# Pizza Seminar

## Department of Mathematics, University of Chicago

### Time & Place

12:30pm–1:20pm on Wednesdays, in Ryerson 352 (“The Barn”).

### Contact

Please email pizzaseminar AT math.uchicago.edu if you want to give a talk or if you have comments about the website.

5 June 2019

#### Ishan Banerjee:

Tomorrow I will present a series of sketches and vague descriptions. I will begin by presenting a reasonably complete proof that curves have finitely many isomorphisms. Then we will sketchily describe the classification of surface diffeomorphisms. We will finish this off by presenting some wish-wash on the Mordell and Shafarevich conjectures(the geometric ones).

29 May 2019

As deadlines come closer, introducing myself has become a dadaist experience: "Da-da-da...". I thought I would take this inspiration and put together a collage of (perhaps) unrelated topics I've always wanted to talk about in a pizza seminar. Come for a short homage to the memory of Prof. Zygmund, performing experiments with "la ola", and some interesting problems of x-rays in finite groups. You won't need a lifesaver for la ola if you eat enough pizza.

22 May 2019

15 May 2019

#### Aaron Chen: How to cut pizzas and split rent

We will talk about various fair division problems, including cake (pizza in our case) cutting and rent splitting. Fairly cutting pizza(s) may be left as an exercise for interested audience members.

8 May 2019

#### Weinan Lin: PizzaTeX

Have you had trouble typesetting complicated tables/category diagrams/big tikz graphs in latex? Do those chunks of codes make your latex complier too slow every time you compile? Do you plan to learn some programming languages (html/js, python, etc) but haven’t started yet because it might take too much time? Do you want to have some pizza?
I want to share my solutions to these problems via Jupyter Notebook. People can also share their knowledges and suggestions about latex during Q&A.

1 May 2019

24 April 2019

#### Karl Schaefer: How to get rich without the Axiom of Choice

Everyone knows how to get rich using the Axiom of Choice: Just use Banach-Tarski to duplicate whatever material you want. In this talk, I'll show you how you can get rich constructively, without the Axiom of Choice. All we'll need is a little bit of ergodic theory and an uncountable amount of money.

17 Apr 2019

#### Noah Taylor + Ronno Das: Mathematical Games and Metamagical Themas

We will demonstrate some mathematical card tricks and provide "mathematical" explanations. There will be mentions of symmetric groups and probability but no sleight of hand will be involved.

10 Apr 2019

#### Daniil Rudenko: Around Monsky's Theorem

I will talk about equidissections of polygons and Monsky theorem. It states that a [square] can not be cut into an odd number of triangles of equal areas. It is fun and elementary topic, which still seems to be very mysterious to me.

3 Apr 2019

#### Eric Stubley: Campanology 101

You'll learn about the history and mathematics behind change ringing, which is the art of performing (subgroups of) symmetric groups on bells. I'll draw some pictures, compute some permutations, and (with some assistance) ring some bells.

13 Mar 2019

#### Adan Medrano Martin del Campo: Music meets topology

There might be topology, there might be music, and there might be pizza*.
* There will not

6 Mar 2019

#### Ben Seeger: Some random random series of functions

I want to talk about some random series, but, as of the writing of this abstract, I don't yet know which ones.

27 Feb 2019

#### Dylan Quintana: The 🎩 Problem

Don't you 🎩e it when you forget w🎩 color your 🎩 is and 🎩h to guess based on w🎩 🎩s your friends are wearing? Let's 🎩ch a plan t🎩 minimizes the odds of embarrassment in this everyday scenario. (Here's a tip: It helps somew🎩 if the number of friends you 🎩h is two less than w🎩ever.

20 Feb 2019

#### Isabella Scott: Forcing for the Working Mathematician

Everyone knows forcing as a tool for proving independence results, but working mathematicians aren't interested in independence results. In this talk, I'll prove some well-known (and not terribly complicated) theorems in a complicated way, using forcing.

13 Feb 2019

#### Anthony Wang: Combinatorial Games

I talk about two player, perfect information games where the players alternate moves. I introduce the games of green hackenbush and blue-red hackenbush, and the concepts of nimbers and surreal numbers.

6 Feb 2019

#### Liam Mazurowski: The Lusternik Schnirelmann Category

Lusternik (noun) A Soviet mathematician famous for his work in topology and differential geometry
Schnirelmann (noun) A Soviet mathematician who worked on number theory, topology, and differential geometry
Category (noun) A collection of objects and morphisms satisfying certain axioms

30 Jan 2019

23 Jan 2019

#### Nat Mayer: The Twin Paradox in a Compact Universe

Do you coexist simultaneously with your future self? This isn't (just) a question your old stoner roommate from college asked you, it's physics! Beginning with a brief primer on special relativity, we will wrestle with the famous twin paradox. Is it really a paradox? Spoiler: no. But with just a bit of nontrivial topology, we will see just how weird spacetime can get.

16 Jan 2019

#### Drew Moore: Amarillo Slim’s Hustle

Want to get better at poker? Well maybe this isn’t the talk for you. Want to get better at simple toy games that mimic certain aspects of poker? Well then oh boy get ready. This Wednesday, we’ll play and solve some simple betting games.

30 Oct 2013

#### Asilata Bapat: Moment maps and Morse theory

A group action on a space (manifold) usually gives us some insight into properties of the space (manifold). The situations get even better if the group, the action, or the manifold is decorated with adjectives such as compact, Lie, algebraic, smooth, symplectic, Hamiltonian, etc. Morse theory is also another powerful tool to extract information about the topology of smooth manifolds. In this talk I will introduce the notion of a moment map for a group action on a symplectic manifold and give a brief survey of what it can be used for. I will also give a quick introduction to Morse(–Bott) theory and talk about connections with the moment map.

23 Oct 2013

#### Nick Salter and Tim Black: Conway’s Tangle Trick

If we just told you straight up that this week’s pizza seminar talk was going to be about math and magic tricks, you’d probably think it would be about as cool and/or interesting as this guy: But what if we told you that we’d be talking about a magic trick invented by this guy: And that this was some of the math involved: Then you’d probably want to come.

