# Mathematics Department

## Colloquium

Most Thursday afternoons during the academic year, the Reed College Department of Mathematics hosts a math talk. The talks are directed to our mathematics majors but are usually accessible on a variety of levels.

### 2018-19 Schedule

#### Fall

4:40-5:30pm in Eliot 314 (unless marked otherwise). Directions to Reed.

Aug 30 Meeting with MajorsTopics to be discussed:Faculty evaluation procedureSenior thesis projectGraduate schools…and, whatever else comes up! Photos will be taken of junior & senior mathematics majors for the department bulletin board. All students welcome. Refreshment will be provided. AI is Coming.....David Krueger, University of MontrealHow can we build advanced artificial intelligence (AI) systems that behave as we intend and expect? ("AI alignment") How can we know whether the AI we've built is safe? ("AI safety")  What will happen if it's not?  Although nobody knows if or when we will develop human-level artificial general intelligence (AGI), recent progress in machine learning (ML), in particular deep learning (DL) and reinforcement learning (RL) have led to a massive surge of interest in AI, with billions of dollars going into research with the explicit aim of building AGI.  This has coincided with increased concern over the transformative social impacts of AI technology, most notably the possibility of human extinction ("AI-Xrisk").  I'll give background on machine learning and AI alignment, safety, and Xrisk; and I'll talk a bit about my research in these areas. All are welcome. Free and open to the public.  Refreshment will be provided Summer Research PresentationsReed students will present overviews of research in mathematics, statistics, and computer science they completed during summer 2018: Ira Globus-Harris and Marika Swanberg - Differentially private analysis of variance Simon Couch and Zeki Kazan - Differentially private non-parametric hypothesis testing Maxine Calle - Putting the "k" in curvature: k-plane constant curvature conditions Henry Blanchette - Citations and collaborations among computer systems publications Summer Research PresentationsReed students will present overviews of research in mathematics, statistics, and computer science they completed during summer 2018: Zichen Cui - Minimally intersecting filling pair origamis David Tamas-Parris and Livia Xu - Generators and relations for the equivariant Barratt-Eccles operad Yevgeniya Zhukova - A look at the functors Ext and Tor Nick Chaiyachakorn - De Rham cohomology is singular cohomology: de Rham's theorem Bitcoins and BlockchainsAdam Groce, Reed CollegeCryptocurrencies like Bitcoin attempt to create an electronic "object" that functions as the equivalent of cash, allowing people to carry out electronic transactions without the need for an intermediary (like a bank or credit card company).  This talk will explain what these cryptocurrencies are and the blockchain technology that makes them work.  The focus will be on understanding the really innovative cryptography that makes them function, but we'll also talk about the economic and policy issues that they raise.  A good technical understanding of how cryptocurrencies really work is crucial if one wants to separate their true promise from the groundless hype. Spaces of Commuting MatricesJosé Manuel Gómez, Universidad Nacional de ColombiaIn this talk I will introduce the space of commuting matrices associated to different groups of matrices both with real and complex coefficients. We will consider different explicit examples. We will pay particular attention to the problem of computing the number of path-connected components in such spaces. At the end of the talk I will try to explain the geometric relevance of such spaces and the geometric significance of the number of path-connected components. Inversion generating functions for signed pattern avoiding permutationsNaiomi Cameron, Lewis and Clark CollegeWe consider the classical Mahonian statistics on the set Bn(Σ) of signed permutations in the