## Invited Speakers - A Closer Look

Karen Lange
Wellesley College

Different Problems, Common Threads: Computing the Difficulty of Mathematical Problems

Wednesday, January 15, 2020, 11:10 a.m.- 12:00 p.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

Mathematics is filled with existence theorems such as "every vector space has a basis". Such statements do not address how one goes about finding the known-to-exist object. The prior theorem naturally leads to the question "given a vector space, can we compute a basis for it?". The answer to this "Basis Problem" is no, so we say that the problem is not "computable".

We can further ask just how far from computable the Basis Problem is and what other problems have the same computational power. A natural way to compare the algorithmic difficulty of two problems is to determine whether having the ability to solve one allows for the solution of the other. Under this problem-reduction approach, two problems have the same computational power if each can be used to solve the other.

In this talk, we will explore the key ideas behind taking a "computable" perspective on mathematics and how this compares to an existence" one. We will illustrate the problem-reduction approach using theorems from across mathematics. The overall structure of problem---difficulty is extremely rich and helps to illuminate what makes problems "tick". Moreover, this approach has strong connections to calibrating exactly which axioms are needed to prove the original existence theorems.

Chandler-Gilbert Community College

Mazes, Riddles, Zombies, and Unicorns!

Wednesday, January 15, 2020, 2:15 p.m.- 3:05 p.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

Can mathematics be an engaging endeavor worthy of academic pursuit? Can students be involved in meaningful learning experiences? Having observed many classrooms, it seems that the answers are a heartbreaking “No.” Reflect on what often happens in classrooms. Teachers may: unwittingly communicate that mathematics is so mind numbing and senseless, that they must “jazz it up”, tricking students to engage in mathematical activity; hijack thinking by telling students what to know without allowing them opportunities to grapple with ideas; write notes while students passively copy. There is no expectation or opportunity to make sense, to reason, to understand, or to engage in authentic mathematical thinking and discovery.

Reflect on what could happen in classrooms. Imagine experiences where students are: using manipulatives to represent real-world situations; building procedural fluency with algorithms by developing conceptual understanding; struggling productively to make sense of ideas while modeling with mathematics; collaborating as they make their reasoning visible while solving challenging problems.

I will highlight these issues and provide effective strategies for involving students in authentic mathematical thinking and discovery as they joyfully engage in joyful mathematics!

Mohamed Omar
Harvey Mudd College

The Art and Craft of Problem Design

Wednesday, January 15, 2020, 3:20 p.m.- 4:10 p.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

How does one pick the right research problem to work on? How can we create assignment problems that are not decent nor good, but great? How does one make innovative problems for math competitions? These are questions that have been central throughout the speaker’s career, and common threads between them have had surprising influences on each other. Come hear how the art and craft of problem design plays a key role in a mathematical career.

Nancy Reid
University of Toronto

### AMS Josiah Willard Gibbs Lecture

In Praise of Small Data: Statistical and Data Science

Wednesday, January 15, 2020, 8:30 p.m.- 9:20 p.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

The over-promotion of Big Data'' has perhaps settled down, but the data are still there, and the rapid development of the new field of data science is a response to this. As more data become available, the questions asked become more complex, and big data can quickly turn into small data. Statistical science has developed an arsenal of methods and models for learning under uncertainty over its 200-year history. Some thoughts on the interplay between statistical and data science, their interactions with science, and the ongoing relevance of statistical theory, will be presented, and illustrated through a number of examples.

Della Dumbaugh
University of Richmond

Prospering Through Mathematics

Thursday, January 16, 2020, 9:00 a.m.- 9:50 a.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

Solomon Lefschetz, Emil Artin, Gertrude Cox and Rudy Horne all enjoyed successful careers in mathematics. These mathematicians, separated by mathematical discipline, nationality, time, institution and background also all faced---and overcame---hardships. They met the challenges of their human lives, in part, through their mathematical work at particular institutions. But what if we turn this lens around? What if we consider how mathematics created the space for these mathematicians to find success? Mathematics in particular allowed them to prosper, to the point where they had plenty to give to the next generation. Their lives prompt us to take up the natural next question of “how?” How can we use the space of mathematics to help shape students and colleagues into better human beings? And how can we accomplish this lofty goal with increasing demands on our time and rosters of students who may only take a single math class? This talk explores the professional experiences and personal lives of mathematicians to underscore the power of mathematics, not just as a career path, but as a place to grow into a full human being. It also outlines effective strategies for intentionally identifying the power of the discipline to direct students and colleagues toward meaningful lives.

Birgit Speh
Cornell University

### AWM-AMS Noether Lecture

Branching Laws for Representations of a Non Compact Orthogonal Group

Thursday, January 16, 2020, 10:05 a.m.- 10:55 a.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

Finite dimensional representations of classical groups were first treated systematically by Hermann Weyl in his famous book "The Classical Groups: Their Invariants and Representations" 80 years ago. In this book he classified the irreducible representations $\Pi:\ SO(n)\to\rm{Aut}(V )$ for a finite dimensional vector space V. H. Weyl also considered the restriction of an irreducible representation to a subgroup and proved that the restriction of a finite dimensional representation is a direct sum of finite dimensional representations. In 1938, famous branching rules were proved describing the multiplicity dim $\rm{Hom}_{SO(n−1) }(\pi, \Pi)$ for the restriction of an irreducible representation $\Pi$ of $SO(n)$ to an irreducible representation $\pi$ of $SO(n-1)$ by F. Murnaghan.

