SPUR/RSI Lecture Series

Aaron Pixton Lecturing 2018 — Aaron Pixton's talk in the 2018 SPUR/RSI Lecture Series

This lecture series is organized by SPUR/RSI faculty advisor David Jerison. These summer lectures are aimed at our SPUR and RSI students.

2023 Lectures

July 6	Lisa Sauermann	The Erdös-Ginzburg-Ziv Problem	The Erdös-Ginzburg-Ziv Problem is a classical extremal problem in discrete geometry. For given numbers m and n, it asks the following question: What is the minimum number s such that among any s points in the n-dimensional integer lattice Z^n there are m points whose centroid is also a lattice point? Surprisingly, a wide range of different algebraic techniques can be used to approach this problem for different values for m and n. This talk will give an overview over these techniques and the known bounds for this problem.
July 12	Tanya Khovanova	Math in Puzzle Hunts	What is a puzzle hunt? In this talk, I will answer this question and go over three very mathematical puzzle hunt puzzles.
July 19	Bill Minicozzi	Minimal surfaces	Minimal surfaces are mathematical models for soap films: surface tension pulls the soap film tight, minimizing its area. I will give an overview of the subject, assuming very little background knowledge, and discuss some connections with other fields and some interesting open questions.
July 26	John Urschel	From moments to matrices	Moments of a function give concrete information about the shape of the associated graph and have been thoroughly studied in a wide variety of fields. In this talk, we will discuss a number of classic moment problems and their connection to practical problems in numerical linear algebra.

2022 Lectures

July 6	Ankur Moitra	Rethinking Robustness	What does it mean for a learning algorithm to be robust? We could hope to tolerate adversarial corruptions, but this makes many simple learning tasks computationally intractable. We could assume a particular stochastic model for noise, but then our algorithms might be overtuning to this one distribution, which was only meant to be a simplifying assumption. In this talk we will revisit some classic learning problems, like learning half-spaces, in models that blend worst-case and average-case noise. We will give simple twists on old algorithms that allow them to enjoy strong provable robustness guarantees. Finally we explore some intriguing connections between robustness and fairness.
July 13	Lauren Williams (Harvard U)	Combinatorics of hopping particles and positivity in Markov chains	The asymmetric simple exclusion process (ASEP) is a model for translation in protein synthesis and traffic flow; it can be defined as a Markov chain describing particles hopping on a one-dimensional lattice. I will explain how the ASEP is connected to combinatorics and orthogonal polynomials, and will also mention some questions and observations regarding positivity in Markov chains.
July 20	Joern Dunkel	Mathematical problems in biology	Recent advances in high-resolution live-imaging and sequencing techniques are offering unprecedented insights into the dynamics of living systems at the intracellular, multicellular and population level. The resulting high-dimensional data pose interesting mathematical challenges as they need to be compressed and translated into interpretable low-dimensional models that can help reveal relevant biological, biophysical and biomedical mechanisms. In this talk, I will discuss several examples related to recent experiments by our collaborators, ranging from the topological classification of disordered multicellular systems to dynamical systems inference for zebrafish embryos, worm locomotion and fish schools.
July 27	Scott Sheffield	What is a random surface?	This talk will be a somewhat more accessible version of my recent ICM talk about random surfaces (see online video, notes, slides, code, and animations). I will introduce some especially natural random surfaces and explain how they are defined, what they are used for and how one can start to explore the subject with computer experimentation.

2021 Lectures

July 7	Larry Guth	Focusing waves and combinatorics of lines	Solutions of the wave equation model sound waves, light waves, etc. Focusing refers to the amplitude getting very large in a small region of space. Estimating how much waves can focus is a problem in real analysis. Over the last 25 years, mathematicians have approached this problem using ideas from combinatorics about the intersection patterns of lines. The story involves parts of math that sound rather far from PDE, such as some topology and some finite fields. In this talk, we will begin with a gentle introduction to waves and the wave equation, and then describe how ideas from some of these other fields come into play.
July 21	Alexei Borodin	Domino tilings of the Aztec diamond	The talk is a survey of one of the most beautiful solvable probabilistic models and its relations with other fields - interacting particle systems, random interfaces in 3d, and random matrices.
August 4	Lisa Sauermann	On the cap-set problem and the slice rank polynomial method	In 2016, Ellenberg and Gijswijt made a breakthrough on the famous cap-set problem, which asks about the maximum size of a subset of $\mathbb{F}_3^n$ not containing a three-term arithmetic progression. They proved that any such set has size at most $2.756^n$. Their proof was later reformulated by Tao, introducing what is now called the slice rank polynomial method. This talk will explain Tao's proof of the Ellenberg-Gijswijt bound for the cap-set problem, and discuss some related problems.