16 Oct 2013

#### Ian Frankel: Points, Lines and Planes

We will be talking about low-dimensional affine and projective geometry. In particular, we will interest ourselves in the axiomatic characterizations of projective and affine geometries, how Desargues’s theorem on perspectivity of triangles relates to the classification of projective geometries in higher dimensions, and what goes wrong in dimension $2$.

9 Oct 2013

#### Daniel Le: Reciprocity Laws, from Diophantus to Langlands

Let $f$ be a polynomial with integer coefficients. How many solutions does $f = 0$ have modulo each prime? The answer is given by reciprocity laws (some established and some conjectural). I’ll begin with several examples before discussing the connection between this question and Galois representations, $L$-functions, and automorphic forms.

5 Jun 2013

29 May 2013

22 May 2013

15 May 2013

8 May 2013

#### Matthew Wright: The Axiom of Determinacy

The axiom of choice is nice and all, but it lets us do some pretty terrible things (Banach–Tarski, anyone?). Wouldn’t it be nice to have some way to prevent these pathologies from happening? A few weeks ago, Jonny mentioned the axiom of determinacy, which does exactly that … with games! I’ll be talking about what exactly the axiom is, how it gets rid of a lot of the pathologies that choice gives us, and how we can use it to get a handle on just how simple these pathological sets could possibly get.

1 May 2013

#### George Sakellaris: Geometry of Numbers, Dirichlet’s Approximation and Lagrange’s Theorem

How do properties of convex subsets of $\mathbb{R}^n$ interact with number theory? How about trying to find integer points in convex sets; for example, $0$ is the only integer point contained in the open cube $(-1,1)^n$ (this is not so hard to prove, actually). Now, what property on the volume of such a set could ensure that this set contains at least one integer point? The answer, given by Minkowski’s theorem (and this is harder), proven in 1889, initiated what is today called the “Geometry of Numbers”. We will give two proofs of this theorem, one geometric and one analytic, and discuss how this can be applied to a couple of number theory problems: Dirichlet’s approximation theorem and Lagrange’s four square theorem.

24 Apr 2013

#### Sergei Sagatov: Oranges, Sphere Packings and Kissing Numbers

Imagine oranges stacked at a grocery store. Is this the most space-efficient stacking pattern? This is the famous Kepler conjecture, originally formulated in 1611, and in 1900 reformulated as part of Hilbert’s eighteenth problem. Intuition says yes, but the proof remained elusive for some centuries, until Thomas Hales gave a computer-assisted proof in 1998. Since this question is much too difficult for us to address, we turn to a related but much simpler problem. Each orange in the interior of the stack touches $12$ other oranges. Is it possible to arrange $13$ oranges around a central one in such a way that they just touch the central orange? This is the famous problem of thirteen spheres, and was a subject of a discussion between Isaac Newton and David Gregory in 1694. Newton believed $13$ oranges was impossible. Gregory thought otherwise. Following a proof by John Leech, we answer the question in the negative: $12$ is the maximal number. We will then briefly discuss a higher-dimensional generalization, the so-called kissing number problem.

17 Apr 2013

#### Jonny Stephenson: What the $L$? A broad-brush non-technical history of set theory

By and large, mathematicians like to work with sets. We’re comfortable with them. We might even think of them as objects that have some real existence.

To make the idea of a set precise, we have a system of axioms called ZFC, which models the behaviour we expect of sets. However, this leads to some problems: the system (if it’s consistent) has many different models. Questions like “does the continuum hypothesis hold?” have answers that depend on which model we consider.

For a long time, this sort of phenomenon was seen as the price we have to pay for our formalism. However, we’d prefer to choose one specific model that’s acceptable to everyone, so that we don’t have to worry about that sort of thing — and (as of this decade) it seems like it might be possible.

Set theory began in 1873, and is still happening. Let’s pay a visit.

10 Apr 2013

#### Katie Mann: Roads and Wheels

Have you ever seen a square wheeled bicycle? If not, here’s a very serious picture. When I saw this, I wondered if you could make a regular-$n$-gon wheeled bicycle that would roll just as smoothly, given the right road. What about an arbitrary polygon? What if I give you a crazy periodic curve for a road — can you build me a wheel to roll along it? Will the wheel be unique? Will you still travel at constant speed if you pedal at a constant rate? Luckily, we know math, so we can find the answers to these pressing questions.

Prerequisite: watching some youtube videos of people riding square wheeled bikes and tricycles.

3 Apr 2013

#### Shuyang Cheng: Tropical algebraic curves

Tropical algebraic curves are relatives of classical (complex, say) curves constructed over the so-called tropical semi-ring. They are combinatorial objects, hence much easier to understand than the classical ones, but most of the classical results survive in the tropicalized theory, often acquiring some cute combinatorial interpretation. In this talk I will state and explain the tropical versions of some classical theorems, and hopefully no background in algebraic curves is required.

13 Mar 2013

#### Alex Wright: Rational billiards

We will start by bouncing a ball in a polygon, proceed through some physics and dynamics, and end up with rigid curves in moduli space giving new and special polygons. There will be the opportunity to win fame and fortune by answer a question about triangles that you could have understood in grade school.

6 Mar 2013

#### Max Engelstein: Travelling Efficiently

Apparently some people have real jobs and this involves having to do things efficiently. We’ll look at one example of how this is sometimes hard; the travelling salesman problem. Interestingly, the problem becomes easier (that is to say there is a known solution) when we pass from the discrete to the continuous. This is the “Analyst’s Travelling Salesman” problem and in solving it we will see some conformal geometry, geometric measure theory and harmonic analysis (of course no knowledge of any of these things will be assumed).

27 Feb 2013

#### Zhouli Xu: Why do (some) people care about the homotopy groups of spheres?

In this talk, I will start with the definition and basic properties of the homotopy groups of spheres, and then talk about their connections with Poincaré conjectures, cobordisms and derived algebraic geometry (if time allows).

20 Feb 2013

#### Galyna Dobrovolska: Fast approximate matrix multiplication and limit points of secant varieties

I will explain A. Schoenhage’s algorithm that allows to multiply two $3$ by $3$ matrices using only $21$ multiplication operations, resulting in a matrix with entries within any given number $c$ from the entries of the actual answer. In order to explain this algorithm, I will show how to reinterpret it as a search for limit points on secant varieties of a Segre embedding.