hyperoctahedral group Bn which avoid all patterns in Σ, where Σ is a set of patterns of length two. In 2000, Simion gave the cardinality of Bn(Σ) in the cases where Σ contains either one or two patterns of length two and showed that |Bn(Σ)| is constant whenever |Σ| = 1, whereas in most but not all instances where |Σ| = 2, |Bn(Σ)| = (n+1)!. We answer an open question of Simion by providing bijections from Bn(Σ) to Sn+1 in these cases where |Bn(Σ)| = (n+1)!. In addition, we extend Simion’s work by providing a combinatorial proof in the language of signed permutations for the major index on Bn(21, ̄2 ̄1) and by giving the major index on Dn(Σ) for Σ = {21, ̄2 ̄1} and Σ = {12, 21}. The main result of this paper is to give the inversion generating functions for Bn(Σ) for almost all sets Σ with |Σ| ≤ . An Introduction to the Bernoulli Numbers, from Pythagoras to PresentEllen Eischen, University of OregonConsider these basic questions: What can we say about whole number solutions to polynomial equations? What about finite sums of powers of whole numbers? Infinite sums of powers of fractions? What about factorizations into primes? In the setting of certain interesting families of examples, fractions called "Bernoulli numbers" unify these seemingly unrelated questions. After an introduction to the Bernoulli numbers, we will explore related developments for these intertwined problems, which lead to central challenges in number theory and beyond. Preperiodic points in complex and arithmetic dynamicsJohn Doyle, Louisiana Tech University The study of complex dynamical systems was begun about a century ago, and interest was renewed in the 1980's by work of Douady, Hubbard, and others. Noting analogies between dynamical systems and various objects in algebraic geometry and number theory, Morton and Silverman began to develop an arithmetic theory of dynamical systems in the early 1990's. I will discuss preperiodic points for polynomial maps, motivated by the problem of counting the number of such points in both the complex and arithmetic settings, and I will survey various results on questions of this type. Using Topological Invariants to Distinguish ObjectsCourtney Thatcher, University of Puget SoundA common question asked in topology is whether or not two objects are the same. One way to try to answer this is by looking at the intrinsic properties of the objects such as the number of pieces, how many holes it contains, and how it is twisted. These properties, known as topological invariants, are shared by spaces that are considered the same, but they may not be able to distinguish between spaces that are different. In this talk, we will take a closer look at a particular invariant known as the Euler characteristic and see how it is used to distinguish 2-dimensional objects (the classification of surfaces). Some additional invariants and classification problems will also be presented. Prime numbers and their biasesStephan R. Garcia, Pomona CollegeWe survey some classical and modern results about prime numbers.  In particular, we highlight remarkable biases displayed by prime pairs that were recently discovered by Pomona undergraduates. Scissors congruence and Hilbert's Third ProblemDaniel Dugger, University of OregonIn 1900 Hilbert posed a (now well-known) list of 23 problems that he thought should guide future research for the next century.  The third problem concerned whether it was possible to chop up a certain tetrahedron (into a finite number of pieces) and reassemble them to form a cube.  The story behind this strange question dates back to Euclid and the early foundations of geometry, and in some form continues to the present day.  This was the first of Hilbert's problems to be solved, by Max Dehn.  The solution hinged on being able to define and manipulate a new type of "number system".  I will describe this story from the beginning and give a sketch of Dehn's work.  The talk will assume little in the way of prerequisites; it can probably be understood by anyone who has taken high school geometry.