2020 Lectures

July 1	Peter Hintz	Scattering theory and resonances	What do you hear when you strike a bell or pluck a guitar string? A note which has a certain frequency and fades out at a certain rate. (If you listen carefully, you hear a superposition of such notes.) Scattering theory is the study of "open systems", such as these two, in which energy can escape from bounded regions off to infinity; resonances describe the characteristic frequencies (and decay rates) of oscillation of open systems. I will explain what this means in mathematical terms and present examples ranging from toy models to dynamical black holes and the gravitational waves emitted by them.
July 8	Nike Sun	Sharp transitions in computational problems	I will talk about randomized computational problems, such as the random k-SAT problem, or the maximum independent set problem on random graphs. Many of these problems exhibit "phase transition" phenomena, where the behavior of the system changes drastically under a very small change in the model parameters. I will survey some of the progress on understanding these phenomena, which has brought together ideas from statistical physics, computer science, combinatorics, and probability.
July 15	Bjorn Poonen	The abc conjecture	The abc conjecture is an innocent-looking statement about integer solutions to the equation a+b=c that would have profound implications in number theory. I will explain what it says, why it is interesting, and why there is controversy about whether it has been proved.
July 22	Pavel Etingof	Iterating sine, equivalence classes of variable changes, and finite groups with few conjugacy classes	I will talk about the following three problems: What happens if you repeatedly press the "sine" button on your calculator? What are equivalence classes of variable changes y=f(x) where f(0)=0 and f'(0)=1? How to construct "the most noncommutative" finite groups, i.e. ones with as few conjugacy classes as possible? Are you wondering what these problems are doing in the same talk? Come and find out!

2019 Lectures

June 26	Anette (Peko) Hosoi	Luck and the Law: Quantifying the Role of Chance in Fantasy Sports and Other Activities	Fantasy sports have experienced a surge in popularity in the past decade. One of the consequences of this recent rapid growth is increased scrutiny surrounding the legal aspects of the games, which typically hinge on the relative roles of skill and chance in the outcome of a competition. While there are many ethical and legal arguments that enter into the debate, the answer to the skill versus chance question is grounded in mathematics. In this talk I will analyze data from daily fantasy competitions and propose a new metric to quantify the relative roles of skill and chance in games and other activities. This metric is applied to FanDuel data and to simulated seasons that are generated using Monte Carlo methods; results from real and simulated data are compared to an analytic approximation which estimates the impact of skill in contests in which players participate in a large number of games. We then apply this metric to professional sports, fantasy sports, cyclocross racing, coin flipping, and mutual fund data to determine the relative placement of all of these activities on a skill-luck spectrum.
July 3	Tristan Collins	The Ricci Flow and Applications	I will introduce the Ricci flow on a Riemannian manifold, which is an approach to "simplifying" or "uniformizing" the geometry of a manifold modeled on the classical heat equation. I will discuss some applications, including the Geometrization conjecture (after Perelman), and connections to algebraic geometry through the minimal model program.
July 10	Tanya Khovanova	How to Write and Not to Write a Math Paper	As a head mentor for RSI and PRIMES for several years, I reviewed and commented on more than 200 math papers. I have a huge collection of mistakes and misunderstandings on how to write a math paper. I will share the most important and common mistakes with you. I will also give tips on how to write a good math paper.
July 17	Andrew Lawrie	Soliton Collisions in Geometric Wave Equations	Free waves propagating in a vacuum disperse, spreading out and decaying as time evolves. More complicated dynamical behavior can emerge if there are nonlinear effects, as is the case in many natural systems. For example, classical scalar field theories such as the sine-Gordon equation and the phi-4 model admit soliton solutions. Solitons are coherent solitary waves — their profile is independent of time. In this talk we will discuss the dynamics of solutions that behave in one time direction like a sum of dynamically decoupling solitons. We will then examine the flow of these nonlinear waves in the opposite time direction, where the solitons move towards each other and collide.
July 24	David Jerison	What can we say about the shapes of eigenfunctions and the distribution of eigenvalues?	We will introduce Schrödinger operators and the Schrödinger evolution equation. Then we will describe the Anderson model, introduced by Philip Anderson in the 1950s, in which the Schrödinger operators are certain finite dimensional matrices. Anderson discovered that sometimes the corresponding eigenvectors are highly localized. This turns out to be crucial to the way that semiconductors work and is very important to the design of LEDs. We will discuss famous unsolved problems in pure mathematics and physics associated with Anderson localization and the adjacent problem of predicting the shapes of the eigenvectors and the distribution of eigenvalues.