13 Feb 2013

#### Katie Mann: Why have sex?* *an information theory approach.

You know what’s great about knowing math? You can prove (yes, pretty much prove) that sexual reproduction kicks butt — on a grand evolutionary scale — compared to the far less fun asexual alternative.

Starting from some really elementary assumptions, we’ll get a model for information acquisition through evolution (whatever that means), plug in some elementary probability theory, and conclude that sex generally wins over no sex, especially in unstable environments. Applied math FTW.

Disclaimers:
1. Any probability theory involved will be easy enough for a geometer to understand.
2. You might learn something about the physical world, or at least how science understands it.
3. This talk is largely modeled after a book chapter by the same name in Mackay’s “information theory, inference and learning algorithms”. Highly recommended.

6 Feb 2013

#### Nick Salter: The Nielsen–Thurston Classification

Let $S$ be a closed surface, and let $f$ be any homeomorphism of $S$ to itself. Is there a classification of the types of behavior that $S$ can exhibit? Of course not — the homeomorphism group is absurdly large, and there are homeos doing arbitrarily crazy things in arbitrarily small neighborhoods. But what if you allow yourself to adjust your homeomorphism by an isotopy? The remarkable answer here, known to Nielsen and first proved by Thurston, is that there exists a complete classification of homeomorphisms up to isotopy, and that any $f$ is isotopic to one of only three specific possibilities. I’ll explain this result, which will take us on a brief tour of the mapping class group and of Teichmüller space, and discuss some applications to geometry and topology.

30 Jan 2013

#### Asilata Bapat: A pictorial introduction to toric varieties

I will give an introduction to toric varieties and describe how to build them by drawing pictures. The structure of these varieties has been extensively studied. In fact we can extract a lot of information about a toric variety just by staring at its corresponding picture. This makes toric varieties particularly convenient to compute with.

The talk will feature some algebra, various* kinds of geometry, and lots of pictures.

*at least two

23 Jan 2013

#### Daniel Le: $P$-adic zeta functions

Special values of zeta functions are conjectured to have deep arithmetic meaning. Euler discovered that special values of the Riemann zeta function satisfy mysterious congruences. I’ll explain these congruences using measure theory! (I might be overselling the analysis involved.)

16 Jan 2013

#### Yiwei She: Seeing the Light

This very second $10^14$ (plus or minus a few dozen trillion) photons are entering your eyes, letting you read this abstract. In this pizza seminar I will talk about the mathematics behind these particles, Schrödinger equation. Along the way we’ll look at the history and development of this theory as well as the interpretations of what the theory predicts.

9 Jan 2013

#### Henry Scher: Hahn Banach in Welfare Economics

Welfare economics is a branch of economics associated with aggregating individual preferences into a social preference. Sometimes that’s impossible to do nicely, as in Arrow’s Theorem (see Yiwei’s talk last year), but sometimes we can get exactly what we want — and for that, we need geometric Hahn–Banach.

5 Dec 2012

#### Andrew Geng: Musical Tuning and Continued Fractions

The interval between two pitches sounds nice when the ratio of their frequencies is a ratio of small integers. An octave is a ratio of $2:1$, so to get the $12$-note chromatic scale we just split that evenly — a half-step is a ratio of $2^{1/12}:1$. Seven half-steps make a perfect fifth, which is a ratio of $2^{7/12}:1 = 3:2$… we wish. Sadly, you can even hear the difference! But I heard that continued fractions lead to good approximations by rationals, so maybe we can do better with a different number than $12$.

28 Nov 2012

#### Preston Wake: Real numbers that aren’t (too) well approximated by rationals

In this talk, we’ll think about the equation $x^n+2y^n=1$ (and others like it) and ask “how many integer solutions $x$ and $y$ does it have?” We’ll explore two different methods for attacking this type of problem: the first uses the fact that algebraic numbers aren’t very well approximated by rationals (what?!), and the second uses geometry, somehow.

21 Nov 2012

#### John Wilmes: The toppling ideal and its minimal free resolution

The chip firing game offer a fascinating bridge between combinatorial and algebraic objects. In this talk, I will introduce the chip firing game on a directed multigraph and discuss connections with lattice ideals. In the undirected case, there is a fabulous interplay between combinatorial data in the graph and algebraic information in the corresponding “toppling ideal.” In particular, we can use the graph structure to give an explicit minimal free resolution.

14 Nov 2012

#### Henry Chan: Simplicial Sets and Classifying Spaces

Simplicial objects play an important role in fields such as algebraic topology, algebraic geometry. Even Drinfel’d was amazed by how fundamental this thing is for some of his work. I will do some basic definitions and use them to construct classifying spaces of groups we know and love. This is not rocket science!

7 Nov 2012

#### Bena Tshishiku: Associations of a non-associative algebra

John Baez describes the octonions as “that crazy old uncle nobody lets out of the attic”. In this talk we’ll define this eccentric non-associative algebra and describe its connection to Bott periodicity and exceptional Lie groups.

31 Oct 2012

#### Wouter van Limbeek: Monstrous Moonshine

To celebrate Halloween, I will tell a story about the Monster group. This starts with the important observation (by John McKay) that $196884 = 196883 + 1$.

Here, of course, $196883$ is the smallest dimension of a faithful representation of the Monster group, and $196884$ is the first nontrivial coefficient of a certain modular function associated to $\mathrm{SL}(2,\mathbb{Z})$. This was generalized by Conway and Norton to other fun identities such as $864299970 = 842609326 + 21296876 + 2 \times 196883 + 2 \times 1$.

Based on this, Conway and Norton formulated the ‘Monstrous Moonshine conjecture’ (later proven by Borcherds), which explains the relation between the Monster group and modular forms on $\mathrm{SL}(2,\mathbb{Z})$. I will show where this connection comes from, and we’ll discover that the Monster group is quite the beautiful creature after all!

24 Oct 2012

#### Shuyang Cheng: Schur–Weyl–Howe duality

Schur functors could be defined for a general Karoubian tensor category, but the most interesting and classical application is still that of constructing new group representations from old ones. In this setup there exists an almost purely combinatorial correspondence between $S_n$-rep and (part of) $\mathrm{GL}_k$-rep known as the Schur–Weyl duality. In this talk I will talk about a natural generalization of this phenomenon first studied by Roger Howe. There will be some exciting applications.

