The Analysis and Applications Seminar will have an organizational meeting on Tuesday, January 28, at 12:30pm in Math 402. If you would like to speak this semester or nominate someone to speak this semester, please attend this meeting or write to the organizers Shankar Venkataramani and Minh Kha.  If you would like to subscribe to seminar announcements, please send an email to: appliedmath@math.arizona.edu

The Quantitative Biology Colloquium will have an organizational meeting on Tuesday, January 28, at 4:00 pm in Math 402. If you would like to speak this semester or nominate someone to speak this semester, please attend this meeting or write to the organizers Joe Watkins or Tim Secomb.  If you would like to subscribe to seminar announcements, please send an email to: appliedmath@math.arizona.edu

Texts on ordinary differential equations typically present the picture of a vector field in the plane. At a zero of a vector field something interesting happens; eigenvalues indicate the nearby behavior of the vector field. This talk presents the corresponding story for differential forms in the plane. There is a simple and useful picture of a differential form, but it is quite different from that for a vector field. There are also numerical invariants at a zero of a differential form, but they belong to quite another chapter of linear algebra. Nevertheless, they have an interesting practical interpretation.

The Calculus Concept Inventory (CCI) was developed in 2013 to measure basic constructs from differential calculus. The Calculus 1 Concept Inventory (C1CI) was developed in 2016 to compare students' performance on pre/post tests for Calculus 1 in project DIRACC (Developing and Investigating a Rigorous Approach to Conceptual Calculus). This month I joined a panel of high school, community college, and university calculus instructors (led by a team from Texas Tech University) to combine and revise these two assessments. In this hour we'll see how we fare at the panel's charge: combine the two exams and rephrase the questions so that they can be used as a PRE and post assessment in Calculus 1.

With the explosive production of digital data and information, data-driven methods, deep neural networks (DNNs) in particular, have revolutionized machine learning and scientific computing by gradually outperforming traditional hand-craft model-based algorithms. While DNNs have proved very successful when large training sets are available, they typically have two shortcomings: First, when the training data are scarce, DNNs tend to suffer from overfitting. Second, the generalization ability of overparameterized DNNs still remains a mystery despite many recent efforts.

When does a candidate have the approval of a majority? How does the geometry of the political spectrum influence the outcome? What does mathematics have to say about how people behave? When mathematical objects have a social interpretation, the associated results have social applications.  We will show how some classical and new mathematics can be used to understand voting in "agreeable" societies.  This talk also features research with undergraduates.

Sperner's Lemma is a combinatorial analog of a famous theorem in topology: the Brouwer fixed point theorem. In this talk, I will trace recent connections, generalizations, and applications of Sperner's lemma to the Nash equilibrium theorem in economics, problem of fair division, and the game of Hex.

Pre-trained transformer networks combine a neural self-attention mechanism with a training objective that is compatible with large unlabeled datasets. Models such as BERT and XL-NET have followed this approach to produce new state-of-the-art results across a wide range of natural language processing tasks. But why exactly these models are so successful is not well understood. In this talk, I will present work that looks more closely at the internals of these models and investigates how they acquire and represent linguistic knowledge.

With the advent of computers in the middle of the 20’th century, through
the remarkable computations of Fermi-Pasta-Ulam (mid 50s) and
Kruskal-Zabusky (mid 60s) it was observed numerically that nonlinear
equations modeling wave propagation asymptotically exhibit as
superposition of “traveling waves” and “radiation”. This has become
known as the “soliton resolution conjecture”. This conjecture, roughly
speaking, says that any reasonable solution to a disperive equation 
eventually resolves into a superposition of a radiation component plus a
finite number of "nonlinear bound states"  or "solitons".  After an
informal introduction to dispersive equations, I will discuss the
soliton resolution and long-time dynamics for the modified Korteweg-de
vries (mKdV) equation with initial conditions in some weighted Sobolev
spaces combing ideas from integrable systems and partial differential
equations (PDEs).

This talk introduces a nonparametric framework for analyzing physiological sensor data collected from wearable devices. The idea is to apply the stochastic process notion of occupation times to construct activity profiles that can be treated as monotonically decreasing functional data.

Whereas raw sensor data typically need to be pre-aligned before standard functional data methods are applicable, activity profiles are automatically aligned because they are indexed by activity level rather than by follow-up time. We introduce a nonparametric likelihood ratio approach that makes efficient use of the activity profiles to provide a simultaneous confidence band for their mean (as a function of activity level), along with an ANOVA type test. These procedures are calibrated using bootstrap resampling. Unlike many nonparametric functional data methods, smoothing techniques are not needed. Accelerometer data from subjects in a U.S. National Health and Nutrition Examination Survey (NHANES) are used to illustrate the approach. The talk is based on joint work with Hsin-wen Chang (Academia Sinica, Taipei).