2018 Lectures

June 28	Gigliola Staffilani	The Many Faces of Dispersive Equations	In this talk I will survey some recent and not so recent cool properties of the nonlinear Schrodinger equation, probably the most important equation in quantum physics. I will talk about dispersion, solitons, scattering, Strichartz estimates, Vinogradov Mean Value theorem, energy transfer and non-squeezing theorems… Some of these topics look really far removed from a physical set up, but if you are curious to learn about the connection come to the talk! Very little knowledge of analysis is assumed from the audience.
July 5	Steven Johnson	So You Think You Know How to Take Derivatives?	The familiar chain rule makes it straightforward to differentiate any composition of differentiable functions. Problems in machine learning, engineering, and other many fields, however, involve very complicated intermediate calculations (e.g. integrating an ODE or solving a huge systems of equations) where the ordinary chain rule is impractical. Instead, modern large-scale optimization uses a "reverse" or "adjoint" method for differentiation that can lead to many surprising results.
July 12	Tanya Khovanova	How to Write and not to Write a Math Paper	As a head mentor for RSI and PRIMES for several years, I reviewed and commented on more than 200 math papers. I have a huge collection of mistakes and misunderstandings on how to write a math paper. I will share the most important and common mistakes with you. I will also give tips on how to write a good math paper.
July 19	Yufei Zhao	Pseudorandom Graphs and the Green-Tao Theorem	The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. I will explain some of the main ideas of the proof from a graph theoretic perspective, with a focus on the role of pseudorandomness in the proof.
July 26	Aaron Pixton	Divergent Series on Lattices	Although the series $1 + 2 + 3 +$ ... diverges, zeta function regularization gives this sum the curious value of $-1/12$. I will discuss two ways of making sense of summing similar divergent series such as ... $+ -2 + -1 + 0 + 1 + 2 +$ ... that are doubly infinite rather than singly infinite. I will then explain how to generalize to sums over lattices of higher rank, as well as how to interpret parts of this as giving a summation formula for certain finite sums.

2017 Lectures

June 28	John Bush	Hydrodynamic Quantum Analogs	A decade ago, Yves Couder and coworkers discovered that droplets walking on a vibrating fluid bath exhibit several features previously thought to be exclusive to the microscopic, quantum realm. These walking droplets propel themselves by virtue of a resonant interaction with their own wavefield, and so represent the first macroscopic realization of a pilot-wave system of the form proposed for microscopic quantum dynamics by Louis de Broglie in the 1920s. New experimental and theoretical results in turn reveal and rationalize the emergence of quantization and quantum-like statistics from this hydrodynamic pilot-wave system in a number of settings.
July 5	Tanya Khovanova	How to Write and Not to Write a Math Paper	As a head mentor for RSI and PRIMES for several years, I reviewed and commented on more than 200 math papers. I have a huge collection of mistakes and misunderstandings on how to write a math paper. I will share the most important and common mistakes with you. I will also give tips on how to write a good math paper.
July 11	Elchanan Mossel	The Math Behind Some Games	Playing games often lead to interesting mathematical models and problems. I will discuss two examples of such problems. While no background is required for the talk, the audience is encouraged to play MineSweeper and Mafia before the talk.
July 12	Semyon Dyatlov	Open Quantum Maps	I present recent progress on the spectral gap question in quantum chaos using the relatively simple model of open quantum baker's maps. These are families of matrices of size going to infinity which quantize the classical open baker's map, one of the standard examples of a strongly chaotic system. The proofs rely on elementary discrete harmonic analysis and algebra.
July 26	Jared Speck	Degenerate Behavior in Solutions to Hyperbolic PDE	Roughly speaking, a hyperbolic PDE is an evolution-type PDE whose solutions exhibit finite speed of propagation. A fundamental issue permeating the study of nonlinear hyperbolic PDEs is that, aside from equations with special structure, initially smooth solutions can break down in finite time. The last decade has borne witness to a flurry of research activity that, for various nonlinear hyperbolic PDEs with ties to geometry and physics, has yielded i) proofs of stable breakdown, ii) sharp information about the nature of the breakdown, and iii) a detailed description of the mechanisms driving it. One type of breakdown is the formation of a singularity, but it is now known that other types of breakdown are also possible. The most significant advancements have occurred in the setting of more than one spatial dimension, where new ideas are needed to complete the proofs compared to the case of one spatial dimension. In this talk, I will survey some of these results, discuss their connections to geometry, and speculate on the future of the field.