17 Oct 2012

N/A

10 Oct 2012

#### Max Engelstein: Outwitting the Greeks with Origami

The ancient greeks struggled for centuries with straightedge and compas constructions. Amongst their greatest challenges was trisecting an arbitrary angle (a task which was much later proved impossible with straightedge and compass). In this talk we strike a blow against the western hegemony and use origami to simply trisect arbitrary angles. Along the way we’ll talk about some Galois theory and do some folding. Origami paper will be provided.

3 Oct 2012

#### Ben Fehrman: Black–Scholes: Determining the Price of a Wager

We will determine the price you pay to bet on the stock market. Specifically, we will determine the price of so-called European options. Simply, after say a week, the value of a stock will either go up or go down, and you can bet on it. The price you pay for the opportunity is determined by the Black–Scholes equation, which we will derive following an informal discussion of Brownian motion, stochastic calculus and stock market gambling.

30 May 2012

#### Jared Bass: Combinatorial Games

A combinatorial game is a two-player game of no chance and perfect information. In analyzing these games we stumble upon many beautiful objects, for example the surreal numbers, which is a field (that is, it would be if it were a set) including both the real numbers and the ordinals. But combinatorial games have many more surprises in store for you. As do I.

23 May 2012

#### John Wilmes: Regularity and removal lemmas

A classic theorem of number theory due to Szemerédi states that every set of natural numbers of “positive density” contains arbitrarily long arithmetic progressions. The case of three-term arithmetic progressions follows quickly from the innocent-seeming triangle removal lemma, an approach which was recently generalized to $k$-term arithmetic progressions. In this talk, I will discuss aspects of a proof of Szemerédi’s theorem, and connections to graph theory. Szemerédi’s regularity lemma, which puts a “pseudorandom” structure on arbitrary graphs, will be central to everything.

16 May 2012

#### Valia Gazaki: Diophantine equations and ideal class groups

Does the equation $y^3=5+x^2$ have integer solutions? The answer is no, but if you try to do it by hand, most probably you won’t be succesful. The reason is that the ring of integers of $\mathbb{Q}(\sqrt{-5})$ is not a PID.

In this talk I will discuss the classical problem of number theory, about how far the ring of integers of algebraic number fields are from being Unique Factorization Domains. Namely, I will define the ideal class group of a Dedekind domain and then we will see that in the case of the ring of integers of an algebraic number field this group is always finite. The fact that the order is $2$ in the case of the quadratic field $\mathbb{Q}(\sqrt{-5})$, will play the crucial role in the nonsolvability of the previous equation.

9 May 2012

#### Postdoc Panel: Tuca Auffinger, Jon Chaika, Angélica Osorno, Artem Pulemotov

Postdocs Tuca Auffinger, Jon Chaika, Angélica Osorno, and Artem Pulemotov will share their wisdom about the job search process, adjusting to being a postdoc, life after the postdoc, and whatever else the audience wants to know.

This is a great opportunity for both younger grad students and those nearing graduation to learn about the academic career path, and to get their questions answered. We hope to see everyone there!

2 May 2012

#### Simion Filip: Renormalization

Renormalization is a technique that’s widely used in dynamical systems, statistical physics, probability, PDEs or anything else that involves some analysis. I will start by giving a heuristic for how the method works and then I will work out several examples from very different areas. Diophantine approximation, lattice models, the logistic family and the Central Limit Theorem are some examples that I hope to cover.

25 Apr 2012

#### Jacob Perlman: A rigorous treatment of big numbers, small numbers, equalish, and other such vagaries.

Did you know that most things you work with are too big? Simply too big, far too much, and a great deal too many. By introducing some axioms for clearing up this infinite clutter we spend much of our time embroiled in, internal set theory provides a framework in which many of the concepts and proofs that we use all the time are cleaning and more intuitive; examples will hopefully be persuasive. Sadly there are some costs, mostly in the form of ridiculous sounding consequences; examples will hopefully be controversial and borderline offensive. I will conclude with vague gesturing at a proof that this is not a bunch of hooey.

Warning: this talk may contain some set theory.

18 Apr 2012

#### Wouter van Limbeek: The geometry of nilpotent groups: A theorem of Milnor and Wolf

Given a finitely generated group $G$, one can build the Cayley graph of $G$ and use the word metric to make it into a metric space. Geometric group theory studies how the algebraic properties of $G$ influence and are influenced by the geometry of the Cayley graph. I will prove a classical theorem by Milnor and Wolf, classifying the (virtually) nilpotent groups among all solvable groups by their geometry.

(Now seems a good time to shamelessly advertise my FFSS talk on Thursday [4:15 pm in E308] in which I will talk about Gromov’s amazing generalization, recognizing the virtually nilpotent groups among all finitely generated groups by their geometry.)

11 Apr 2012

Algebraic objects are often given “biased” definitions — defined in terms of one or two operations satisfying relations. I will discuss monads and operads, two tools which allow discuss algebraic objects in an “unbiased” way, in which every possible operation is treated identically by the formalism. Examples and applications will hopefully be plentiful.

4 Apr 2012

#### Asilata Bapat: Graph colouring and Möbius inversion

What is the number of proper $k$-colourings of a given graph, as a function of $k$? A perhaps non-obvious fact is that this is a polynomial, called the chromatic polynomial of the graph. I will explain a beautiful proof of this fact using Möbius inversion (which we will learn along the way).

As a bonus, I may talk about one or two other examples of slick proofs that use Möbius inversion.

28 Mar 2012

#### Shuyang Cheng: Weyl’s law for closed surfaces

Let $X$ be a closed surface of constant curvature. Then the Weyl’s law states that $N(t)\sim (t\cdot Area(X))/4\pi$, where $N(t)$ is the counting function of Laplacian eigenvalues on $X$. In this talk I’ll sketch an algebraic proof based on a simple version of the trace formula.

7 Mar 2012

#### Yiwei She: Arrow’s Impossibility Theorem, or the dirty secret of democracy

In a year of elections worldwide, we may ask ourselves, does it really matter who we vote for? We use current events in Russia as an illuminating example.

