## Invited Speakers - A Closer Look

#### Prestrained Elasticity: Curvature Constraints and Differential Geometry with Low Regularity

Wednesday January 6, 2016, 10:05 a.m.-10:55 a.m., Ballroom 6BC, Washington State Convention Center

Marta Lewicka, University of Pittsburgh

This lecture is concerned with the analysis of thin elastic films exhibiting residual stress at free equilibria. Examples of such structures and their actuations include: plastically strained sheets; specifically engineered swelling or shrinking gels; growing tissues; atomically thin graphene layers, etc. These and other phenomena can be studied through a variational model, pertaining to the non-Euclidean version of nonlinear elasticity, which postulates formation of a target Riemannian metric, resulting in the morphogenesis of the tissue which attains an orientation-preserving configuration closest to being the metric's isometric immersion.

In this context, analysis of scaling of the energy minimizers in terms of the film's thickness leads to the rigorous derivation of a hierarchy of limiting theories, differentiated by the embeddability properties of the target metrics and, a-posteriori, by the emergence of isometry constraints with low regularity. This leads to questions of rigidity and flexibility of solutions to the weak formulations of the related PDEs, including the Monge-Ampere equation. In particular, we observe that the set of $C^{1,\alpha}$ solutions to the Monge-Ampere is dense in $C^0$ provided that $\alpha<1/7$, whereas rigidity holds when $\alpha>2/3$.

Wednesday January 6, 2016, 11:10 a.m.-12:00 p.m., Ballroom 6BC, Washington State Convention Center

Xiao-Li Meng, Harvard University

Statisticians are increasingly posed with thought-provoking and often paradoxical questions, challenging our qualifications for entering the statistical paradises created by Big Data. Questions addressed in this talk include 1) Which one should I trust: a 1% survey with 60% response rate or a self-reported administrative dataset covering 80% of the population? 2) With all the big data, is sampling or randomization still relevant? 3) Personalized treatments---that sounds heavenly, but where on earth did they find the right guinea pig for me? The proper responses are respectively 1) “It depends!," because we need data-quality indexes, not merely quantitative sizes, to determine; 2) “Absolutely!," and indeed Big Data has inspired methods such as counterbalancing sampling to combat inherent selection bias in big data; and 3) “They didn't!," but the question has led to a multi-resolution framework for studying statistical evidence for predicting individual outcomes. All proposals highlight the need, as we get deeper into this era of Big Data, to reaffirm some time-honored statistical themes (e.g., bias-variance trade-off), and to remodel some others (e.g., approximating individuals from proxy populations verses inferring populations from samples).

### AMS Colloquium Lectures

#### Quasirandom Sets, Quasirandom Graphs, and Applications

Wednesday January 6, 2016, 1:00 p.m.-2:00 p.m., Ballroom 6BC, Washington State Convention Center

W. Timothy Gowers, University of Cambridge, UK

See lecture notes here.

In this lecture I shall discuss a few applications of discrete Fourier analysis on finite Abelian groups. I shall also talk about quasirandom graphs, explaining what they are and why they are useful. The two topics are closely related, and I shall explain why. Finally, as a way of motivating certain generalizations of Fourier analysis to be discussed in the second and third lectures, I shall give examples of problems that do not yield to the basic technique discussed here.

#### Arithmetic Progressions of Length 4, Quadratic Fourier Analysis, and 3-Uniform Hypergraphs

Thursday January 7, 2016, 1:00 p.m.-2:00 p.m., Ballroom 6BC, Washington State Convention Center

In this lecture I shall say something about quadratic (and higher-order) Fourier analysis, which relates to notable results such as Szemer\’edi's theorem and the Green-Tao theorem. I shall also discuss a notion of quasirandomness for hypergraphs and show that it relates to quadratic Fourier analysis in a similar way to the way that quasirandom graphs relate to conventional Fourier analysis.

