Stanford Kiddie Colloquium

Title: Independence in Number Theory
Abstract: I'll cover three examples in which a notion of independence enters number theory.
The first one is related to understanding the function "number of prime factors of n", and here the independent event is "divisibility by a prime".
The second, is in the theory of the Riemann zeta function. Here independence arises through the oscillation of the harmonics p^{-it} as p ranges through the primes and t in through an interval.
In both cases the understanding that some number theoretic objects behave "independently" allows us to prove some pretty cool theorems, the proofs of which I will sketch.
If there's time, I'll discuss my last example, which is the conjectural relation between random matrix theory and the zeta function. There is also some independence phenomena happening there, but it is not as well understood and hence no nice theorems.

Title: Expressing polynomials as determinants.
Abstract: There are two very closely related problems: expressing a convex semi-algebraic set in R^n by the semi-definiteness of a linear pencil, and writing a polynomial as the determinant of a (possibly large, ideally semi-definite) matrix whose entries are linear functions. In this talk I present how these are related, why it has been of great interest in Convex Optimization and in Control Theory, and the state of the art. In particular, there is an elementary (albeit tricky) construction that works when the boundary is sufficiently nice, and an algebrogeometric construction that works for bivariate polynomials. Pure mathematicians need not be scared, since both approaches are highly impractical.

Title: A Toy Example of Modularity
Abstract: I'll be talking about the circle of ideas that connects modular forms, elliptic curves, and Galois representations. Perhaps the most famous of these connections is the modularity theorem, that associated a modular form to every elliptic curve. To illustrate why these connections are useful, I will be presenting three different explanations for a combinatorial congruence on the power series of an infinite product. A combinatorial problem has a pedestrian explanation in terms of modular forms, a conceptual explanation in terms of the geometry of an elliptic curve, and a third equivalent explanation in terms of a representation of the absolute Galois group of Q. The talk will be example based, so little previous knowledge is necessary.

Title: The Sternberg linearization theorem: dangerous - lucrative - microlocal analysis
Abstract: Given a vector field on Euclidean space which vanishes at the origin, when can you choose a local diffeomorphism for which the pushforward of the vector field agrees with a linear vector field in a neighborhood of the origin? In this talk I will give a sufficient condition for this to be answered in the affirmative (the Sternberg linearization theorem), and provide parts of a proof, due to Edward Nelson. I will also advertise microlocal analysis as much as possible and give some idea of how the ideas of the Sternberg linearization theorem relates to my recent research.

Title:
Abstract:

Title: My Favorite Undergraduate Math Theorem a.k.a. An Introduction to Hodge Theory for the Analytically Inclined
Abstract: I will start with a review of the classical Hodge Decomposition Theorem for differential forms on a Riemannian manifold. Then we will re-interpret and strengthen the result for a concrete family of 3-manifolds, to make explicit a truly beautiful connection between the topology, geometry and analysis of a manifold. Algebraic geometers attend this talk at their own risk.

Title: Zeros in Integer Linear Progressions
Abstract: I'll discuss the question of zeros in sequences of integers defined by linear recurrences, and use these questions to motivate the uses of p-adic analysis. Time permitting, I'll prove a beautiful theorem of Mahler, Skolem, and Lech describing the possible zero sets of integer linear recurrences.

Title: Global Arithmetic Dynamics
Abstract: Over the past century or so, number theorists have discovered a lot of cool facts about rational and integral points on elliptic curves. We will discuss some of these results and see how to interpret them as statements about certain special dynamical systems. We then generalize them into (usually) very difficult conjectures about more general classes of dynamical systems.

Title: Evasion paths in mobile sensor networks
Abstract: Imagine that disk-shaped sensors wander in a planar domain. A sensor can't measure its location but does know when it overlaps a nearby sensor. We say that an evasion path exists in this sensor network if a moving evader can avoid detection. A theorem of Vin de Silva and Robert Ghrist gives a necessary condition, depending only on the time-varying connectivity graph of the sensor network, for an evasion path to exist. Can we sharpen this theorem? We'll consider examples that show the existence of an evasion path depends not only on the network's connectivity data but also on its embedding.