29 Feb 2012

#### Daniele Rosso: RSK Correspondence and Growth Diagrams

The Robinson–Schensted–Knuth correspondence is a classical combinatorial result that everyone should know. It has a nice symmetry property which is not at all obvious at first glance from the usual algorithm. Surprisingly, there’s another algorithm, due to Fomin, which uses so called ‘growth diagrams’, which gives the same correspondence and for which symmetry becomes obvious.

More precisely, what I am actually going to do is that I will draw some boxes, put numbers in them and then move them around and magic will happen!

22 Feb 2012

#### Fedor Manin: What is a building?

In some sense, buildings are a higher-dimensional generalization of trees. We’ll use the example of trees to get a clear, mostly combinatorial presentation of the definition. Then we’ll talk about spaces that happen to be buildings, notably the Tits boundary of a symmetric space.

15 Feb 2012

#### Francis Chung: Adventures in the land of isospectral planar domains, or, Can you hear the shape of a drum?

Mark Kac famously posed the question above in a 1966 article, although similar questions had been asked before. John Milnor showed almost right away that the answer is no, not if you listen to $16$-dimensional torus drums, but for the rest of us, the question wasn’t resolved until 1992. I’ll explain what on earth the question means, and how it got answered in the end.

8 Feb 2012

#### Matthew Wright: Why it’s possible to give a pop quiz, and what that has to do with the incompleteness of arithmetic

Gödel’s incompleteness theorems say that any nice enough and strong enough logical system is incomplete (that is, has statements that are not provably true or false), and that no such system can prove its own consistency. We’ll be showing how to prove these by taking common paradoxes and formalizing them using Kolmogorov complexity: Berry’s paradox (“the smallest natural number than can’t be described in fewer than thirty words”) gives us a way to prove the first incompleteness theorem, while the surprise examination paradox (“the pop quiz can’t be on Friday, because then it wouldn’t be a surprise…”) lets us prove the second incompleteness theorem. We’ll then turn things around and see what the incompleteness of arithmetic has to say about resolving the paradoxes.

1 Feb 2012

#### Bena Tshishiku: Ramsey theory and topological dynamics

What’s your favorite experience of the integers? Perhaps you enjoy its additive structure or that it contains solutions to $x^2+y^2=z^2$. Ramsey theory asks, “Are such properties preserved under finite partition?” We discuss some Ramsey theoretical results and an approach through ergodic theory.

25 Jan 2012

#### Daniel Le: Four lines in space

an introduction to algebraic geometry

Given four skew lines in $3$-space, how many lines intersect all four? The answer is two*! Problems of this nature are called enumerative problems in algebraic geometry and often require sophisticated calculations. This one is simple enough that I will give two proofs of this fact: one using elementary algebraic geometry and one using intersection theory on a Grassmannian. No rings or schemes required!

*sort of.

18 Jan 2012

#### Daniel Studenmund: The Fundamental Theorem of Projective Geometry (FTPG)

Don’t you love thinking about lines? I do. The FTPG is all about lines. We will prove the baby case, which says that a bijection of R^n taking lines to lines must be linear. (Go figure.) Then we’ll take a gander at the general statement of the FTPG, with perhaps some indication its proof on the side. Time permitting, we will define “buildings” and state Tits’ big generalization of the FTPG.

11 Jan 2012

#### Kate Turner: Euler Calculus and an application to Target Enumeration

We have all done measure theory courses and know how to integrate with respect to a measure. In this pizza seminar we instead will consider integrating with respect to the Euler characteristic. We will see that there is a well defined calculus theory if we restrict to “constructible functions” - we even get Fubini’s theorem. This Euler calculus will then be applied to the problem of target enumeration for both stationary and moving targets (this means counting how many targets from a function that tells us at each point how many targets we can see.)

4 Jan 2012

#### Jessica Lin: Entropy in Dynamical Systems

This talk is meant to be an introduction to the notion of entropy in dynamical systems. We will define topological and metric entropy and discuss their relationship to other dynamical concepts (in particular, Lyapunov exponents). Geometric and probabilistic examples will be provided. This talk will be a short tasting of dynamical systems, ergodic theory, geometry, probability, and pizza of course.

30 Nov 2011

#### Wouter van Limbeek: Hyperbolic Groups

In past pizza seminars we have seen the hyperbolic plane. Because it turns out the hyperbolic plane is pretty nice to work with, we define a group to be hyperbolic if it looks like the hyperbolic plane (in some sense). I’ll give some examples, and show a few of the remarkable implications for the group structure.

23 Nov 2011

#### Rolf Hoyer: Some Stable Homotopy Theory

Hopefully, all of you care about invariants of topological spaces. I intend to motivate and introduce the stable homotopy category, which can be more informally be called “the place where cohomology theories live”. I will say some words about ordinary cohomology, which you may or may not remember from the first year, and then I will gloss over decades of work by forgetting one of the Eilenberg–Steenrod axioms. This will give a conceptual outline of the general picture, after which I will attempt to convince you as to why such a perspective is useful.

16 Nov 2011

#### Preston Wake: Dilogarithm: the Kevin Bacon of math topics

Dilogarithm is like a logarithm, except somewhere in a definition of logarithm, you change a $1$ to a $2$. You might think, “that’s a completely arbitrary thing to do, why would you do that?” Well, the answer is that lots of smart people have done it before. Dilogarthims have been used by Milnor and Thurston to study hyperbolic geometry, by Bloch to study $K$-theory, by Coleman to study explicit reciprocity laws and $L$-functions, by Kirillov to study physics, and by Deligne and Beilinson to study motives.

In this talk, I’ll explain how to get from $K$-theory to hyperbolic $3$-manifolds and Mostow rigidity to zeta functions of quadratic imaginary fields — all through dilogarithm.

9 Nov 2011

#### Ben Fehrman: Spun Out

Basic physical concepts will be recalled and applied to the motion of a spinning top. The motion is readily observed to consist of three periodic cycles. The cycles are shown to be intrinsic to the motion, not the result of external torque. That is, a top spins in space as it does here.

Time permitting Noether’s theorem and Poincare recurrence will be discussed in this setting.