I shall also discuss the more general question of what one would ideally like from a generalization of Fourier analysis. Quadratic Fourier analysis has enough of the desired properties to be a useful technique, but there are certain properties that it lacks, at least in its current form, and there are therefore interesting challenges for future research.

Some parts of this lecture will be hard to understand by people who have not attended the first lecture, but I will try to recap the most important ideas. This lecture will, however, not be necessary for following the third.

#### Fourier Analysis on General Finite Groups

Friday January 8, 2016, 1:00 p.m.-2:00 p.m., Ballroom 6BC, Washington State Convention Center

The first two lectures in this series will be about Fourier analysis and generalizations that apply to scalar-valued functions on finite Abelian groups. This one will be about how it can be generalized in two further directions: to non-Abelian groups and to matrix-valued functions. An obvious example of a matrix-valued function on a group is a representation, and indeed basic representation theory plays a central part in these generalizations. I shall give examples of how non-Abelian Fourier analysis can be used to solve interesting problems at the intersection of combinatorics and group theory. I shall also mention connections between some of these problems and the notion of quasirandom graphs from the first lecture.

#### Singing Along with Math: The Mathematical Work of the Opera Singer Jerome Hines

Wednesday January 6, 2016, 2:15 p.m.-3:05 p.m., Ballroom 6BC, Washington State Convention Center

T. Christine Stevens, American Mathematical Society

For over forty years, Jerome Hines (1921-2003) sang principal bass roles at the Metropolitan Opera in New York and in opera houses around the world. He was also a math major who retained a lifelong interest in mathematics. During the 1950's Hines published five papers in Mathematics Magazine that were based on work that he had done as a student, and he later produced several lengthy mathematical manuscripts about cardinality and infinite sets. I will discuss some of Hines' mathematical work, as well as the way in which his undergraduate experience at UCLA converted him from a student with no particular liking for mathematics into an aspiring mathematician. I also hope to explore the question of what mathematics meant to Hines and why, in the midst of demanding musical career, he felt it important for him to develop and publish his mathematical ideas.

#### Mathematics and Policy: Strategies for Effective Advocacy

Wednesday January 6, 2016, 3:20 p.m.-4:10 p.m., Ballroom 6BC, Washington State Convention Center

Katherine D. Crowley

One day in the United States Senate, a team of political staffers took a spontaneous break from writing legislation to request combinatorial proofs on demand of their favorite mathematical identities from their mathematician colleague (me). As the barrage of job demands implored us to disperse moments later, our legislative director chided me for sneaking in the final answer by induction. What is the level of understanding of mathematics among those who craft our national policies? What impact does a mathematician have in a seat at the table of debate over our country's most pressing challenges? How can mathematicians inform policy, and how can policy support mathematics? I will discuss the elements of effective advocacy for our discipline.

### AMS Josiah Willard Gibbs Lecture

#### Graphs, Vectors, and Matrices

Wednesday January 6, 2016, 8:30 p.m.-9:30 p.m., Ballroom 6BC, Washington State Convention Center

Daniel Alan Spielman, Yale University

I will explain how we use linear algebra to understand graphs and how recently developed ideas in graph theory have inspired progress in linear algebra.

Graphs can take many forms, from social networks to road networks, and from protein interaction networks to scientific meshes. One of the most effective ways to understand the large-scale structure of a graph is to study algebraic properties of matrices we associate with it. I will give examples of what we can learn from the Laplacian matrix of a graph.

We will use the graph Laplacian to define a notion of what it means for one graph to approximate another, and we will see that every graph can be well-approximated by a graph having few edges. For example, the best sparse approximations of complete graphs are provided by the famous Ramanujan graphs. As the Laplacian matrix of a graph is a sum of outer products of vectors, one for each edge, the problem of sparsifying a general graph can be recast as a problem of approximating a collection of vectors by a small subset of those vectors. The resulting problem appears similar to the problem of Kadison and Singer in Operator Theory. We will sketch how research on the sparsification of graphs led to its solution.