2016 Lectures

June 29	Pavel Etingof	Representation of Quivers	A quiver is just an oriented graph. A representation of a quiver is an assignment of a vector space (over some field) to each vertex and of a linear operator between corresponding spaces to each edge. Here is an interesting question: for which (connected) quivers does one have finitely many non-isomorphic representations in each dimension? The answer is a remarkable collection of graphs, called Dynkin diagrams (of ADE type), which also arise in many other areas of mathematics, such as, for example, Lie groups and algebraic geometry.
July 6	Tanya Khovanova	How to Write and Not to Write a Math Paper	As a head mentor for RSI and PRIMES for several years, I reviewed and commented on more than 200 math papers. I have a huge collection of mistakes and misunderstandings on how to write a math paper. I will share the most important and common mistakes with you. I will also give tips on how to write a good math paper.
July 6	Henry Cohn	The Solution of the Cap Set Problem	How large a subset of $\left(Z/3Z \right)^n$ can you find that contains no lines? This is a beautiful and important problem in combinatorics. For $n = 4$, it may be more familiar in the form “In the card game Set, what is the largest collection of cards that does not contain a set?” Until recently (specifically, May 13), this problem seemed hopelessly difficult. The best upper bound known was roughly $3^n/n^{1+ε}$, which is only marginally better than the trivial $3^n$, and achieving that factor of $n^1+ε$ required great ingenuity and effort. In May, however, Ellenberg and Gijswijt proved an upper bound of $2.76^n$, which is an enormous qualitative improvement. Unlike most breakthroughs in mathematics, the Ellenberg-Gijswijt proof is short and elementary. In this talk, we'll see how the entire argument works, using nothing more sophisticated than the rank of a matrix.
July 20	Scott Sheffield	Universal Randomness in 2D	I will introduce several universal and canonical random objects that are (at least in some sense) two dimensional or planar, along with discrete analogs of these objects. In particular, I will introduce probability measures on the space of paths, the space of trees, the space of surfaces, and the space of growth processes. I will argue that these are in some sense the most natural and symmetric probability measures on the corresponding spaces. I will then describe several surprising relationships between these canonical objects. Many of these ideas have been historically motivated by physics —especially string theory, conformal field theory, and statistical mechanics. On the math side, we will also draw inspiration from combinatorics, complex dynamics, stochastic processes, functional analysis, and representation theory.
July 27	Philippe Rigollet	Statistical and Computational Tradeoffs	High dimensional statistics come not only with statistical challenges but also with computational ones. A major breakthrough on this front happened for sparse linear regression with the development of computationally efficient methods that achieve near-optimal performance guarantees. In the context of sparse PCA (principal component analysis), the story is somewhat different: there is a statistical price to pay for computational efficiency. To shed light on this phenomenon, we will (i)Establish minimax rates for sparse PCA (ii) Provide efficient methods for sparse PCA using convex optimization (iii) Prove computational lower bounds using polynomial time reductions.We’ll use tools from linear algebra, probability, convex analysis, graph theory and computational complexity.

