October 6, 2017
Rafe Jones, Carleton College
Irreducibility and polynomial iteration
In this talk I’ll explore the question of whether a polynomial with rational coefficients that is irreducible over the rationals must remain so under iteration, that is, when the polynomial is composed with itself repeatedly. Spoiler alert: the answer is no. I’ll then discuss to what extent the iterates can factor. The polynomial will be called eventually stable if the iterates do not factor “too much”: if the number of irreducible factors of the nth iterate remains bounded as n grows. Any polynomial, not just an irreducible one, is a candidate to be eventually stable. I’ll state some far-reaching conjectures about eventual stability and give a proof in a special case. The principal method is a generalization of the Eisenstein criterion.
October 13, 2017
Jedrzej Sniatycki, University of Calgary
On geometric structures in variational problems
The Legendre transformation associates a Hamiltonian system to a variational problem with a single independent variable and a regular first order Lagrangian. The aim of this lecture is to discuss geometric structures associated to other variational problems. In the case of first order problems with several independent variables, they include De Donder-Weyl theory, multisymplectic structure, and transition to infinite dimensional Hamiltonian structure.
For higher order problems with one independent variable, the Legendre-Ostrogradski transformation leads to constrained Hamiltonian systems. I shall illustrate this technique by discussing the problem of elastica. I shall conclude the talk with discussion of the controversy regarding higher order variational problems with several independent variables.
October 27, 2017
Mathew Felton-Koestler, Ohio University
What is socially just mathematics teacher education?
I will discuss my attempts to answer this question in my teaching and scholarship. I have taught mathematics content and methods courses for future elementary and middle school teachers at three institutions and I have worked to integrate equity and social justice in all of my courses. I do so in a variety of ways, but a central part of my work involves designing lessons that connect mathematics to social and political issues. I’ll explore some of the challenges of defining and enacting socially just mathematics teacher education.
November 3, 2017
Quoc Tran-Dinh, University of North Carolina at Chapel Hill
Some recent second-order methods in convex optimization applications
Many statistics and machine learning applications can be cast into a composite convex minimization problem. Well-known examples include LASSO-type, SVM and inverse covariance estimation. These problems are well studied and can efficiently be solved by several state-of-the-arts. Recent development in first-order, second-order, and stochastic gradient-type methods has brought a new opportunity to solve many other classes of convex optimization problems in large-scale settings. Unfortunately, so far, such methods require the underlying models to satisfy some structural assumptions such as Lipschitz gradient and restricted strong convexity, which may be failed to hold or may be hard to check.
In this talk, we demonstrate how to exploit an analytical structure hidden in convex optimization for developing solution methods. Our key idea is to generalize a powerful concept so-called “self-concordance” introduced by Y. Nesterov and A. Nemirovskii to a broader class of convex functions. We show that this structure covers many applications in statistics and machine learning. Then, we develop a unified theory for designing numerical methods. We illustrate our theory through Newton-type and proximal Newton-type methods. We note that the proposed theory can further be applied to develop other methods as long as the underlying model is involved with a “generalized self-concordant structure”. We provide some numerical examples in different fields to illustrate our theoretical development.
November 17, 2017
David Zureick-Brown, Emory University
Beyond Fermat's last theorem
I'll discuss lots of generalizations of FLT (and some underlying intuitions)—for instance, for integers a,b,c ≥ 2 satisfying 1/a + 1/b + 1/c < 1, Darmon and Granville proved the single generalized Fermat equation xa + yb = zc has only finitely many coprime integer solutions; conjecturally something stronger is true: for a,b,c ≥ 3 there are no non-trivial solutions and for (a,b,c) = (2,3,n) with n ≥ 10, the only solutions are the trivial solutions and (±3,-2,1) (or (±3,-2,±1) when n is even).
January 26, 2018
Anita Wager, Vanderbilt University
What makes number talks so joyful?
In a plenary session at the NCTM meetings last year, Amy Parks and I talked about how in considering joy as the zeroth principle in mathematics, we might provide a new perspective on access, equity, and empowerment. I am going to build on this idea by unpacking ways in which Number Talks support joyful engagement in mathematics for children, teachers, and teacher educators. We will discuss how some of the core ideas of Number Talks that promote equity and access are the very aspects of Number Talks that are so joyful.