#### Spring

4:40-5:30pm in Eliot 314 (unless marked otherwise). Directions to Reed.

Jan 31 Please note change in time.Please note change in location.The Shape of the Quantum RealmSpyridon Michalakis, California Institute of TechnologyTime: 4:30 PMLocation: Vollum Lecture Hall At the turn of the century, a list of thirteen significant open problems at the intersection of math and physics was posted online by the president of the International Association of Mathematical Physics. But with problems such as Navier-Stokes on the list, quick progress seemed unreasonable. Indeed, a decade later, with only one problem partially solved and despite the progress yielding two Fields Medals, the list was all but forgotten. Then, in 2008, as a young mathematician at Los Alamos National Lab, I was tasked with solving the second problem on the list, which asked for a rigorous explanation of the Quantum Hall effect, an important phenomenon in physics with applications to quantum computing and beyond. The solution, which would involve a deep connection between topology and quantum physics, would come a year later. I want to share that journey with you, focusing on insights gained along the way about the relationship of mathematics to physics. How to obtain parabolic theorems from their elliptic counterpartsBlair Davey, City College of New York Experts have long realized the parallels between elliptic and parabolic theory of partial differential equations. It is well-known that elliptic theory may be considered a static, or steady-state, version of parabolic theory. And in particular, if a parabolic estimate holds, then by eliminating the time parameter, one immediately arrives at the underlying elliptic statement. Producing a parabolic statement from an elliptic statement is not as straightforward. In this talk, we demonstrate a method for producing parabolic theorems from their elliptic analogues. Specifically, we show that an $L^2$ Carleman estimate for the heat operator may be obtained by taking a high-dimensional limit of $L^2$ Carleman estimates for the Laplacian. Other applications of this technique will be discussed. Quantum Learning from Symmetric Oracles (or: How to multiply two matrices when you only know one of them)Jamie Pommersheim, Reed CollegeThe study of quantum computation has been motivated, in part, by the possibility that quantum computers can preform certain tasks dramatically faster than classical computers. Many of the known quantum-over-classical speedups, such as Shor's algorithm for factoring integers and Grover's search algorithm, can be framed as oracle problems or concept learning problems.  In one model of concept learning, a student wishes to learn a concept from a teacher by asking questions, called queries, to which the teacher responds. In the interest of efficiency, the student wishes to learn the concept by making as few queries as possible. For any such concept learning problem, there is a corresponding quantum concept learning problem. In the quantum version, the student is allowed to ask a superposition of queries – mathematically, a linear combination of queries – and the teacher answers with the corresponding superposition of the responses.  After making this idea precise, we will examine several concept learning problems and their quantum analogues.  We will discuss recent joint work with former Reed student Daniel Copeland, in which we show how tools from representation theory can be used to precisely analyze any quantum learning problem with sufficient symmetry. What is a perfectoid space and what is it good for?Matthew Morrow, Institut de Mathématiques de Jussieu-Paris Rive GaucheThe subject of p-adic arithmetic geometry is concerned with number theory which is close to’’ a fixed prime number p. There has been incredible progress in recent years, largely thanks to perfectoid spaces: their creation and the development of the subsequent theory was one of the reasons that Peter Scholze was awarded a Fields Medal last year. In the talk I will attempt to explain some of the main ideas around perfectoids and how we use them. Machine Translation: 350 years of progress and new challenges in the connectionist ageJonathan May, University of Southern CaliforniaMethods and mechanisms for algorithmically converting between human languages have been pursued since at least the 17th century, and translation was one of the first applications devised for computers in the post-war era. The availability of large corpora of translation examples and methods for extracting and wrangling corpus statistics in the 1990s led to a proliferation of higher quality output that could be used by anyone. Today's neural network-driven MT systems are far more fluent and flexible than their predecessors but also far more data-hungry, and have opened the door to a series of new challenges and opportunities: Building good-quality translation systems when our training examples are unreliable, when we only have a small number of examples, or when we have no examples at all Building translation systems that behave more like human translators Understanding what our new systems are learning, and characterizing the limits of their learning potential In this talk I'll provide a selective history of machine translation, introduce core methods and ideas of the field, and discuss work that I and others have done to address these new challenge areas. New Points on CurvesDino Lorenzini, University of Georgia Let X/Q be the plane curve defined by a polynomial f(x,y) with coefficients in the field Q of rational numbers. Thus a point (a,b) in the complex plane is on this plane curve iff f(a,b)=0. A rational point on this curve is a point (a,b) where both a and b are rational numbers. It is easy to construct points on this curve which are not rational. For instance, choose any rational number b, and then solve the equation f(x,b)=0 to obtain a point (a,b) on the curve where most often the coordinate a is not a rational number. To any point (a,b), we associate the smallest field extension of Q generated by a and b, and when this field extension Q(a,b) is a finite dimensional Q-vector space, we call its dimension d (the degree of this field over Q). With this definition, a rational point has degree d=1.  In the case of the Fermat curve x^n+y^n=1 with n>2, setting b=2 produces a point (a,b) on that curve with a= n-th root of (1-2^n). The degree of L:=Q(a,b) is then n. So the Fermat curve has points of degree 1 and of degree n. Can one find other points on this curve with degree 1