Applied Mathematics Seminar, Spring 2017

## Mondays 4:00 PM - 5:00 PM, LCB 222

• This seminar can be taken for credit: Students can get 1-3 credits by registering to the Applied Math Seminar class Math 7875 Section 010 for Spring 2016. Students should talk to the seminar organizer before taking it for a credit. Grading is based on attendance and giving a talk by presenting an applied-mathematics paper (not necessarily your own). Student talks will be appropriately labeled to distinguish them from visitor talks. The seminar organizer is available to review your slides, for dry-runs etc.
• Please direct questions or comments about the seminar (or its class) to Yekaterina Epshteyn (epshteyn (at) math.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizer for information about how to subscribe to this list.

January 9. Special Applied Math Seminar.
Speaker: Ricardo Alonso, Department of Mathematics, PUC-Rio
Title: The 1-D dissipative Boltzmann equation
Abstract: We discuss elementary properties of the 1-D dissipative Boltzmann equation. In particular, we show the optimal cooling rate of the model, give existence and uniqueness of measure solutions, and prove the existence of a non-trivial self-similar profile.

January 13 (Friday). Special Applied Math/Statistics/Stochastics Seminar
Note Time, 4:15pm - 5:15pm and Place LCB 222.

Speaker: Dane Taylor,Department of Mathematics, University of North Carolina-Chapel Hill
Title: Optimal layer aggregation and enhanced community detection in multilayer networks
Abstract: Inspired by real-world networks consisting of layers that encode different types of connections, such as a social network at different instances in time, we study community structure in multilayer networks. We study fundamental limitations on the detectability of communities by developing random matrix theory for the dominant eigenvectors of matrices that encode random networks. Specifically, we study modularity matrices that are associated an aggregation of network layers. Layer aggregation can be beneficial when the layers are correlated, and it represents a crucial step for discretizing time-varying networks (whereby time layers are binned into time windows). We explore two methods for layer aggregation: summing the layers' adjacency matrices and thresholding this summation at some value. We identify layer-aggregation strategies that minimize the detectability limit, indicating good practices (in the context of community detection) for how to aggregate layers, discretize temporal networks, and threshold pairwise-interaction data matrices.

January 18 (Wednesday). Special Applied Math/Statistics/Stochastics Seminar.
Note Time, Wednesday 4pm and Place JWB 335.

Speaker: Wenjing Liao, Department of Mathematics, Johns Hopkins University
Title: Multiscale adaptive approximations to data and functions near low-dimensional sets
Abstract: High-dimensional data are often modeled as samples from a probability measure in $R^D$, for $D$ large. We study data sets exhibiting a low-dimensional structure, for example, a $d$-dimensional manifold, with $d$ much smaller than $D$. In this setting, I will present two sets of problems: low-dimensional geometric approximation to the manifold and regression of a function on the manifold. In the first case we construct multiscale low-dimensional empirical approximations to the manifold and give finite-sample performance guarantees. In the second case we exploit these empirical geometric approximations of the manifold to construct multiscale approximations to the function. We prove finite-sample guarantees showing that we attain the same learning rates as if the function was defined on a Euclidean domain of dimension $d$. In both cases our approximations can adapt to the regularity of the manifold or the function eve when this varies at different scales or locations. All algorithms have complexity $C n\log (n)$ where $n$ is the number of samples, and the constant $C$ is linear in $D$ and exponential in $d$.

January 30. Special Applied Math Seminar.
Speaker: Michele Coti Zelati, Department of Mathematics, University of Maryland College Park
Title: Stochastic perturbations of passive scalars and small noise inviscid limits
Abstract: We consider a class of invariant measures for a passive scalar driven by an incompressible velocity field on a periodic domain. The measures are obtained as limits of stochastic viscous perturbations. We prove that the span of the H1 eigenfunctions of the transport operator contains the support of these measures, and apply the result to a number of examples in which explicit computations are possible (relaxation enhancing, shear, cellular flows). In the case of shear flows, anomalous scalings can be handled in view of a precise quantification of the enhanced dissipation effects due to the flow.

February 3 (Friday). Special Applied Math/Statistics/Stochastics Seminar. Note Time is 3pm - 4pm and Room is LCB 219.
Speaker: Dan Shen, Department of Mathematics and Statistics, University of South Florida
Title: PCA Asymptotics & Analysis of Tree Data
Abstract: A general asymptotic framework is developed for studying consistency properties of principal component analysis (PCA). Our framework includes several previously studied domains of asymptotics as special cases and allows one to investigate interesting connections and transitions among the various domains. More importantly, it enables us to investigate asymptotic scenarios that have not been considered before, and gain new insights into the consistency, subspace consistency and strong inconsistency regions of PCA and the boundaries among them. In addition, we studied the asymptotic properties of these sparse PC directions for scenarios with fixed sample size and increasing dimension (i.e. High Dimension, Low Sample Size (HDLSS)). Second we develop statistical methods for analyzing tree-structured data objects. This work is motivated by the statistical challenges of analyzing a set of blood artery trees, which is from a study of Magnetic Resonance Angiography (MRA) brain images of a set of 98 human subjects. The non-Euclidean property of tree space makes the application of conventional statistical analysis, including PCA, to tree data very challenging. We develop an entirely new approach that uses the Dyck path representation, which builds a bridge between the tree space (a nonEuclidean space) and curve space (standard Euclidean space). That bridge enables the exploitation of the power of functional data analysis to explore statistical properties of tree data sets.

February 24 (Friday). LCB 222, 4:15pm - 5:30pm.
Speaker: Gunilla Kreiss, Department of Information Technology, Uppsala University
Title: Error analysis for finite difference methods: What is the relation between the truncation error in a single point, and the global error?
Abstract: In many high order accurate finite difference methods for partial differential equations cases the local truncation error is significantly larger at a few points near boundaries than at interior points. The effect can be analyzed by Laplace transform techniques. In this talk we shall discuss how results from one space dimension can be extended to higher space dimensions. We will in particular consider the second order wave equation.

March 6
Speaker: Graeme Milton, Department of Mathematics, University of Utah
Title: The elasticity tensors of 3d printed materials
Abstract: TBA

March 20
Speaker: Saverio Spagnolie, Department of Mathematics, University of Wisconsin-Madison
Title: TBA
Abstract: TBA

March 22 (Wednesday).
Note Time, Wednesday 1pm - 2pm and Place JFB B-1.

Speaker: Nick Trefethen, Department of Mathematics, University of Oxford
Title: TBA
Abstract: TBA

April 10.
Speaker: Mark Allen, Department of Mathematics, BYU
Title: TBA
Abstract: TBA

May 15
Speaker: Dmitriy Leykekhman, Department of Mathematics, University of Connecticut
Title: TBA
Abstract: TBA

Seminar organizer: Yekaterina Epshteyn (epshteyn (at) math.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148