**Topological Modeling of Complex Data**

*Wednesday, January 10, 2018, 11:10 a.m.- 12:00 p.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Gunnar Carlsson**, Stanford University

One of the fundamental problems faced by science and industry is that of making sense of large and complex data sets. To approach this problem, we need new organizing principles and modeling methodologies. One such approach is through topology, the mathematical study of shape. The shape of the data, suitably defined, is an important component of exploratory data analysis. In this talk, we will discuss the topological approach, with numerous examples, and consider some questions about how it will develop as mathematics.

**Quintessential Quandle Queries**

*Wednesday, January 10, 2018, 2:15 p.m.- 3:05 p.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Alissa Crans**, Loyola Marymount University

Motivated by questions arising in starkly different contexts, quandles have been discovered and rediscovered over the past century. The axioms defining a quandle, an analogue of a group, simultaneously encode the three Reidemeister moves from knot theory and capture the essential properties of conjugation in a group. Thus, on the one hand, quandles are a fruitful source of applications to knots and knotted surfaces; in particular, they provide a complete invariant of knots. On the other, they inspire independent interest as algebraic structures; for instance, the set of homomorphisms from one quandle to another admits a natural quandle structure in a large class of cases. We will illustrate the history of this theory through numerous examples and survey recent developments.

**Information, Computation, Optimization: Connecting the Dots in the Traveling Salesman Problem **

*Thursday, January 11, 2018, 9:00 a.m.- 9:50 a.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**William Cook**, University of Waterloo

Few math models scream impossible as loudly as the traveling salesman problem. Given $n$ cities, the TSP asks for the shortest route to take you to all of them. Easy to state, but if ${\cal P} \neq {\cal NP}$ then no solution method can have good asymptotic performance as $n$ goes off to infinity. The popular interpretation is that we simply cannot solve realistic examples. But this skips over nearly 70 years of intense mathematical study. Indeed, in 1949 Julia Robinson described the TSP challenge in practical terms: ``Since there are only a finite number of paths to consider, the problem consists in finding a method for picking out the optimal path when $n$ is moderately large, say $n = 50$." She went on to propose a linear programming attack that was adopted by her RAND colleagues Dantzig, Fulkerson, and Johnson several years later.

Following in the footsteps of these giants, we use linear programming to show that a certain tour of 49,603 historic sites in the US is shortest possible, measuring distance with point-to-point walking routes obtained from Google Maps. We highlight aspects of the modern study of polyhedral combinatorics and discrete optimization that make the computation feasible. This is joint work with Daniel Espinoza, Marcos Goycoolea, and Keld Helsgaun.

**Changing Mathematical Relationships and Mindsets: How All Students Can Succeed in Mathematics Learning**

*Thursday, January 11, 2018, 11:00 a.m.- Noon* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Jo Boaler**, Stanford University

This talk and discussion will consider how important new brain science can change students’ ideas and approaches to mathematics, change students’ mathematics pathways dramatically, and promote equity in mathematics classrooms. We will hear about research in neuroscience and education, watch classroom videos and consider mathematics transformations for school and college students.

**Tensor Decomposition: A Mathematical Tool for Data Analysis**

*Thursday, January 11, 2018, 11:10 a.m. - 12:00 p.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Tamara G. Kolda**, Sandia National Laboratories

Tensors are multiway arrays, and tensor decompositions are powerful tools for data analysis. In this talk, we demonstrate the wide-ranging utility of the canonical polyadic (CP) tensor decomposition with examples in neuroscience and chemical detection. The CP model is extremely useful for interpretation, as we show with an example in neuroscience. However, it can be difficult to fit to real data for a variety of reasons. We present a novel randomized method for fitting the CP decomposition to dense data that is more scalable and robust than the standard techniques. We further consider the modeling assumptions for fitting tensor decompositions to data and explain alternative strategies for different statistical scenarios, resulting in a _generalized_ CP tensor decomposition.

**Algebraic Structures on Polytopes**

*Thursday, January 11, 2018, 2:15 p.m. - 3:05 p.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Federico Ardila**, San Francisco State University

Generalized permutahedra are a beautiful family of polytopes with a rich combinatorial structure and strong connections to optimization. We study their algebraic structure: we prove they are the universal family of polyhedra with a certain ``Hopf monoid" structure. This construction provides a unifying framework to organize and study many combinatorial families: 1. It uniformly answers open questions and recovers known results about graphs, posets, matroids, hypergraphs, and simplicial complexes. 2. It reveals that three combinatorial reciprocity theorems of Stanley and Billera--Jia--Reiner on graphs, posets, and matroids are really the same theorem. 3. It shows that permutahedra and associahedra ``know" how to compute the multiplicative and compositional inverses of power series. The talk will be accessible to undergraduates and will not assume previous knowledge of these topics.

