete list of irreducible complex representations of Sn.

Title: The Müntz-Szász Theorem: Digging holes in the Weierstrass Approximation Theorem
Abstract: The classical Weierstrass Theorem says that every continuous function on a compact interval can be uniformly approximated by linear combinations of monomials {1, x, x^2, ...}. But what if we remove some of the monomials? Does the collection of those with prime powers have dense span? What if the powers are perfect squares? In this talk, we'll use complex analysis to prove an elegant necessary and sufficient condition for denseness.

Title: Bo(Z/2)(Z/2) Peri0diZ000Zy
Abstract: Bott periodicity is a striking pattern in mathematics in which various important infinite sequences unexpectedly repeat themselves with period eight. First discovered by Raoul Bott in the fifties in the context of the homotopy groups of the classical Lie groups, periodicity is now known to play a fundamental role in many seemingly disparate areas of mathematics, including homotopy theory, K theory, elliptic operator theory, non-commutative differential geometry, and theoretical physics. In this talk, we'll first introduce Bott periodicity in the context of Clifford algebras and their representations, an arena where periodicity arises in a particularly concrete way. Clifford algebras can be viewed as a natural continuation of the sequence: real numbers, complex numbers, quaternions. After doing some examples and classifying Clifford algebras, we'll give an informal tour (without proofs) of some of the many guises of the periodicity theorem. I'll assume no background beyond some basic algebra and topology.