Title: The Ring of motives
Abstract: Many invariants of spaces (polyhedra, manifolds, varieties) are invariant under certain operations of cutting and pasting; for example, volume and Euler characteristic. We'll discuss a formalization of this notion, the ring of motives, in various cases. We'll compute the structure of this ring in some cases (for example, manifolds with a group action), and use this computation to reduce some topological calculations to combinatorics.

Title: The Transcendence of e
Abstract: Everyone knows that e and pi are transcendental. There are even relatively short proofs of these facts. They seem quite unnatural. I will explain how a sane mathematician would come up with them, and the relation of these techniques to the continued fraction expansion of e.

Title: On the Structure of Groups Which Act Freely on Spheres
Abstract: Finite groups can be studied as groups of symmetries in different contexts. For example, they can be considered as groups of permutations or as groups of matrices. In topology we like to think of groups as transformations of interesting topological spaces, which is natural extension of classical problem of describing symmetries of geometric spaces. First interesting case would be spherical space form problem which is to classify manifolds with sphere (up to homotopy or homeomorphism) as universal cover. The problem was first stated by Hopf in 1925. Smith could prove that if a finite group acts freely on a homotopy sphere then all sylow subgroups should be cyclic. Cartan and Eilenberg made an easy observation that every such groups should have periodic cohomology. Milnor in 1957 showed not every periodic group can act freely on any sphere and then development began to accelerate until Madsen-Thomas-Wall completely solved spherical space form problem in 70's. I will leisurely talk about Smith theory and Milnor's idea.

Title: Optimal Transport and a Synthetic Notion of Ricci Curvature
Abstract: I'll discuss the basics of optimal transport and the associated Wasserstein metrics. I'll also discuss how this gives a synthetic notion of lower Ricci bounds on metric measure spaces, how it is preserved under Gromov-Hausdorff convergence. Time permitting, I'll discuss some further results, potentially including but not limited to the relationship with Ricci flow, how to prove the isoperimetric inequality with optimal transport, how to view certain PDE's as gradient flows in Wasserstein space, as well as functional inequalities.

Title: Gromov's Knot Distortion
Abstract: Gromov defined the distortion of an embedding of S^1 into R^3 and asked whether every knot could be embedded with distortion less than 100. There are (many) wild embeddings of S^1 into R^3 with finite distortion, and this is one reason why bounding the distortion of a given knot class is hard. I will discuss recent work which shows that there exist knots which require arbitrarily large distortion. For example, torus knots require large distortion (work of the speaker), as do the (knotted, connected) ramification sets of ramified covers M --> S^3 where M is an arithmetic hyperbolic 3-manifold (work of Gromov and Guth). I will also mention some natural conjectures about the distortion, for example that the distortion of the (2,p)-torus knots is unbounded.

Title: Entire Functions from C to shining C
Abstract: In 1926 Rolf Nevanlinna proved that any function holomorphic on all of C is uniquely identified by the pre-images (ignoring multiplicity) of any four complex numbers. This is but one of many striking consequences of the very rich theory of value distributions, that began with Picard's theorem, was greatly pushed forward and generalized by Nevanlinna's fundamental theorems, reinterpreted by Ahlfors in terms of covering spaces, and further developed by many others.

Title: No Strings Attached
Abstract: Though a relatively young subject, I will give an overview of the history of string topology. String topology is concerned with studying the topology of the free loop space and in particular algebraic structures on its homology coming from splitting and merging of loops.

Title: The Prime Generating Sequence That Couldn't
Abstract: Define a sequence of integers by
a_0 = 3,
a_1 = 0,
a_2 = 2, and then recursively by
a_{n+3} = a_n + a_{n+1}.
Calculate out a few terms, or a few thousand, and you'll notice a curious pattern: the n-th term a_n is divisible by n exactly when n is prime!
The first counter-example is n = 271441, for which a_n has over thirty thousand digits. I'll explain why this coincidence holds so often, why it had to fail sometime, and why it takes so long to find a counterexample. The proofs will mostly just involve graphs, paths, polynomials, and matrices, with some finite-state automata for fun.