### MAA Invited Address - SPEAKER unable to speak.  In his place, Francis Su (Harvey Mudd College) will give a talk on the same topic.

#### Fair Division

Thursday January 7, 2016, 9:00 a.m.-9:50 a.m., Ballroom 6BC, Washington State Convention Center

Steven Brams, New York University

Ideas about fair division, including “I cut, you choose," can be traced back to the Bible. But since the discovery 20 years ago of an $n$-person algorithm for the envy-free division of a heterogeneous divisible good, such as cake or land, interest in fair division has burgeoned. Besides envy-freeness, properties such as equitability, efficiency, and strategy-proofness have been studied, and both existence results and algorithms to implement them will be discussed (some implementations will be shown to be impossible). More recent work on algorithms for the fair allocation of indivisible items, and trades among properties, will be presented. Applications, including those to dispute resolution, will be discussed.

NEW ABSTRACT BY FRANCIS SU: I'll give an overview of the problem of "fair division", whose ideas trace back to antiquity but was perhaps first posed as a mathematical challenge by Steinhaus in 1948.  How do you "cut" a "cake" "fairly"?  All these words must be made precise and that is where mathematics comes in.  I'll show how the problem has attracted ideas from many areas: measure theory, graph theory, game theory, and combinatorial topology, and---a sign that this is an interesting problem---just plain old ingenuity.

### AWM-AMS Noether Lecture

#### The Power of Noether's Ring Theory in Understanding Singularities of Complex Algebraic Varieties

Thursday January 7, 2016, 10:05 a.m.-10:55 a.m., Ballroom 6BC, Washington State Convention Center

Karen E. Smith, University of Michigan

In one of the tremendous innovations of twentieth century mathematics, Emmy Noether introduced the rigorous definition of commutative rings and their homomorphisms. One of her main motivating examples was the ring of polynomial functions on a complex algebraic variety. The algebraic study of these rings can have deep geometric consequences for the corresponding variety. In this talk, I hope to explain one example of this phenomenon: namely, how reduction to prime characteristic can give us insight into the singularities of the corresponding algebraic variety. Of course, I will need to convince you that we gain something powerful in reducing modulo $p$, since we have given up all the tools of analysis in doing so. What we gain is the Frobenius operator on the ring, which raises elements to their $p$-th powers, and is a ring homomorphism in characteristic $p$. I hope to explain how the Frobenius operator is helpful in understanding the singularities. As an application, I will describe some work with Angelica Benito, Jenna Rajchgot and Greg Muller on the singularities of varieties that arise in the theory of cluster algebras in combinatorics.

#### Chaotic Billiards and Vibrations of Drums

Thursday January 7, 2016, 2:15 p.m.-3:05 p.m., Ballroom 6BC, Washington State Convention Center

Steve Zelditch, Northwestern University

There are two ways to play' on a drum, which we allow to be shaped in any way, for instance as a standard circular drum-head, or as a stadium-shaped drum-head. First, one may play billiards on it, shooting a ball in a straight line that bounces off the sides by the law of equal angles. For a circular drum-head the billiard trajectories are completely predictable, but for the stadium-shaped drum they are chaotic and unpredictable. Second, the drum-head may vibrate in one if its normal modes. To visualize these modes, one sprinkles sand on the drum and watches the sand accumulate on the nodal set, where the drum is not vibrating (Chladni). My talk is concerned with the question, how are billiard trajectories related to nodal lines? What do the nodal lines look like as the frequency of vibration tends to infinity? In particular, what happens if the billiards are chaotic'?

#### Conjugacy Classes and Group Representations

Thursday January 7, 2016, 3:20 p.m.-4:10 p.m., Ballroom 6BC, Washington State Convention Center

David Vogan, Massachusetts Institute of Technology

The conjugacy classes in a group carry a lot of nice information in an easy-to-understand package: conjugacy classes of permutations are classified by their cycle decomposition, and conjugacy classes of matrices by (more or less!) their eigenvalues.