February 2, 2018
Richard Moy, Willamette University
Factoring polynomials: You mean there's more than the quadratic formula?
Most students learn how to factor polynomials in high school using techniques such as the quadratic formula. However, factoring polynomials becomes more complicated when they have large degree. And what if a polynomial doesn't factor, i.e. it is irreducible? How does one go about showing that a particular polynomial is irreducible? In this talk, we will address these questions and more! (Audience participation will be required. A knowledge of modular arithmetic is recommended though not required.)
February 9, 2018
Benjamin Dalziel, Oregon State University
The mathematics and biology of cities
Humanity recently crossed a particular landmark—over half of us now live in cities. Cities are complex, and, at times, beautiful. Through a range of perspectives, from mathematics to biology, this talk will invite you to consider the surprising ways cities reflect, and determine, the lives of their inhabitants, from viruses, to humans, to ideas.
March 9, 2018
Nathan Kutz, University of Washington
Data-driven discovery of governing equations and physical laws
The emergence of data methods for the sciences in the last decade has been enabled by the plummeting costs of sensors, computational power, and data storage. Such vast quantities of data afford us new opportunities for data-driven discovery, which has been referred to as the 4th paradigm of scientific discovery. We demonstrate that we can use emerging, large-scale time-series data from modern sensors to directly construct, in an adaptive manner, governing equations, even nonlinear dynamics and PDEs, that best model the system measured using modern regression and machine learning techniques. We can also discover nonlinear embeddings of the dynamics using Koopman theory and deep neural network architectures. Recent innovations also allow for handling multi-scale physics phenomenon and control protocols in an adaptive and robust way. The overall architecture is equation-free in that the dynamics and control protocols are discovered directly from data acquired from sensors. The theory developed is demonstrated on a number of canonical example problems from physics, biology and engineering.
April 13, 2018
Mauro Maggioni, Johns Hopkins University
Learning and geometry for stochastic dynamical systems in high dimensions
We discuss geometry-based statistical learning techniques for performing model reduction and modeling of certain classes of stochastic high-dimensional dynamical systems. We consider two complementary settings. In the first one, we are given long trajectories of a system, e.g. from molecular dynamics, and we estimate, in a robust fashion, an effective number of degrees of freedom of the system, which may vary in the state space of then system, and a local scale where the dynamics is well-approximated by a reduced dynamics with a small number of degrees of freedom. We then use these ideas to produce an approximation to the generator of the system and obtain, via eigenfunctions of an empirical Fokker-Planck equation (constructed from data), reaction coordinates for the system that capture the large time behavior of the dynamics. We present various examples from molecular dynamics illustrating these ideas.
In the second setting we only have access to a (large number of expensive) simulators that can return short paths of the stochastic system, and introduce a statistical learning framework for estimating local approximations to the system, that can be (automatically) pieced together to form a fast global reduced model for the system, called ATLAS. ATLAS is guaranteed to be accurate (in the sense of producing stochastic paths whose distribution is close to that of paths generated by the original system) not only at small time scales, but also at large time scales, under suitable assumptions on the dynamics. We discuss applications to homogenization of rough diffusions in low and high dimensions, as well as relatively simple systems with separations of time scales, and deterministic chaotic systems in high-dimensions, that are well-approximated by stochastic diffusion-like equations.
May 4, 2018
Marek Elżanowski , Portland State University
Connection and curvature in solid crystals with defects
The kinematic model of continuous defective crystals in which a state of a smooth solid crystalline structure is defined by prescribing on a body manifold three linearly independent vector fields (a frame), focuses mainly on objects which are invariant under elastic deformations (diffeomorphisms). Such objects, in particular, the dislocation density tensor S, and its derivatives, are viewed as providing a measure of crystal’s defectiveness.
