November 15, 2019
Per-Olof Persson, University of California–Berkeley
High-order discontinuous Galerkin methods for fluid and solid mechanics
It is widely believed that high-order accurate numerical methods, for example discontinuous Galerkin (DG) methods, will eventually replace the traditional low-order methods in the solution of many problems, including fluid flow, solid dynamics, and wave propagation. The talk will give an overview of this field, including the theoretical background of the numerical schemes, the efficient implementation of the methods, and examples of real-world applications. Topics include high-order compact and sparse numerical schemes, high-quality unstructured curved mesh generation, scalable preconditioners for parallel iterative solvers, fully discrete adjoint methods for PDE-constrained optimization, and implicit-explicit schemes for the partitioning of coupled fluid-structure interaction problems. The methods will be demonstrated on some important practical problems, including the inverse design of energetically optimal flapping wings and large eddy simulation (LES) of wind turbines.
November 22, 2019
Jamie Juul, University of British Columbia
Arboreal Galois representations
The main questions in arithmetic dynamics are motivated by analogous classical problems in arithmetic geometry, especially the theory of elliptic curves. We study one such question, which is an analogue of Serre's open image theorem regarding ℓ-adic Galois representations arising from elliptic curves. We consider the action of the absolute Galois group of a field on pre-images of a point α under iterates of a rational map f (that is, points that eventually map to α as we apply f repeatedly). These points can be given the structure of a rooted tree in a natural way. This determines a homomorphism from the absolute Galois group of the field to the automorphism group of this tree, called an arboreal Galois representation.
December 6, 2019
Andrew Bridy, Yale University
Structure and randomness in functional graphs of polynomials over finite fields
Let f be a polynomial with integer coefficients. For a finite field Fp, we form a (directed) graph that describes the action of f on Fp by drawing a vertex for each element of Fp and drawing a (directed) edge between the vertices x and y if f(x)=y. For certain special polynomials like f(x)=xn, the graphs are very structured and easy to describe. For most polynomials, various aspects of their functional graphs resemble the graphs of functions chosen at random. We investigate this relationship and prove that, for some families of polynomials, the number of cycles of any length behaves in a way that is as "random" as possible. This is joint work with Derek Garton.
January 10, 2020
L. Ridgway Scott, University of Chicago
Automated modeling with FEniCS
The FEniCS Project develops both fundamental software components and end-user codes to automate numerical solution of partial differential equations (PDEs). FEniCS enables users to translate scientific models quickly into efficient finite element code and also offers powerful capabilities for more experienced programmers. FEniCS and other automated software are catalyzing a change for PDEs similar to the one that Matlab did for linear algebra.
FEniCS uses the variational formulation of PDEs as a language to define models. We will explain the variational formulations for simple problems and then show how they can be extended to simulate fluid flow. The variational formulation also provides a firm theoretical foundation for understanding PDEs. We argue that combining the theory with practical coding provides a way to teach PDEs, their numerical solution, and associated modeling without requiring extensive mathematical prerequisites. We demonstrate that this approach requires no background in PDEs or finite elements, only multi-variate calculus.
FEniCS also provides a productive platform for research. We will present examples where it has been used to answer questions that would have required months of programming using traditional techniques.
January 24, 2020
Jay Ver Hoef, National Oceanic and Atmospheric Administration
Why all statistical models should be spatial
This talk will contain some philosophical musings on modeling data, and why I think that we should always use spatial models. I begin with a quick review of spatial statistics, drawing connections from probability, inference, and the linear model to Popper's philosophy, Occam's razor, and Neyman-Pearson hypothesis testing. I present the idea that independence is not an appropriate null model on deciding whether or not to adopt a spatial model. However, there are technical issues for spatial statistics, for both very small sample sizes, and very large sample sizes, that have stymied their use. For a long time, sample sizes on the order of 100 to 1000 have been practical limits. For small sample sizes the problems centered on poor estimates of spatial autocorrelation. However, these problems can be mitigated by marginalization of parameter estimates. I show the connection between the t-distribution and MCMC sampling, and how those ideas can be extended to spatial models with small sample sizes. I verify their performance through simulations. For larger data, there are now a plethora of methods. I review some of those methods, and illustrate one that I am developing based on data partitioning. I show how to model tens of thousands of samples in mere minutes, verify the method with simulations, and illustrate it for stream network data. In summary, many technical issues have been solved for spatial models, from small to large sample sizes, and it is time for statisticians and scientists to adopt the more complicated spatial models as default.
February 7, 2020
Kimball Martin, University of Oklahoma
Galois orbits of modular forms
Modular forms are fundamental objects in number theory and arithmetic geometry. There is a natural decomposition of modular forms into Galois orbits, which tells us about rationality properties of modular forms. We will explain some conjectures and results about the sizes of these Galois orbits. This is intimately related to the existence of geometric objects such as elliptic curves and abelian varieties, as well as to zeroes of L-functions.