The group of automorphisms of a compact manifold is a (infinite dimensional) Lie group. In the simplest case of a circle (i.e. closed string), there are well-known structural similarities with the group of automorphisms of a vector space. In the latter case, it is known how to express the group in terms of generating (root) subgroups and relations (basically due to Steinberg, corresponding to the Serre relations for the Lie algebra); these relations are aptly described by Kac as analytic continuations of the relations for the symmetric group (the Weyl group). The point of the talk is to explain that for (various skeletal) subgroups of the group of automorphisms of a circle, the relations disappear (in terms of heuristics, because the Weyl group is trivial). This involves some rudimentary algebraic geometry.

Prerequisites: knowledge of Lie theory would be helpful (to understand the motivation), but knowledge of group theory is all that is needed for understanding the basic results.

Various aspects of standard model particle physics might be explained by a suitably rich algebra acting on itself, as suggested recently by Furey (2015). This talk discusses the statistical behavior of large causal tree diagrams that combine freely independent elements in such an algebra. It is shown that some of the familiar limiting distributions in random matrix theory (namely the Marchenko-Pastur law and Wigner's semicircle law) emerge in this setting as limits of normalized sums-over-paths of non-negative elements of the algebra assigned to the edges of the tree. These results are established in the setting of non-commutative probability. Trees with classically independent positive edge weights (random multiplicative cascades) were originally proposed by Mandelbrot as a model displaying the fractal features of turbulence.   The novelty of the present approach is the use of non-commutative (free) probability to allow the edge weights to take values in an algebra. Potential applications in theoretical neuroscience related to Alan Turing's famous "Imitation Game" paper are also discussed.

Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs using basis functions which are exact solutions of the Partial Differential Equation (PDE), making explicit use of information about the ambient medium. But wave propagation problems in inhomogeneous media is modeled by PDEs with variable coefficients, and in general no exact solutions are available. Generalized Plane Waves (GPWs) are functions that have been introduced, in the case of the Helmholtz equation with variable coefficients, to address this problem: they are not exact solutions to the PDE but are instead constructed locally as high order approximate solutions. We will discuss the origin, the construction, and the properties of GPWs. The construction process introduces a consistency error, requiring a specific analysis.

The first half of the talk will be devoted to the problem of time-domain wave scattering by elastic obstacles. In this situation, part of the incident wave is scattered off the obstacle and part of it excites a perturbation that propagates throughout it according to its physical properties. From the mathematical point of view, this results in two  equations coupled through transmission conditions at the interface of the solid. In order to deal efficiently with the scattered wave in the unbounded domain, I will propose a formulation that couples boundary integral equations at the interface with partial differential equations inside the scatterer. To exemplify the analysis I will consider the case of a linear elastic body and will show extensions to bodies with more complex physical properties.

The second part of the talk pertains an application coming from plasma physics. In axially symmetric magnetic confinement devices, the equilibrium between the magnetic and hydrostatic forces can be formulated in terms of a free boundary problem involving a semi-linear elliptic equation for a scalar potential posed in free space. Given a reactor configuration and the location and intensities of the external coils, the total magnetic field and the location of the plasma have to be determined.

In order to deal computationally with the unbounded domain, an artificial boundary containing the reactor is introduced and the exterior problem is reformulated as an integral equation. The resulting couple integro-differential system must be then discretized and solved. The location and properties of the plasma boundary are heavily dependent on the behavior of the external parameters, such as the  location and current intensities in the coils, material properties of the reactor, etc. all of which are subject to variability. A satisfactory solution must therefore also address and quantify the uncertainty due to the stochasticity in the problem parameters.  

This work has been done in collaboration with Antoine Cerfon (New York University), George Hsiao and Francisco-Javier Sayas (University of Delaware), Nestor Sanchez and Manuel Solano (Universidad de Concepción, Chile), and Howard Elman and Jiaxing Liang (University of Maryland, College Park). 

Exponential random graphs are powerful in the study of modern networks. By representing a complex global configuration through a set of tractable local features, these models seek to capture a wide variety of common network tendencies. This talk will look into the asymptotic structure of weighted exponential random graphs and formulate a quantitative theory of phase transitions. The main techniques that we use are variants of statistical physics. Based on joint work with multiple collaborators.

The current penalized regression methods for selecting predictor variables and estimating the associated regression coefficients in the Cox model are mainly based on partial likelihood. In this paper, an empirical likelihood method is proposed for the Cox model in conjunction with appropriate penalty functions when the dimensionality of data is high. Theoretical properties of the resulting estimator for the large sample are proved. Simulation studies suggest that empirical likelihood works better than partial likelihood in terms of selecting correct predictors without introducing more model errors. The well-known primary biliary cirrhosis data set is used to illustrate the proposed empirical likelihood method.  

This is joint work with Dongliang Wang and Tong Tong Wu.