2 Nov 2011

#### Jacob Perlman: Trick-or-treaters walking house to house finds the optimal shape for a bag of candy OR optimal transport and the isoperimetric inequality

Sometimes there is a bunch of candy somewhere (like in other peoples houses), and sometimes you want this candy to be elsewhere (like in your house), but moving candy isn’t easy. Finding the best way to get your candy from their house to yours is a question of optimal transport. Thinking about what this optimal transport is can be hard, but thankfully just its existence (which we will assert without proof) can be very useful. As an example, we will prove the isoperimetric inequality using nothing but some mathematical facts you might find lying around your kitchen, with sufficient generality to extend trivially to the anisotropic isoperimetric inequality, which is fun to say.

26 Oct 2011

#### Max Engelstein: Analytic Capacity: or how I learned to stop worrying and love Geometric Measure Theory

The talk will center around the problem of when you can extend a bounded holomorphic function to a removed compact set. If that sounds boring to you don’t worry; the talk is really about geometric measure theory and how it is super cool! There will be Hausdorff measure, rectifiable sets and maybe (if everyone behaves) tangent measures!! If that still sounds boring to you still don’t worry; at least there will be pizza.

19 Oct 2011

#### Shuyang Cheng: The Cauchy–Crofton formula

The Crofton formula says the length of a plane curve is equal to the “expected” number of its intersections with a random straight line. In the talk I’ll include some applications and generalizations.

12 Oct 2011

#### Daniel Schäppi: Combinatorial Games

Combinatorial games are a special type of two-player game defined by J.H. Conway. Wikipedia has this to say on the matter: “Combinatorial game theory (CGT) is a branch of applied mathematics and theoretical computer science that studies sequential games with perfect information, that is, two-player games which have a position in which the players take turns changing in defined ways or moves to achieve a defined winning condition.”

Now, the obvious question is: what does this have to do with category theory? If you want to find out, come to Pizza seminar tomorrow!

5 Oct 2011

#### Katie Mann: Schmidt games and using them to win at Math

By the end of my talk, you should be able to understand — better yet, appreciate — a recent result of our own Howie Masur and Jon Chaika, along with Yitwah Cheung. The talk is about Schmidt games, what it means to win them, how you can beat me at them (yes you can), and how they relates to badly approximable real numbers, hyperbolic space (!), and (!!) Teichmuller geodesics. Despite the words “Teichmuller geodesics” above, this talk should be non-scary.

P.S. The paper in question is: http://arxiv.org/abs/1109.5976.

P.P.S. I don’t recommend you try to read it before pizza seminar.

28 Sep 2011

#### Simion Filip: How to solve the quintic?

I will try to explain how one can find the roots of a general polynomial of degree five. We shall use tools from dynamical systems, algebraic geometry, representation theory and analysis. The talk will be very accessible (I promise).

1 Jun 2011

#### Postdoc panel

Postdocs William Lopes, Angélica Osorno, and Vlad Vicol will share their wisdom about the job search process, adjusting to being a postdoc, life after the postdoc, and whatever else the audience wants to know.

This is a great opportunity for both younger grad students and those nearing graduation to learn about the academic career path, and to get their questions answered. We hope to see everyone there!

25 May 2011

#### Matt Thibault: How to construct expander graphs

An expander graph is a highly connected sparse finite graph; examples which come to mind are nerve cells in the brain, a robust computer network, and social networks. The existence of expander graphs follows easily from a probabilistic standpoint, but explicit constructions have required Kazhdan’s Property $T$ or the Ramanujan–Petersson conjecture. In my talk, I will introduce expander graphs and Ramanujan graphs. Then I will proceed to talk about the construction of expander graphs from Kazhdan’s Property $T$. You can expect to see plenty of graphs.

18 May 2011

#### Anna Marie Bohmann, Emily Riehl, Tom Church: Other people’s theses

Anna Marie Bohmann discusses stirring things up on surfaces, a la Spencer Dowdall. Pretty pictures are included.

Emily Riehl discusses how to do equivariant algebraic topology for all groups at once, a la Anna Marie Bohmann.

Tom Church discusses model structures with extra structure — no prerequisites necessary — a la Emily Riehl.

11 May 2011

#### Katie Mann: Why don’t we understand anything about groups acting on the plane?

We know a fair amount about groups of homeomorphisms of the line. The plane is a different story. In my talk, I’ll give some examples of seemingly elementary questions that we have no idea how to solve, as well as some that have been solved only very recently. If you’re lucky, I’ll also give a proof of one thing we do know about homeomorphisms of the plane, the Brouwer plane translation theorem. It was proved in 1912 and it’s still a cutting edge tool in the study of group actions on the plane.

4 May 2011

#### Jenny Wilson: The Littlewood–Richardson Coefficients

Though the Littlewood Richardson Coefficients were introduced in 1934 as a means of computing characters of representations of the symmetric group, these coefficients have since appeared in the answers to questions in surprisingly diverse areas of mathematics. We will look at some of the places they have arisen, as well as a method of calculating these coefficients. Time permitting, we will survey their relevance to the recently completed work of Knutsen, Tao, and Woodward on determining the possible eigenvalues of a sum of Hermitian matrices.

This talk will have very few proofs, but lots of little boxes.

27 Apr 2011

#### Jacob Perlman: What’s a mathematician gotta do to get a constant around here?

It is the dream of every young mathematician, gazing wearily out the window in ninth grade, to someday have their own constant, like Euler, Archimedes, Conway, or Euler–Mascheroni (the more pasta like of Euler’s two constants). Well I still don’t have a constant and neither do you, but Khinchin does! It’s about 2.685. Why is that number better than any number you were the first one to think of? You can either google it or come to pizza seminar. Since only one of those comes with pizza, I think the best course of action is clear.

20 Apr 2011

#### Ben Fehrman: Motion by Mean Curvature

The motion of a surface by its mean curvature will eventually be discussed. Indeed, the evolution may be defined for an arbitrary compact set. Foremost, however, is to determine why anybody talks about it. It will be shown that the pressure differential across a membrane determines its mean curvature and, in particular, that soap films form minimal surfaces. Applications of mean curvature flow to image processing will be mentioned.