February 14, 2020
Michelle Stephan, The University of North Carolina at Charlotte
Preliminary thoughts on the critical STEM consciousness of middle grade students
Today’s students will be working in the most data rich world we have ever seen. Technology has made it easy to take millions of pieces of data and analyze them quickly and affordably to make important decisions about people’s lives. Several mathematicians, social scientists, and psychologists have written about the role that mathematics has played in disenfranchising already-marginalized groups. Mathematics educators who are concerned with equity and inclusion have been calling for mathematics curricula that place social justice at the center of mathematics teaching and learning. The few studies that exist typically focus on using local community contexts to engage students in using mathematics to “read and re-write” their world. Spurred by these articles, we have become interested in how we might leverage big data and other mathematical concepts to help students learn to reason ethically with mathematics so that they make moral decisions before implementing the results of their mathematical work. In other words, if we increase middle and high school students’ critical STEM consciousness, can this prevent future unjust uses of mathematics (and other STEM disciplines) so that the world does not need re-writing? We created a series of interview tasks that elicit students’ current views of the role that mathematics and data play in making impactful decisions as well as what ethical principles they draw on when they do so. I will present the preliminary results of interviews of three 8th grade students and outline a trajectory of future research.
February 21, 2020
Shabnam Akhtari, University of Oregon
Representation of integers by binary forms
We will survey some classical facts in Number Theory on approximation of algebraic numbers by rationals. We will talk about how these wonderful results in the field of Diophantine approximation lead to interesting theorems about Diophantine equations. For example, we will see that if F(x,y) is an irreducible binary form with integer coefficients of degree at least 3, then the equation F(x,y) = 1 can have only finitely many solutions in integers x,y. We will conclude by stating more recent results about counting the number of integer solutions of such equations.
February 28, 2020
Youyi Fong, Fred Hutchinson Cancer Research Center
Fast grid search and bootstrap-based inference for two-phase polynomial regression
Two-phase polynomial regression models (e.g. Robison, 1964; Fuller, 1969; Gallant and Fuller,1973; Zhan et al., 1996) are a generalization of two-segmented regression models (e.g. Hinkley 1971), in which the linear segments are replaced by polynomial functions. The point at which phase transition happens is often called threshold or change point and is estimated as part of the model. Such models are widely used in many applied fields today to model nonlinear relationships. Estimation of two-phase polynomial regression models is a non-convex, non-smooth optimization problem. Finding the true maximum likelihood estimator requires grid search, which is very slow if done in a brute force way. Following our previous works on two-segmented regression models estimation (Elder and Fong 2019), we develop fast grid search algorithms for threshold linear regression models with higher order trends and demonstrate their performance. We further develop model-robust confidence intervals for model parameters, as well as pointwise and simultaneous confidence bands for mean functions through Efron bootstrap. We conduct Monte Carlo studies to demonstrate the performance of the proposed methods, and illustrate the application of the models using several real datasets. This is joint work with Hyunju Son.
March 6, 2020
Kamel Lahouel, Johns Hopkins University
Revisiting the tumorigenesis timeline with a data-driven generative model
Cancer is driven by the sequential accumulation of genetic and epigenetic changes in oncogenes and tumor suppressor genes. The timing of these events is not well understood. Moreover, it is currently unknown why the same driver gene change appears as an early event in some cancer types and as a later event, or not at all, in others. These questions have become even more topical with the recent progress brought by genome-wide sequencing studies of cancer. Focusing on mutational events, we provide a mathematical model of the full process of tumor evolution that includes different types of fitness advantages for driver genes and carrying-capacity considerations. The model is able to recapitulate a substantial proportion of the observed cancer incidence in several cancer types (colorectal, pancreatic, and leukemia) and inherited conditions (Lynch and familial adenomatous polyposis), by changing only 2 tissue-specific parameters: the number of stem cells in a tissue and its cell division frequency. The model sheds light on the evolutionary dynamics of cancer by suggesting a generalized early onset of tumorigenesis followed by slow mutational waves, in contrast to previous conclusions. Formulas and estimates are provided for the fitness increases induced by driver mutations, often much larger than previously described, and highly tissue dependent. Our results suggest a mechanistic explanation for why the selective fitness advantage introduced by specific driver genes is tissue dependent.
Special online edition via Zoom
3:15PM Friday, May 8, 2020,
Joseph H. Silverman, Brown University
How Quantum Computers Will Break the Internet, and What Mathematicians Are Doing About It
Abstract: What do internet commerce, online banking, and updates to your phone apps have in common? All of them depend on modern public key cryptography for security. You've probably heard of the RSA cryptosystem, which is used by many internet browsers. Less well known is a digital signature scheme called ECDSA, based on elliptic curves, that is used by many applications including cryptocurrencies. All of these cryptographic systems are doomed if/when someone builds a full-scale operational quantum computer. It hasn't happened yet (as far as we know), but there are vast resources being thrown at the problem, and slow-but-steady progress is being made. So the search is on for cryptographic algorithms that are secure against quantum computers. In this talk I'll discuss a mix of math and history and prognostication centered around the themes of quantum computers and public key cryptography. No background in either of these subjects will be assumed.