# Colloquia

## Lijiang Wu 2/4/2015

Non-local Interaction Equations in Heterogeneous Environment With Boundary

Date: 2/4/2015

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Lijiang Wu

Abstract:We discuss biological aggregation in a heterogeneous environment and on noncovex, nonsmooth domains. We model the heterogeneous environment as a Riemannian manifold with boundary and develop gradient flow approach in the space of probability measures on the manifold endowed with Riemannian 2-Wasserstein metric. We use the gradient flow structure to show the well-posedness of a class of nonlocal interaction equations. We discuss how heterogeneity of the environment leads to new dynamical phenomena. We also present a result on generalizing the well-posedness of weak measure solutions to a class of nonlocal interaction equations on nonconvex and nonsmooth domains. We use particle approximations, solve the discrete ODE systems and pass to the continuum limit by stability property. The novelty here is that under mild regularity conditions on space(i.e. prox-regularity), we can show the well-posedness of the ODE systems and the stability properties with explicit dependence on the geometry of the space(prox-regular constant). The talk is based on collaborations with J. A. Carrillo and D. Slepcev

## Seyfi Turkelli 1/14/2015

Expander graphs and

arithmetic applications

Date: 1/14/2015

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Professor Seyfi Turkelli

Abstract: In this talk, I will first introduce expander graphs, gonality of curves and the arithmetic objects that we are interested in. Then, I will talk about several recent expansion results and their applications to number theory.

## RUME Seminar 12/4/2014

Viewing teaching and learning mathematics in K-12 schools through the lens of the Common Core State Standards

Date: 12/4/2014

Time: 4:00PM

Place: 315 Armstrong Hall

Johnna Bolyard, Associate Professor of Mathematics Education

Matthew Campbell, Assistant Professor of Mathematics Education

Abstract:

The Common Core State Standards for Mathematics put forth the recommended mathematical content and discipline-specific proficiencies to guide K-12 education. In the midst of what has become a hot-button political issue around assessment, school funding and accountability, and government control in schools, the Standards themselves provide the next step in a long line of research and policy on students' mathematical development and the educational environments that promote that development. In this talk we will provide some background on the Common Core Standards using examples from elementary and high school, highlighting both the content of the Standards as well as the focus on "mathematical practice". We will discuss implications of these new standards on K-12 mathematics teaching, teacher development and support, and on students' preparedness for college mathematics and mathematics-focused careers.

## David Swigon 11/6/2014

Dynamics of price and wealth in a

multi-group asset flow model

Date: 11/6/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Professor David Swigon

Abstract: A recently developed model of asset flow dynamics allows one to use

the tools of nonlinear dynamics to study various market conditions and

trading scenarios. I will present the results of two such studies. In the

first we focused on the stability of market equilibria in cases in which

investor groups follow commonly used trading strategies, such as

fundamentalist or trend-based. We show that a market comprised of

fundamental traders is always stable while the presence of trend-based (also

called momentum) traders destabilizes market equilibria and potentially

leads to market bubbles or flash crashes. In the second study we analyzed

the constant rebalanced portfolio (CRP) strategy, in which investor divides

his wealth equally between different types of assets and maintains those

proportions constant as the price changes. We show that CRP strategy is

optimal in that it minimizes the potential losses incurred during price

fluctuations in the market. We also show that any other trading strategy can

be taken advantage by other investors in the market and lead to a loss of

wealth.

## Richard Coultas 11/5/2014

K-Crossing

Free Antichains

Date: 11/5/2014

Time: 3:45PM-4:45PM

Place: 315 Armstrong Hall

Professor Richard Coultas

Abstract:

A set S of vectors in Z^w is an antichain if no vector in S has all coordinates less than or equal to the coordinates of some other vector in S, and is k-crossing free if a certain condition on the maximum difference between corresponding coordinates of different vectors is satisfied. This paper considers how large a k-crossing free antichain of vectors in Z^w can be. The problem is solved for w=3, and some partial results are given for w>3.

## C.S. Aravinda 9/30/2014

Dynamics of geodesic

conjugacies

Date: 9/30/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Professor C. S. Aravinda

abstract:

The question of whether a time-preserving geodesic conjugacy

determines a closed, negatively curved Riemannian manifold up to an

isometry is one of the central problems in Riemannian geometry. While an

answer to the question in this generality has yet remained elusive, we

discuss some available affirmative results.

## Dr. Dejan Slepcev 9/26/2014

Introduction to optimal transportation and Wasserstein gradient flows

Two Different Times.

1. Date: 9/26/2014

1. Time: 10:00AM-10:50AM

1. Place: Mountaineer Room, Mountainlair

2. Date: 9/26/2014

2. Time: 2:35PM-3:25PM

2. Place: Mountaineer Room, Mountainlair

Professor Dejan Slepcev

abstract:

I will cover some of the basics of the theory of optimal transportation and gradient flows in spaces of measures endowed with Wasserstein metric. The goal it to provide the background material for the geometric approaches to the PDE the workshop focuses on. In the first part I will introduce the notion of optimal transport, the Monge and Kantorovich formulations and discuss some properties of the optimal transportation maps. I will then discuss the geometry of the space of probability measures, in particular the McCann interpolation, its connection to pressureless Euler equation and the Benamou-Brenier characterization of the Wasserstein distance.