13 Apr 2011

#### Mona Merling: Algebraic $K$-Theory

Last quarter, Anna Marie tried to convince you what an extraordinary construction topological $K$-theory is. Now I will try to convince you how extraordinary algebraic $K$-theory is also. There will be lots of algebra, topology, number theory, category theory, and at some point even a limit (not a categorical one) will appear to make the analysts happy.

4 Apr 2011

#### Emily Riehl: All concepts are Kan extensions

Every spring Peter May tries to teach all of category theory in three lectures. This year I plan to do him one better and teach all of category theory in a single lecture*. To be fair, my task is rather simplified by the famous observation of MacLane that “all concepts are Kan extensions.”

In part I, I’ll give the simple definition of left and right Kan extensions and then state an incredibly useful formula** which defines Kan extensions pointwise. After the break, I’ll cover all concepts — or at least limits/colimits, adjunctions, and the Yoneda lemma.

*And I won’t even go over time.

**Completely seriously this is the only reason I know how to define geometric realization, or extension of scalars, or induced representations, or derived functors.

30 Mar 2011

#### Jessica Lin: The Mathematics of the Mexican Wave: An Introduction to Mean Field Games

Suppose you are at a soccer match and some group in the audience begins to start the wave. As another member of the audience, your decision to stay seated and stand up is based on trying to optimize your position in the wave, as well as considering the positions of those around you. This is a canonical model for a mean field game, which is a system of partial differential equations meant to describe an optimization problem depending on the behavior of multiple (and eventually infinitely many) agents. This talk will be an introduction to mean field games, which some of you may note is a common topic Pierre-Louis Lions comes to talk about. In this talk, I will describe Hamilton–Jacobi equations, Fokker–Planck equations, and the coupling that happens between them in a mean field game.

9 Mar 2011

#### Fedor Manin: Is the Mandelbrot set computable?

We’ll discuss the title question as well as related questions such as “what does that even mean?” and “who cares?” In the process we’ll define computation over a general ring, with the usual theory of computation as a special case. No previous knowledge of computability necessary.

2 Mar 2011

#### Rolf Hoyer: Hopf Invariant One

The Hopf invariant of a map from $S^{2n-1}$ to $S^n$ can be used to study related questions in algebra (When can $\mathbb{R}^n$ be a division algebra?) and topology (When does $S^{n-1}$ have a trivial tangent bundle?). I will give an overview of this relationship, and sketch a proof due to Adams and Atiyah of the fact that $n=1$, $2$, $4$, $8$ are the only possible values. This will use only the basic structure of complex $K$-theory. This realization helped motivate the development of modern stable homotopy theory.

23 Feb 2011

#### Anna Marie Bohmann: Topological $K$-theory: a generally extraordinary construction

Topological $K$-theory was the first “extraordinary” cohomology theory to be discovered, and it remains a fundamental tool in algebraic topology today. I will introduce topological $K$-theory, starting from concrete considerations, and discuss Bott periodicity. For those of you who recall Niles Johnson’s applications of $K$-theory to astronomy in a pizza seminar of several years back, please note that my talk will contain a similiar application to $K$-theory to lexicography.

16 Feb 2011

#### Spencer Dowdall: Geodesics on the modular curve — What’s their deal?

The infamous modular curve is the quotient of the upper half plane by the modular group $SL(2,\mathbb{Z})$. We’ll consider geodesics on this space and try to understand their dynamical behavior. It turns out this has deep (read ‘elementary but cool’) connections to continued fraction expansions. Using this connection we’ll answer questions like: Are there geodesics that stay in a compact set? What about closed geodesics, or geodesics that return to a compact part infinitely often? Can you find a dense geodesic? All this and more at pizza seminar this week!

9 Feb 2011

#### Preston Wake: Division Algebras: What’s Their Deal?

When you learn abstract algebra, they talk about division algebras, but in real life, no one’s ever heard of any (except quaternions). I’ll talk about why there aren’t many division algebras; there’ll be some Brauer Groups, a few polynomials, and a splash of Galois cohomology.

2 Feb 2011

#### Jessica Lin: Semigroup methods in PDE (Cancelled)

Semigroup theory is a classical, abstract tool used to solve differential equations. This talk is meant to be an introduction to the subject. Topics will include basic definitions, properties, and the Hille–Yosida theorem. If time permits, I will discuss applications of semigroup methods to nonlinear PDE.

26 Jan 2011

#### Laurie Field: The isoperimetric inequality

I’ll talk about the Brunn–Minkowski Inequality (which looks like the usual Minkowski inequality, but for volumes of sets) and its relation to the isoperimetric inequality. If time permits, I’ll also talk about the “length of a potato”.

19 Jan 2011

#### Francis Chung: How to become invisible* through the power of mathematics*** *in one wavelength of light** **may not be visible light ***may need more, strictly speaking, than just mathematics**** ****I mean, it’s not like thinking about differential geometry can make you invisible.***** *****and yeah, invisible is a bit of an exaggeration. Let’s go with ‘hard to see’.****** ******This is going to be about more than just putting a bag on your head, I swear.

I think the title pretty much says it all.

12 Jan 2011

#### Charles Staats: Some applications of model theory to algebra

I will first give a very brief introduction to model theory. In particular, I will try to make it clear that model theory is mathematics, not meta-mathematics (although it certainly has applications to the latter, which I may not address at all). Then, I will show how the model theory of fields can be applied to prove statements like the following: Any injective polynomial map from $\mathbb{C}^n$ to $\mathbb{C}^n$ is surjective. If $k$ is an algebraically closed field, $K$ an algebraically closed extension, and $A$ a $k$-algebra, then [statement] holds for $A$ iff it holds for $A \otimes K$. (“Statement” can be “$A$ is an integral domain”, “$A$ is integrally closed”, …)

Disclaimer: I am not a model theorist, nor any other sort of logician, and have never taken a course or read a book on model theory.

5 Jan 2011

#### Evan Jenkins: Quantum invariants of $3$-manifolds

Last year, I gave a pizza seminar talk on why people who care about statistical mechanics should care about braided monoidal categories. To my horror, I discovered that not everybody here cares about statistical mechanics! So as a second attempt, I’m going to explain why people who care about $3$-manifolds should care about braided monoidal categories. (Don’t tell me you don’t care about $3$-manifolds!)

