# Colloquia

## 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.

## Dr. Xin He 7/10/14

Some Graph-Based Models

and Algorithms in Genomics

Date: 7/10/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Dr. Xin He

Abstract:

Graph theory has a number of applications in bioinformatics and genomics

research. In this talk, I will discuss two recent problems that can

formulated in graphical theoretical terms. Next generation sequencing (NGS)

technology greatly reduces the cost of DNA sequencing. Such NGS data

typically consists of a large number of short "reads" of length a few

hundred base pairs or less, and how to assemble these reads to the full

genomic sequences is a challenging combinatorial problem. I will review

some of the algorithms that have been proposed for this problem, which are

essentially finding Hamiltonian or Eulerian path in a graph. In practice,

the problem is made harder by the errors in sequencing, repeats in genomic

sequences, and so on. In the second part of my talk, I will discuss a

problem of mapping genes that contribute to human diseases. Existing

studies tend to analyze one gene at a time through association studies,

which test if the variation of DNA sequences in a gene correlates with the

status of a disease (whether a person is affected or not). Genomic data, on

the other hand, creates a map of how genes are related to each other. I

will discuss some of our attempts at integrating association data (at the

level of individual genes) and gene network data. Such an approach

effectively identifies highly related gene groups (dense subgraphs) that

are most likely to contribute to diseases.

## Professor Jerrold Griggs 5/14/2014

Symmetric Venn Diagrams and

Symmetric Chain Decompositions

Date: 5/14/2014

Time: 3:00PM-4:00PM

Place: 315 Armstrong Hall

Jerrold Griggs

Abstract: Here

## Professor Gheorghe Craciun 5/2/2014

Date: 5/2/2014

Time: 2:30PM-3:30PM

Place: 315 Armstrong Hall

Gheorghe Craciun

Abstract: Complex interaction networks are present in all areas of biology, and

manifest themselves at very different spatial and temporal scales. Persistence,

permanence and global stability are emergent properties of complex networks, and

play key roles in the dynamics of living systems.

Mathematically, a dynamical system is called persistent if, for all positive

solutions, no variable approaches zero. In addition, for a permanent system, all

variables are uniformly bounded. We describe criteria for persistence and permanence

of solutions, and for global convergence of solutions to an unique equilibrium, in a

manner that is robust with respect to initial conditions and parameter values.

We will also point out some connections to classical problems about general

dynamical systems, such as the construction of invariant sets and Hilbert's 16th

problem.

## Professor Xujing Wang 5/1/14

Date: 5/1/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Xujing Wang

Abstract:

Living systems are characterized by complexity in structure and emergent

dynamic orders. In many aspects the onset of a chronic disease resembles a phase

transition in a complex dynamic system: quantitative changes accumulate largely

unnoticed until a critical threshold is reached, which causes abrupt qualitative changes

of the system. This insight can help us identify the key factors driving the

disease, and the critical parameters that are predictive of disease onset. In this

study we investigated this ideas in a real example, the insulin-producing

pancreatic islet ?-cells and the onset of type 1 diabetes. Within each islet, the ?-cells are

electrically coupled to each other. This intercellular coupling enables

the ?-cells to synchronize their insulin release, thereby generating the multi-scale

temporal rhythms in blood insulin that are critical to maintaining blood glucose

homeostasis. Using percolation theory we show how normal islet function is

intrinsically linked to network connectivity. Percolation is a geometric phase transition of

network structure that has profound impact to the function and dynamics of the

network. We found that the critical amount of ?-cell death at which the islet cellular

network loses site percolation, is consistent with laboratory and clinical

observations of the threshold loss of ?-cells that causes islet functional failure. In

addition, numerical simulations confirm that the islet cellular network needs to be

percolated for ?-cells to synchronize. Furthermore, the interplay between site

percolation and bond strength predicts the existence of a transient phase of islet functional

recovery after onset and introduction of treatment, potentially explaining

a long time mystery in the clinical study of type 1 diabetes: the honeymoon

phenomenon. Based on these results, we hypothesized that the onset of T1D may be the

result of a phase transition of the islet ?-cell network. We will further discuss the

potential applications, as well as the importance of such approaches in the study of

complex networks in living systems.

## Professor Martha Alibabli 4/25/2014

Date: 4/25/2014

Time: 3:30PM-4:30PM

Place: 422 Armstrong Hall

Martha Alibabli

Abstract:

Teachers use a range of modalities to communicate in the classroom. In this talk, I focus on how teachers use gestures during instructional communication in mathematics. The bulk of the data are drawn from a corpus of middle school mathematics lessons covering a range of topics. I focus specifically on the role of teachers’ gestures in their communication in two discourse contexts: (1) segments of the lessons in which teachers attempt to connect ideas, concepts, or procedures, and (2) teachers’ responses to "trouble spots" in the classroom discourse (i.e., points where students do not understand or misunderstand the instructional material). The data reveal that gesture is an integral part of classroom communication, and that teachers adjust their gestures depending on lesson content and students’ needs.

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