2015 Lectures

June 24	Bjorn Poonen	Sums of Squares	I will give a survey of some results about representing integers as sums of squares. For instance, I will prove that if $n$ is a positive integer such that the sphere $x^2 + y^2 + z^2 = n$ has a rational point (a point with rational coordinates), then it has an integer point.
July 1	David Jerison	How Smooth Is a Chemically Polished Surface	In 1986 chemists Meakin and Deutch proposed a model for the process of chemical polishing, in which a corrosive fluid eats away at a metal. At step $n$ in this process, the fluid is represented as a set of $n$ points $A_n$ in the square or cubic lattice in two or three dimensions. The metal is represented by the complementary set, that is, all the rest of the lattice points. The blob of fluid grows one particle at a time: a new particle is introduced (say at the origin) in $A_n$ and performs a random walk. The particle stops the first time it exits $A_n$. One should think of this exit time as the moment that the new corrosive particle meets the metal and eats a bit of it away. By definition, the set $A_{n+l}$ is the set $A_n$ union this new site. Thus, the sequence $A_n$ is random. Nevertheless, with probability 1 it has a deterministic limiting shape. We'll discuss this shape and also the typical fluctuations from it. Understanding the continuum limit of this discrete problem will require not only probability theory, but also some partial differential equations and Fourier analysis. (The Meakin-Deutch process is known as internal diffusion-limited aggregation or internal DLA.)
July 8	Larry Guth	The No Rectangles Problem	We consider the following problem. We have an $N$ by $N$ square grid, like a checkerboard, where $N$ is a large number. We put checkers on this board, and we have to obey the rule that we don't put checkers on all four corners of a rectangle. Subject to this rule, we want to put as many checkers as we can on the board. What is the best strategy? We'll discuss the following questions. How many checkers can a computer put on the board using a naive algorithm? How much do mathematicians know about this problem? We'll start with simple (high-school level) observations and end with difficult open problems.
July 15	Henry Cohn	What Is the Computational Complexity of Linear Algebra?	How quickly can one multiply two large matrices, or solve a system of linear equations? Surprisingly, these natural questions have never been completely answered. In 1969 Volker Strassen showed how to solve these problems for $n$-by-$n$ matrices in roughly $n^{2.81}$ operations, which was an amazing improvement over the obvious $n^3$ algorithm. In this talk I'll survey what's known and discuss how group theory can lead to powerful algorithms.
July 22	Ankur Moitra	Vertex Sparsification	A fundamental problem in graph algorithms is to sparsify a graph — usually by removing edges — but to do so in a way that approximately preserves its connectivity structure. However, removing edges can only get you so far, and many natural graphs are already sparse. We will examine an alternative notion of sparsification which we call vertex sparsification: Is it possible to remove (arbitrarily many) nodes while still approximately preserving the connectivity structure on the remaining nodes? We will show that the answer is yes, and cast the problem as an exponentially large, zero-sum game in order to reveal its connections to problems in metric embeddings.

2014 Lectures

July 2	Scott Sheffield	If Life Gives You Random Trees, Make Random Surfaces	Brownian motion is, in a sense, the canonical random path. What is the canonical random (simply connected) surface? I will discuss some canonical answers to that question. In particular, I will describe some natural ways to produce a two-dimensional random surface as a topological quotient of a pair of random trees.
July 2	David Vogan	Regular Polyhedra in $n$ Dimensions	In the two‐dimensional plane there is a regular polygon with m sides for every $m$ ≥ 3. In three dimensions there are five regular polyhedra. It isn't easy to see a pattern from those two examples, but all the regular polyhedra in n dimensions were described by the nineteenth century Swiss mathematician Ludwig Schläfli. I'll explain how the great Canadian geometer H.S.M. Coxeter made the description of regular polyhedra into an algebra problem, and solved it.
July 16	William Minicozzi	Geodesics and a Heat Equation for Curves	A curve is a geodesic if it minimizes the length between any two of its points or, more generally, if this holds for all short enough segments. For example, straight lines are the only geodesics in Euclidean space while a great circle is a geodesic on the unit sphere since any segment of length at most pi minimizes length. Around 1900, Poincare asked whether every surface, no matter how it is bent, contains at least one closed geodesic. Birkhoff solved this around 1917 with a clever variational argument that introduced a sort of discrete heat equation for curves. After explaining these classical results, I will discuss how higher dimensional versions of Birkhoff's variational argument have appeared in some recent major results. I will also explain the continuous version of this heat equation - known as the curve shortening flow — and its generalization to higher dimensions.
July 23	Ankur Moitra	Disentangling Gaussians	In a remarkable study in 1894, Karl Pearson introduced the problem of learning the parameters of a mixture of Gaussians given random samples from it. This has been a central problem in statistics (and later machine learning) ever since, but the standard estimators fall well short of what we would like because there are no efficient algorithms to compute them! More recently, this problem has been taken up by theoretical computer scientists. Here I will describe the first algorithms with provable guarantees that work even when the components overlap. I will also describe the broader context into which this works fits, and some of the other exciting problems at the intersection of theoretical computer science, machine learning and statistics.