Lickorish showed that every closed, oriented $3$-manifold can be obtained from a framed link in the $3$-sphere by a process called Dehn surgery. Two such manifolds will be equivalent if and only if the links are related by the so-called Kirby moves. So any invariant of framed links that doesn’t change under Kirby moves gives rise to an invariant of closed, oriented $3$-manifolds. Invariants of framed links live, in a very natural sense, in ribbon categories, which are braided monoidal categories with a bit of extra structure. I will explain what all of these terms mean, and I will sketch the construction, due to Reshetikhin and Turaev, of a $3$-manifold invariant that comes from any particularly nice ribbon category.

1 Dec 2010

#### Jared Bass: How to have fun with your credit card

At one time or another, you’ve probably done some sort of construction with a compass and straightedge like dropping a perpendicular or bisecting an angle. We’ll describe all the things you can construct with these tools, and see what you can construct when you’re missing some of them. And then we’ll see exactly what you can do with your credit card.

24 Nov 2010

#### Preston Wake: The Weil Conjectures

The underlying problem of the Weil Conjectures goes back to some of the oldest questions in number theory. The problem is this: count the number of solutions to a polynomial modulo $p$. For example: $x^2-a=0 \mod p$, or $ax^2-by^2 = 1 \mod p$. How many solutions are there? What happens as I change $p$? What if I ask for several equations to hold simultaneously?

The Weil Conjectures (proven by Dwork (1960), Grothendieck (1965), and Deligne (1974)) are the result of thinking of these questions in the world of algebraic geometry and algebraic topology. We state the Weil Conjectures and discuss some of the ideas used to prove them.

17 Nov 2010

#### Josh Grochow: Matrix multiplication & algebraic geometry — a match made in heaven*

Multiplying two matrices is dirt easy, and we all learned how do to it in time immemorial. But it turns out we learned a stupid** way! There are much better ways to multiply matrices. I will discuss some of these better ways, as well as the significant role algebraic geometry and representation theory play in the computational complexity of matrix multiplication. Time permitting, I will discuss how the very same algebro-geometric and representation-theoretic ideas form the core of the only current approach to resolving much bigger questions in complexity, like permanent versus determinant and P vs NP.

* And by “heaven” I may mean “Volker Strassen’s office in Germany in the 1970s.”

** Slow.

10 Nov 2010

#### Ian Shipman: Lefschetz’ Method

One can profit greatly by making a Morse-theoretic analysis of the topology of algebraic manifolds. However it is possible to introduce an algebraic version of a Morse function to understand some of the most interesting elementary aspects of their topology without leaving the universe of algebraic geometry. Starting essentially from scratch, I’ll introduce Lefschetz pencils, and use their level sets to reveal the famous vanishing cycles. If there is time I’ll argue the Lefschetz hyperplane theorem geometrically.

3 Nov 2010

#### Jacob Perlman: Banach–Tarski or How I Learned to Stop Worrying and Love the Axiom

Ladies and Gentlefolk, you are cordially invited to witness a slight of hand of the most sensational sort: the division of one ball into two of equal size using nothing but Euclidean motions and the axiom of choice. Amaze at the non-constructable sets. Learn to perform this trick yourself, guaranteed to confound at parties.*

*Not guaranteed.

27 Oct 2010

#### Kate Turner: Discrete Morse Theory

You hopefully can remember Ben’s talk on Morse theory three weeks ago. Discrete Morse theory is a version of Morse theory for maps from simplicial complexes (constant on each cell). There are many analogous result in discrete Morse theory to its smooth cousin. This talk will discuss a few of them.

20 Oct 2010

#### Daniele Rosso: An unexpected consequence of the Spectral theorem: the Hoffman Singleton theorem

Let $G$ be a $d$-regular (every vertex has $d$ edges) graph with no cycles of length $4$ or less, with the minimum possible amount of vertices $1+d^2$. Then $d$ is one of the folllowing numbers: $1$, $2$, $3$, $7$ or $57$! If you have a soft spot for Grothendieck’s prime, this is the talk for you! Also, if you need ideas for your research, an open problem will be stated.

Disclaimer: This is shamelessly stolen from Lazlo Babai’s REU class.

13 Oct 2010

#### Timur Akhunov: Quasilinear well-posedness of PDE

When solving an ordinary differential equation (ODE) the basic strategy of constructing solutions uses contraction mapping principle on a suitable complete metric space. This method works as long as we look for local solutions (defined on an interval $[0,T]$) and we are ready to allow $T$ to be small. As a payback for this restriction, we are able to establish a number of good properties for the equation. Not only do we get the existence of a solution, but a solution so obtained is unique in the complete metric space above and depends smoothly on the given data. Collectively, existence, uniqueness and smooth dependence is referred to as (smooth) well-posedness.

This basic strategy works surprisingly well for many partial differential equations, particularly “semi-linear” ones, where non-linear terms are of “lower order”. However, when one tries to establish well-posedness for more more non-linear “quasi-linear” initial value problems, where the “top order” terms depend on the solution, it is still often possible to establish existence, uniqueness and even continuous dependence on the data, but one is ofter forced to give up the smooth dependence on data. As a consequence the contraction mapping principle is not directly suitable to study such problems.

In this talk, I want to illustrate the difficulties with the quasilinear equations and methods of their analysis with a simple example.

6 Oct 2010

#### Ben Fehrman: Morse Theory

The tools of Morse theory provide an extraordinary insight into the global topology of smooth manifolds. After developing the theory, we will observe almost as afterthoughts that every smooth manifold has the homotopy type of a CW-complex. And, that the summed indices of a vector field with isolated zeros is necessarily the manifold’s Euler-characteristic. This talk will emphasize the foundations of finite-dimensional Morse theory.

That is, given a manifold and a generic smooth function we will determine the manifold’s topology through a simple analysis of the function’s critical points. The development is concrete. To emphasize this we will show that a compact manifold admitting a generic smooth function with exactly two critical points is necessarily homeomorphic to the sphere. Time-permitting, applications of these ideas to certain infinite-dimensional spaces or to the development of homology theories will be mentioned if only briefly.

29 Sep 2010

#### Tom Church: The prime-generating sequence that couldn’t

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.