In the second part I will discuss the notion of a gradient flow in the spaces of probability measures, and in particular the characterization of some PDE like the heat equation, porous medium equation and nonlocal-interaction (a.k.a aggregation) equation as such gradient flows. Finally I will discuss the applications of this viewpoint to existence, uniqueness and stability of the solutions.

## Professor Alexandre Xavier Falcão 9/10/2014

Image Segmentation using

The Image Foresting Transform

Date: 9/10/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Professor Alexandre Xavier Falcão

Abstract:

Image segmentation is a challenging task that consists of an image

partition into either superpixels (regions) that include the

delineation of the desired object borders, or define the precise

spatial extent of such objects in the image. The image foresting

transform (IFT) provides a unified framework to the design of

operators based on optimum connectivity between image elements

(pixels, superpixels, or their components). Its applications include

filtering, segmentation, distance transforms, skeletonization, shape

description, clustering, and classification.

This lecture presents a short overview on the IFT in order to discuss

several aspects related to boundary-based, region-based, interactive,

and automated image segmentation. The IFT algorithms rely on the

suitable choice of an adjacency relation and a connectivity

function. The adjacency relation defines an image graph, whose nodes

are the image elements and arcs connect the adjacent ones. The

connectivity function assigns a value to any path in the graph,

including trivial ones formed by a single node. The maxima (minima) of

the trivial connectivity map are called "seeds" --- they compete among

themselves to conquer their most strongly connected nodes such that

the image is partitioned into an optimum-path forest rooted at the

winner seeds. The image operators result from the attributes of the

forest (optimum paths, their connectivity values, root labels).

The lecture shows how to extend the IFT to the design of pattern

classifiers and discusses connectivity functions for boundary-based

and region-based segmentation, seed selection and imposition, oriented

boundaries and regions, and the incorporation of object information

(texture and shape models) into the interactive and automated

segmentation processes. Most IFT-based methods execute in time

proportional to the number of nodes (linear or sublinear time). The

lecture discusses how to correct segmentation errors in sublinear time

(without starting the process from the beginning, even when the image

was processed by other segmentation approach) and how to obtain smooth

object boundaries without loosing consistency in segmentation (i.e.,

the nodes remain connected to their respective roots). Finally, it

concludes with the current research on IFT-based image segmentation.

Short Biography

Alexandre Xavier Falcão is professor at the Institute of Computing,

University of Campinas (UNICAMP), Brazil. He received a B.Sc. in

Electrical Engineering from the Federal University of Pernambuco,

Brazil, in 1988. He has worked in biomedical image processing,

visualization, and analysis since 1991. In 1993, he received a

M.Sc. in Electrical Engineering from UNICAMP. During 1994-1996, he

worked with the Medical Image Processing Group at the Department of

Radiology, University of Pennsylvania, USA, on interactive image

segmentation for his doctorate. He got his doctorate in Electrical

Engineering from UNICAMP in 1996. In 1997, he worked in a research

center (CPqD-TELEBRAS) developing methods for video quality

assessment. His experience as professor of Computer Science started in

1998 at UNICAMP. His main research interests include graph algorithms

for image processing, image segmentation, volume visualization,

content-based image retrieval, mathematical morphology, digital video

processing, remote sensing image analysis, machine learning, pattern

recognition, and biomedical image analysis.

## Dr. Pete Donnell 9/9/14

Monotonicity, nonexpansivity

and chemical reaction networks

Date: 9/9/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Dr. Pete Donnell

Abstract:

There is currently significant research interest into chemical reaction networks (CRNs), largely due to their importance in biochemistry. CRNs are commonly modelled as ODEs. Given an ODE representing a CRN, characterising its possible asymptotic behaviour is in general a difficult task. A significant proportion of real world CRNs exhibit very simple behaviour, for example global convergence to a unique steady state. However, more exotic CRNs such as the Belousov-Zhabotinsky reactions, which can have a stable nontrivial periodic orbit, are not uncommon.

In this talk I will give a brief overview of two areas of theory, monotonicity and nonexpansivity, that can be used to constrain the possible dynamics of an ODE. It is known that a monotone ODE cannot have stable periodic orbits. If a monotone ODE also has a linear first integral, it is possible in some cases to show that it is nonexpansive and that every bounded trajectory converges to a steady state. Examples drawn from chemical reaction network theory will be presented as applications of this result.

## Professor Andreas Brandstädt 9/3/2014

The complexity of the efficient

domination problem

Date: 9/3/2014

Time: 4:00PM-5:00PM

Place: 315 Armstrong Hall

Professor Andreas Brandstädt

Abstract:

Packing and covering problems in graphs and hypergraphs and their relationship

belong to the most fundamental topics in combinatorics and graph algorithms and

have a wide spectrum of applications in computer science, operations research and

many other ﬁelds.

Recently, there has been an increasing interest in graph and hypergraph problems

combining packing and covering properties, and the NP-complete Exact Cover problem

on hypergraphs (asking for a collection of hyperedges covering each vertex exactly

once) is a good example for this. Closely related graph problems are the Efficient

Domination (ED) and Efficient Edge Domination (EED) problems; the ED problem

corresponds to the exact cover problem for the closed neighborhoods of a graph,

and the EED problem on a graph G is the ED problem on its line graph L(G).

We give a survey on previous results, describe some connections to other problems

such as Maximum Weight Independent Set and Minimum Weight Dominating

Set, consider some new cases where the problems are eﬃciently solvable by

using structural properties of graph classes and using closure properties under the

square operation, and we also extend the graph problems to hypergraphs.

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