Math Department Events Listing

We will introduce a familiar problem, that of approximating the Riemann mapping by a piecewise linear map. In particular, we will consider an approach slightly different from the standard circle packing approach, which considers hyperbolic circle packings with boundary horocycles. Instead, we will consider certain types of "folded" triangulations, and consider certain discrete conformal deformations. In the process, we will introduce a natural set of curvatures on these objects. It turns out that these fit in naturally with the notions of curvature on nonconvex sets which are derived from Steiner's formula for convex sets and Weyl's tube formula. We will describe these curvature formulations and see how they fit into our framework for studying approximations of the Riemann mapping. This is mostly work in progress.

We will discuss a control theorem for the p-primary Selmer group of an abelian variety with respect to extensions of the form: maximal pro-p extension of a number field unramified outside a finite set of primes (which do not include any primes dividing p) in which another finite set of primes splits completely. In a case related to the Fontaine-Mazur conjecture, the control theorem will give information about p-ranks of Selmer and Tate-Shafarevich groups.

This meeting is designed especially for junior faculty members who have not yet served as an undergraduate advisor, and for other faculty members who would like an advising update.

Three-dimensional structures of proteins provide the scaffold on which their fine-tuned functionalities are built. The adaptability of the polypeptide chain to take on diverse structures confers its equally diverse functional repertoire. While the overall structure of proteins (the tertiary structure) is usually irregular and complex, it can often be described as a compact arrangement of locally regular and symmetrical elements called secondary structures, the most common of which are the alpha helix and the beta strand. We have developed an approach to exploit the local regularity in such structural elements and thus to represent protein folds efficiently using the differential geometry of space curves. The first part of the talk will describe the problem of constructing a curve model of a known protein structure. We show that the approach is particularly suitable for representing alpha helical folds and we will discuss improvements under development for extending the method to more complex folds.

These space curve models are an abstract representation which makes it possible to explore the diversity of potential protein folds without prior consideration of the amino acid sequence, and a long-term goal is to use this capability for protein design. In the second part of the talk we consider the converse problem of constructing amino acid sequences compatible with a proposed curve model. Finally we will outline an experimental approach whereby these models could be constructed. The gross features of the protein structure will be defined using space curves, combinatorial libraries of compatible protein sequences will be generated, and then these libraries will be screened for a desired function or property, thereby generating the optimally evolved fold for the desired property or function.

I will highlight learning materials for a first course in real analysis that I have developed through interactions with about 150 students in real analysis from 2006 – 2009. The materials lead the students in building and debating the concepts and definitions before they are presented in polished form. Over these eight semesters, metaphors on hawks and penguins emerged, white tigers and moose appeared to quell students' frustrations, and homework problems unfolded into multi-tiered tasks in response to their road-blocks and mistakes. At the University of Texas at Arlington, these materials and teaching strategies have helped undergraduate Analysis I move from a success rate of about 45% up to 81%, in classes made even more challenging by true/false questions deliberately targeted at students' misconceptions. The materials are organized into five components, each with a specific purpose. The Concept Checks are short, focused questions, aimed at potential misunderstandings, to be debated by the students in class. Discovery Exercises are longer sequences of questions that lead the class in formulating definitions, arriving at statements and proofs of theorems, and exploring different viewpoints and applications. The Scaffolded Tasks are multi- tiered investigations that may be given as homework, the Historical Vignettes enhance students’ appreciation and perspective of the ideas, and Capstone Connections highlight relationships between the course material and other areas of mathematics. These activities are being published on the website http://www.uta.edu/faculty/shipman/analysis, supported in part by NSF grant DUE #0837810.

In our graduate careers, we encounter tensors in at least four guises: (1) as presented in abstract-algebra texts; (2) as multilinear functions (e.g. in differential geometry); (3) as multidimensional arrays; (4) as objects which transform in a certain way on change of coordinates (physicists' favorite definition). A bewildered first-year graduate student might well inquire: "Are these four completely different things, all masquerading under the same name?" Fortunately, that bewildered student has survived, more or less intact, into his final year and is able to confidently report that they are in fact ALL THE SAME THING. Wednesday at noon we'll see why, learn how to think about freedom (freeness), and enjoy a bagel or two.

The Lee-Yang theorem is one of the most revisited results in statistical mechanics, especially because of its application on the study of phase-transition phenomena. One consequence of this theorem is to ensure that for any nonzero uniform magnetic field, i.e. h = {h_i}_{i ∈ Z^d}, h_i = h ∈R\{0} for all i in Z^d and β=1/kT, the ferromagnetic (J > 0) Ising model on Z^d has a unique Gibbs measure in the thermodynamic limit, independent of the boundary conditions. In this seminar we will discuss the model without uniformity assumption on the magnetic field h. For such models the Lee-Yang Theorem holds, so it is natural to ask if for those models the Lee-Yang Theorem still imply the absence of phase transition. We will show that the Ising model can have a first-order phase transition even in the case where the magnetic field takes positive values in all sites of the lattice.

Are you looking for something to do this summer? Would you like to improve your math skills while improving your mathematical network? Then join us at our meeting this Wednesday, February 10th, at 5:45pm in the math east lobby. Dr. Young will kick the meeting off by discussing a fun summer experience here in Arizona. Dr. Velez will then talk about how to put together the perfect resume, and about some great REU opportunities. If you already have a resume you can print it out and bring it along- that way you can perfect it!

I will explore a method of estimating the quality of a solution to a stochastic optimization problem. That is, given an optimization problem with random data, and a proposed solution, I will present a method for estimating the distance between the proposed solution and the optimal solution. The talk will include a brief introduction to stochastic optimization, and the two previous results that motivated my study. Some computational results will be given on classic stochastic optimization problems.

Abstract: Under weak conditions of smoothness and mixing, we propose splinebackfitted spline (SBS) estimators of the component functions for nonlinear additive autoregression model that is both computationally expedient for analyzing high dimensional large time series data, and theoretically reliable as the estimator is oracally efficient and comes with asymptotically simultaneous confidence band. Based on newly derived confidence bands, we can test whether a particular parametric form fits a given data set. Simulation evidence strongly corroborates with the asymptotic theory, and a real-date example is used to illustrate our techniques. Key words and phrases: B spline, confidence band, knots, mixing, oracle efficiency.

Iceberg dynamics can be modeled as governed by a force-balance momentum equation, and the presence of sea ice, as well as its properties, can be an important contributing force. In turn, the presence and dynamics of icebergs have an effect on the surrounding sea ice. I will discuss an approach to iceberg modeling that has been implemented in a sea ice model, focusing on the interaction between icebergs and sea ice in the Antarctic. This talk will be accessible to all graduate students, with an abundance of pretty pictures.

We discuss the reconstruction of compactly supported piecewise smooth functions from non-uniform samples of their Fourier transform. This problem is relevant in applications such as magnetic resonance imaging (MRI). We summarize two standard techniques, convolutional gridding and uniform resampling, and address the issue of non-uniform sampling density and its effect on reconstruction quality. We compare these classical reconstruction approaches with alternative methods such as spectral re-projection and methods incorporating jump information.