When the dislocation density tensor is material point independent (uniform), the underlining continuum can be viewed as a Lie group acting on itself. The non-uniformity of the distribution of defects implies, on the other hand, that the underlying continuum can be regarded as a non-trivial homogeneous space of the ambient Lie group G the algebra of which is isomorphic to the Lie algebra of vector fields generated by the given frame.
In this presentation, we shall focus on the non-uniform case discussing the relation between the form of the lattice algebra (the Lie algebra induced by the given frame) and the geometry of the said homogeneous space as characterized by the canonical principal connection.
May 11, 2018
Theodore Chao, The Ohio State University
Teaching in the age of Twitter: Utilizing social media technology as a tool for democratizing mathematics teacher education
Like it or not, social media has revolutionized the teaching profession. Today’s most vibrant mathematics teacher discussions are happening on twitter. A quick look at hashtags such as #iteachmath, #elemmathchat, #MTBoS, and #noticewonder yield a wealth of mathematics teachers engaged in idea sharing and rich conversations about their practice. And while this space seems to be filled with fresh ideas and discussions, we as a community of educational researchers must also be cautious and mindful of how social media operates and the dangers of very public conversations. In this research talk, Dr. Chao shares his emerging work utilizing twitter and other social media technologies to empower pre-service mathematics teachers. What affordances do online discussions create for emerging teachers? What are the dangers of social media for new teachers? How can social media technology be utilized as a tool for supporting equity in the mathematics classroom? And how do we, as a field, defend teachers attacked for positioning mathematics as a controversial sociopolitical topic?
May 18, 2018
Özlem Ejder, Colorado State University
Solving polynomial equations
The study of Diophantine equations, that is, the solutions of polynomial equations, has a history which dates to ancient Greece and beyond. In this talk, we will give fun examples of such equations and their solutions focusing mostly on the cubic equations. An elliptic curve is given by a cubic equation. In this talk, we will give a brief history of elliptic curves, the group structure on the set of rational points and their relations to various structures.
May 25, 2018
Peter Veerman, Portland State University
Social balance and the Bernoulli equation
Since the 1940s there has been an interest in the question of why social networks often give rise to two antagonistic factions. Recently a dynamical model of how and why such a balance might occur was developed. This talk provides an introduction to the notion of social balance and a new (and simplified) analysis of that model. This new analysis allows us to choose general initial conditions, as opposed to the symmetric ones previously considered. We show that for general initial conditions, four factions will evolve instead of two. We characterize the four factions, and give an idea of their relative sizes.
June 1, 2018
Nikolay Bliznyuk, University of Florida
A hierarchical Bayesian spatio-temporal model for multi-pathogen transmission of hand, foot, and mouth disease
Mathematical modeling of infectious diseases plays an important role in the development and evaluation of intervention plans. These plans, such as the development of vaccines, are usually pathogen-specific, but laboratory confirmation of all pathogen-specific infections is rarely available. If an epidemic is a consequence of co-circulation of several pathogens, it is desirable to jointly model these pathogens in order to study the transmissibility of the disease. Our work is motivated by the hand, foot and mouth disease (HFMD) surveillance data in China. We build a hierarchical Bayesian multi-pathogen model by using a latent process to link the disease counts and the lab test data. Our model explicitly accounts for spatio-temporal disease patterns. The inference is carried out by an MCMC algorithm. We study operating characteristics of the algorithm on simulated data and apply it to the HFMD in China data set.
June 8, 2018
Mauricio Sadinle, University of Washington
Nonparametric identified methods to handle nonignorable missing data
There has recently been a lot of interest in developing approaches to handle missing data that go beyond the traditional assumptions of the missing data being missing at random and the nonresponse mechanism being ignorable. Of particular interest are approaches that have the property of being nonparametric identified, because these approaches do not impose parametric restrictions on the observed-data distribution (what we can estimate from the observed data) while allowing the estimation of a full-data distribution (what we would ideally want to estimate). When comparing inferences obtained from different nonparametric identified approaches, we can be sure that any discrepancies are the result of the different identifying assumptions imposed on the parts of the full-data distribution that cannot be estimated from the observed data, and consequently these approaches are especially useful for sensitivity analysis. In this talk I will present some recent developments in this area of research and discuss current challenges.