A noncompact orthogonal group $SO(p, q)$ also has infinite dimensional irreducible representations. Unfortunately the restriction of an infinite dimensional representation of $SO(p, q)$ to a subgroup $SO(r, s)$ is often not a direct sum of irreducible representations. In this talk I will show that for infinite dimensional representations $\Pi$ of $SO(n, 1)$ and infinite dimensional representations $\pi$ of $SO(n − 1, 1)$ we obtain very similar branching laws for dim $\rm{Hom}-{SO(n−1) }(\Pi, \pi)$ although the restriction of the representation is not a direct sum of irreducible representations.

Federico Ardila-Mantilla
San Francisco State University

### MAA Project NExT Lecture on Teaching and Learning

Todxs Cuentan: Difference, Humanity, and Belonging in the Mathematics Classroom

Thursday, January 16, 2020, 11:10 a.m.- 12:00 p.m. Four Seasons Ballroom 1, Lower Level, Colorado Convention Center

Everyone can have joyful, meaningful, and empowering mathematical experiences; but no single mathematical experience is joyful, meaningful, and empowering to everyone. How do we build mathematical spaces where every participant can thrive? Audre Lorde advises us to use our differences to our advantage. bell hooks highlights the key role of building community while addressing power dynamics. Rochelle Gutierrez emphasizes the importance of welcoming students' full humanity. This talk will discuss some efforts to implement these ideas in mathematical contexts, and some lessons learned along the way.

Kenneth A. Ribet
University of California, Berkeley

A 2020 View of Fermat's Last Theorem

Thursday, January 16, 2020, 3:20 p.m.- 4:10 p.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

Abstract TBA

Vilma Mesa
University of Michigan

Instruction and Resources in Post-secondary Mathematics: How their Interplay Shape What We Do in the Classroom

Friday, January 17, 2020, 9:00 a.m.- 9:50 a.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

In this presentation I describe studies I have conducted to investigate how instructors, students, and resources interact in classrooms in order to create opportunities for mathematics learning in post-secondary settings. I showcase the evolution of two apparently independent research strands that together have helped me understand first, the centrality of resource use by instructors and students and its implications for student learning and, second, the complexity of the work that faculty do.

Skip Garibaldi
Institute For Defense Analyses Center for Communications Research, La Jolla

Uncovering Lottery Shenanigans

Friday, January 17, 2020, 11:10 a.m.- 12:00 p.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

Mathematicians since Euler have been interested in studying lotteries. Analyzing the games is made easier because the probabilities involved are clearly defined, yet the very large number of participants and variances in outcomes lead to subtle questions. Focusing on specific games provides an additional source of interest, because each game has individual quirks. When looking at the outcomes of real games, it sometimes appears that people are up to shenanigans. Digging deeper, it may turn out that the apparent anomalies are not nefarious at all, just an illusion resulting from the oceans of tickets that are sold. Sometimes the scheme is real, but, surprisingly, entirely legal. And sometimes it is mostly illegal or an indicator of a larger criminal enterprise. This talk focuses on the latter kind of shenanigans.

Pomona College

### MAA Invited Address - Lecture for Students

On the Scales of One to Infinity: Learning to Listen to Your Mathematics

Friday, January 17, 2020, 1:00 p.m.- 1:50 p.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

Many mathematical constructs can be manifested as sounds! The visual palette is three dimensional, or four if you include color; in some ways, the sonic palette is richer. Our ears can perceive along many axes, including pitch, loudness, timbre, harmonic complexity and time. How could you better understand your mathematical problem by hearing it, as well as seeing it? For example, why is the `harmonic” series called by this name? Can we hear that the harmonic series diverges? Did you know we can “listen” to a dynamical system in order to understand its structure? Some features of common functions are better heard than seen! Together, we will explore the “route to chaos” via graphs and sounds, with live demonstrations.

aBa Mbirika
University of Wisconsin-Eau Claire

Two Research Projects Birthed from Curiosity, Recreation, and Joy

Saturday, January 18, 2020, 10:05 a.m.- 10:55 a.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

This talk will center around two undergraduate research projects that were born from two specific recreational math topics. These topics brought me joy and then suddenly turned into full-blown research. The first topic emerged from a connection between the Fibonacci sequence modulo 10 and astrology. Oh No! Does the speaker believe in astrology!? Don’t worry, this topic will be strictly number theory with, of course, a foundation that gives its connection to astrology (in particular, the zodiac). The second topic arose from noticing the magical and mystic golden ratio appearing as an eigenvalue of a certain tridiagonal real symmetric matrix. Generalizing this matrix to ever-increasing sizes, a wondrous joy is born from the corresponding sequence of characteristic polynomials that emerge. And lo and behold the diagonal entries in Pascal’s triangle appear as the coefficients of these polynomials in an attractively inviting manner. Though the first project is one of number theory, and the second is one of combinatorial linear algebra, a cute connection between the two topics will be revealed at the end of the talk.

Rajiv Maheswaran
Second Spectrum

### MAA-AMS-SIAM Gerald and Judith Porter Public Lecture

The Fantastic Intersection of Math and Sports: Where No One is Afraid of a Decimal Point

Saturday, January 18, 2020, 3:00 p.m.- 3:50 p.m. Four Seasons Ballroom 2, 3, 4, Lower Level, Colorado Convention Center

Stereotypes permeate the perceptions of people who are either athletic or analytical. The disciplines of mathematics and sports have been kept at a distance, yet they share a commonality of deep interest in numbers. In this talk, I’ll detail how the emergence of new kinds of data and novel applications of it are building bridges across two worlds where no one is afraid of a decimal point.

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