2013 Lectures

July 3	Benjamin Elias	Categorification	What if everything you thought you knew was but a shadow, a flat projection of a deeper, richer truth? Chances are you are trapped in a bad science fiction movie, but just possibly you are a mathematician taking part in the recent phenomenon known as categorification theory. I will talk about what categorification is, using as an illustration some categorifications of a familiar object in math: the natural numbers. In the simplest categorification, one replaces the number "5" with a set containing 5 elements. Suddenly, simple statements like "5=5" lead to interesting structures.
July 10	Peter Shor	Factoring Using a Quantum Computer	Quantum computers are hypothetical machines that operate using the principles of quantum mechanics. I will explain the theory behind how quantum computers work. Quantum computers can be used to factor large numbers into primes much more quickly than the known algorithms on classical computers. I will also explain the quantum factoring algorithm.
July 17	Jacob Fox	Szemerédi’s Regularity Method and Patterns in the Primes	A graph consists of a set $V$ of elements called vertices and a set of $E$ of pairs of vertices, called edges. Szemerédi’s regularity method is one of the most powerful techniques in graph theory, combining Szemerédi’s regularity lemma with a counting lemma. The regularity lemma is a rough structural result for all graphs. It was first used in the proof of Szemerédi’s theorem that dense subsets of the integers contain arbitrarily long arithmetic progression. A few years ago, Ben Green and Terence Tao proved the very old conjecture that the prime numbers contain arbitrarily long arithmetic progressions. The main new ingredient in the proof is a relative Szemerédi theorem, which shows that any dense subset of a pseudorandom set of integers contains long arithmetic progressions. I will discuss how recent joint work with David Conlon and Yufei Zhao extending the regularity method to sparse graphs is used to give a strengthening of the relative Szemerédi theorem and gives a simpler proof of the Green-Tao theorem.
July 24	Larry Guth	The No Rectangles Problem	We consider the following problem. We have an $N$ by $N$ square grid, like a checkerboard, where $N$ is a large number. We put checkers on this board, and we have to obey the rule that we don't put checkers on all four corners of a rectangle. Subject to this rule, we want to put as many checkers as we can on the board. What is the best strategy? We'll discuss the following questions. How many checkers can a computer put on the board using a naive algorithm? How much do mathematicians know about this problem? We'll start with simple (high-school level) observations and end with difficult open problems.

2012 Lectures

July 6	Richard Stanley	Pattern Avoidance	A permutation $w =a_1 ··· a_n$ of $1, ... , n$ is said to avoid a permutation $v = b_1 ··· b_k$ of $1, … , k$ if $w$ has no subsequence whose elements are in the same relative order as v. For instance, the permutation 421635 is not 231-avoiding because of the subsequence 463. One of the earliest results in this area is that for any permutation $v$ of 1,2,3, the number of permutations $w$ of $1, ... , n$ that avoid $v$ is the Catalan number $C_n = \frac{1}{n+1}\binom{2n}{n}$. We will survey some of the highlights of pattern avoidance. There are many open problems.
July 11	Haynes Miller	Immersions	We all doodle. If we are neat, the doodles form smooth curves which close up (so they can be repeated endlessly). In more imposing language, we have immersed a circle in the plane. If we are mathematically oriented, we will want to classify these objects. This is the content of the Whitney‐Graustein Theorem. Steve Smale generalized this theorem, and among other things he thought about immersing surfaces in Euclidean spaces. This work led to Smale's discovery of the eversion of the sphere, and a Fields Medal (1966), and has been fundamental in differential topology ever since.
July 18	Abhinav Kumar	Sums of Squares	Which integers can be written as the sum of n squares, where $n$ = 2,3,4,...? And in how many ways? One can also generalize this question, replacing the integers by a field (such as the rationals) or another ring (such as polynomials in one or several variables). This question is connected to algebra and number theory, but also (less obviously) to topology, for instance. I'll describe some of the tools used to attack this classical problem, and some of the beautiful mathematics it produced.
July 25	Michael Artin	Fractions	Ordinary fractions, rational numbers, facilitate integer computation because they allow division. Fractions can also be used in other situations. Rational functions, for example, are fractions of polynomials. But what if multiplication isn't commutative — if a product $vu$ isn't necessarily equal to $uv$? For instance, one can redefine multiplication of polynomials in two variables $x,y$ using the rule $yx = 2xy$. Or, one can work with polynomials such as $xxy+xyx+$..., using no commutation rule at all. Do fractions make sense in these situations? If not, is there a substitute? We'll sketch some of the answers that were found between 1930 and 1980 by people including Ore, Jategaonkar, Makar‐Limanov, and culminating in the amazing work of Cohn.
August 1	Jacob Fox	Ramsey Theory	Ramsey theory refers to a large body of deep results in mathematics whose underlying philosophy is captured succinctly by the statement that "Every large system contains a large well‐organized subsystem." This subject is currently one of the most active areas of research within combinatorics, overlapping substantially with number theory, geometry, analysis, logic, and computer science. This lecture will introduce some of the fundamental results and problems in Ramsey Theory.