The sizes of conjugacy classes measure how noncommutative the group is.
The representations of a group offer much more information, but in less agreeable packaging: it is not so easy to say even what the representations of a permutation group are, for example.

An idea of Kirillov and Kostant from the 1960s seeks to describe (abstract and mysterious) representations in terms of (concrete and geometric) conjugacy classes. I'll recall what their idea looks like; some of its classical successes; and some ways that it fits into modern mathematics.

#### What Makes for Powerful Classrooms---and What Can We Do, Now That We Know?

We now understand the properties of classrooms that produce powerful mathematical thinkers and problem solvers. The evidence comes mostly but not exclusively from K-12. The question for us: What are the implications for the ways we teach post-secondary mathematics?

#### The $SL(2,\mathbb{R})$ Action on Moduli Space

Friday January 8, 2016, 10:05 a.m.-10:55 a.m., Ballroom 6BC, Washington State Convention Center

Alex Eskin, University of Chicago

I will discuss ergodic theory over the moduli space of compact Riemann surfaces and its applications to the study of polygonal billiard tables. There is an analogy between this subject and the theory of flows on homogeneous spaces; I will talk about some successes and limitations of this viewpoint. This is joint work with Maryam Mirzakhani and Amir Mohammadi.

#### How to Keep Your Genome Secret

Friday January 8, 2016, 11:10 a.m.-12:00 p.m., Ballroom 6BC, Washington State Convention Center

Kristin Estella Lauter, Microsoft Research

Over the last 10 years, the cost of sequencing the human genome has come down to around \$1,000 per person. Human genomic data is a gold-mine of information, potentially unlocking the secrets to human health and longevity. As a society, we face ethical and privacy questions related to how to handle human genomic data. Should it be aggregated and made available for medical research? What are the risks to individual's privacy? This talk will describe a mathematical solution for securely handling computation on genomic data, and highlight the results of a recent international contest in this area. The solution uses “Homomorphic Encryption", based on hard problems in number theory related to lattices. This application highlights the importance of a new class of hard problems in number theory to be solved. ### MAA Lecture for Students #### The Fractal Geometry of the Mandelbrot Set Friday January 8, 2016, 1:00 p.m.-1:50 p.m., Ballroom 6A, Washington State Convention Center Robert Devaney, Boston University In this lecture we describe several folk theorems concerning the Mandelbrot set. While this set is extremely complicated from a geometric point of view, we will show that, as long as you know how to add and how to count, you can understand this geometry completely. We will encounter many famous mathematical objects in the Mandelbrot set, like the Farey tree and the Fibonacci sequence. And we will and many soon-to-be-famous objects as well, like the “Devaney" sequence. There might even be a joke or two in the talk. ### AMS Invited Address #### Ancient Solutions to Parabolic Partial Differential Equations Saturday January 9, 2016, 9:00 a.m.-9:50 a.m., Ballroom 6BC , Washington State Convention Center Panagiota Daskalopoulos, Columbia University Some of the most important problems in geometric partial differential equations are related to the understanding of singularities. This usually happens through a blow up procedure near the potential singularity which uses the scaling properties of the equation. In the case of a parabolic equation the blow up analysis often leads to special solutions which are defined for all time$-\infty<t\le T$, for some$T\le+\infty$. We refer to them as ancient if$T<+\infty\$. The classification of such solutions, when possible, often sheds new insight to the singularity analysis.

We will give a survey of recent research progress on ancient solutions to geometric flows such as the Ricci flow, the Mean Curvature flow and the Yamabe flow. Our discussion will also include other models of nonlinear parabolic partial differential equations.

We will address the classification of ancient solutions to parabolic equations as well as the construction of new ancient solutions from the gluing of two or more solitons.

#### A Mathematical Tour Through a Collapsing World

Saturday January 9, 2016, 10:05 a.m.-10:55 a.m., Ballroom 6BC, Washington State Convention Center