**Searching for Hyperbolicity**

*Thursday, January 11, 2018, 3:20 p.m. - 4:10 p.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Ruth Charney**, Brandeis University

While groups are defined as algebraic objects, they can also be viewed as symmetries of geometric objects. This viewpoint gives rise to powerful tools for studying infinite groups. The work of Max Dehn in the early 20th century on groups acting on the hyperbolic plane was an early indication of this phenomenon. In the 1980's, Dehn's ideas were vastly generalized by Mikhail Gromov to a large class of groups, now known as hyperbolic groups. In recent years there has been an effort to push these ideas even further. If a group fails to be hyperbolic, might it still display some hyperbolic behavior? Might some of the techniques used in hyperbolic geometry still apply? The talk will begin with an introduction to some basic ideas in geometric group theory and Gromov's notion of hyperbolicity, and conclude with a discussion of recent work on finding and encoding hyperbolic behavior in more general groups.

**Toy Models**

*Friday, January 12, 2018, 9:00 a.m.- 9:50 a.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Tadashi Tokieda**, Stanford University

Would you like to come see some toys?

`Toy' here has a special sense: an object from daily life which can be found or made in minutes, yet which, if played with imaginatively, reveal behaviors that intrigue scientists for weeks. We will explore table-top demos of several such toys, and extract a mathematical story. Some of the toys will be classical but revisited, others will be original, and all will be surprising to mathematicians/physicists and amusing to everyone else.

**Emergent Phenomena in Random Structures and Algorithms**

*Friday, January 12, 2018, 10:05 a.m.- 10:55 a.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Dana Randall**, Georgia Institute of Technology

Markov chain Monte Carlo methods have become ubiquitous across science and engineering to model dynamics and explore large combinatorial sets. Over the last 20 years there have been tremendous advances in the design and analysis of efficient sampling algorithms for this purpose. One of the striking discoveries has been the realization that many natural Markov chains undergo phase transitions whereby they abruptly change from being efficient to inefficient as some parameter of the system is modified, also revealing interesting properties of the underlying stationary distributions.

We will explore valuable insights that phase transitions provide in three settings. First, they allow us to understand the limitations of certain classes of sampling algorithms, potentially leading to faster alternative approaches. Second, they reveal statistical properties of stationary distributions, giving insight into various interacting models, such as colloids, segregation models and interacting particle systems. Third, they predict emergent phenomena that can be harnessed for the design of distributed algorithms for certain asynchronous models of programmable active matter. We will see how these three research threads are closely interrelated and inform one another.

**Minimal Surfaces, Volume Spectrum, and Morse Index**

*Friday, January 12, 2018, 11:10 a.m.- 12:00 p.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**André Neves**, University of Chicago

In the last years there have been several developments relating minimal surface theory with Morse theory and with the volume spectrum introduced by Gromov in the 70’s. I will survey some these developments and explain how a further study could help in solving some well known open problems in Geometry.

**HOW MANY DEGREES ARE IN A MARTIAN CIRCLE? And Other Human - and Nonhuman - Questions One Should Ask About Everyday Mathematics**

*Friday, January 12, 2018, 1:00 p.m.- 1:50 p.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**James Tanton**, MAA Mathematician at Large

Who chose the number 360 for the count of degrees in a circle? Why that number? And why do mathematicians not like that number for mathematics? Why is the preferred direction of motion in mathematics counterclockwise when the rest of world naturally chooses clockwise? Why are fingers and single digit numbers both called *digits*? Why do we humans like the numbers 10, 12, 20, and 60 particularly so? Why are logarithms so confusing? Why is base *e* the “natural” logarithm to use? What happened to the vinculum? (Bring back the vinculum, I say!) Why did human circle-ometry become trigonometry? Let's spend a session together exploring tidbits from the human - and nonhuman - development of mathematics.

**Transforming Learning: Building Confidence and Community to Engage Students with Rigor**

*Saturday, January 13, 2018, 10:05 a.m.- 10:55 a.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Maria Klawe**, Harvey Mudd College

As the first woman and the first mathematician to become president of Harvey Mudd College, I have been delighted to see our departments transform the teaching of rigorous mathematical content in ways that attract and retain female students in mathematics, computer science, engineering and physics. This talk describes the curricular and classroom transformations that have taken place over the last decade and the significant increases in diversity that have occurred as a result. Just in the last three years we have seen graduating classes in computer science, engineering and physics that were more than 50% female. I hope that attendees will leave energized and inspired to experiment in their own departments.

**Political Geometry: Voting Districts, "Compactness," and Ideas About Fairness**

*Saturday, January 13, 2018, 3:00 p.m. - 4:00 p.m.* *Ballroom 6AB, 3rd Level, San Diego Convention Center*

**Moon Duchin**, Tufts University

The U.S. Constitution calls for a census every ten years, followed by freshly drawn congressional districts to evenly divide up the population of each state. How the lines are drawn has a profound impact on how the elections turn out, especially with increasingly fine-grained voter data available. We call a district *gerrymandered* if the lines are drawn to rig an outcome, whether to dilute the voting power of minorities, to overrepresent one political party, to create safe seats for incumbents, or anything else. Bizarrely-shaped districts are widely recognized as a red flag for gerrymandering, so a traditional districting principle is that the shapes should be "compact"—since that typically is left undefined, it's hard to enforce or to study. I will discuss "compactness" from the point of view of metric geometry, and I'll overview opportunities for mathematical interventions and constraints in the highly contested process of electoral redistricting. To do this requires a rich mix of law, civil rights, geometry, political science, and supercomputing.