Introduction Graph Theory Cells Community Connectedness As A Measure of Robustness Dr. Jason Miller Department of Mathematics Truman State University November 17, 2006 J. Miller Department of Mathematics Truman State University Connectedness As A Measure of Robustness
Connectedness as a Measure of Robustness
Nov 01, 2014
Talk to the Kirksville Chapter of Sigma Xi that describes research on describing the vascular structure of networks of HUVEC cells. I also talk a little bit about Truman's mathematical biology program.
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Introduction Graph Theory Cells Community
Connectedness As A Measure of Robustness
Dr. Jason Miller
Department of Mathematics
Truman State University
November 17, 2006
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
About the Talk
1 Introduction
2 Graph Theory
3 Vascular Networks
4 Research Communities
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
What is Graph Theory?
Fundamental Objects
An abstract graph is made up of
nodes, and
edges that connect nodes.
Example
This is the complete graph on 5nodes. Its nodes are most thoroughlyinterconnected.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
What is Graph Theory?
Fundamental Objects
An abstract graph is made up of
nodes, and
edges that connect nodes.
Example
This is the complete graph on 5nodes. Its nodes are most thoroughlyinterconnected.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
What is Graph Theory?
Fundamental Objects
An abstract graph is made up of
nodes, and
edges that connect nodes.
Example
This is the complete graph on 5nodes. Its nodes are most thoroughlyinterconnected.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
What is Graph Theory?
Fundamental Objects
An abstract graph is made up of
nodes, and
edges that connect nodes.
Example
This is the complete graph on 5nodes. Its nodes are most thoroughlyinterconnected.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internetand edge represent neworking that connects the computers.Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodesrepresent intersections. Analysis of such a graph can illuminatehow vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internetand edge represent neworking that connects the computers.Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodesrepresent intersections. Analysis of such a graph can illuminatehow vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internetand edge represent neworking that connects the computers.Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodesrepresent intersections. Analysis of such a graph can illuminatehow vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internetand edge represent neworking that connects the computers.Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodesrepresent intersections. Analysis of such a graph can illuminatehow vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Applications of Graph Theory
Applications
Graphs is used to illuminate questions in ecology, epidemiology,sociology, business, and computer science.
Example (The Internet)
Consider the graph where nodes represent servers on the Internetand edge represent neworking that connects the computers.Analysis of such a graph can illuminate network traffic problems.
Example (Transportation Flow)
Consider the graph where edges represent a roadways and nodesrepresent intersections. Analysis of such a graph can illuminatehow vehicular flow relates to road configuration.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
1
2
3 4
5
My Interest
Graph connectedness is a measure ofrobustness.
Example (Complete Graph, 5 Nodes)
Complete graphs are robust againstlosing nodes.Lose node #5, and the remainingnodes and edges still form a singlenetwork.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
1
2
3 4
5
My Interest
Graph connectedness is a measure ofrobustness.
Example (Complete Graph, 5 Nodes)
Complete graphs are robust againstlosing nodes.Lose node #5, and the remainingnodes and edges still form a singlenetwork.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
1
2
3 4
5
My Interest
Graph connectedness is a measure ofrobustness.
Example (Complete Graph, 5 Nodes)
Complete graphs are robust againstlosing nodes.Lose node #5, and the remainingnodes and edges still form a singlenetwork.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
1
2
3 4
5
My Interest
Graph connectedness is a measure ofrobustness.
Example (Complete Graph, 5 Nodes)
Complete graphs are robust againstlosing nodes.Lose node #5, and the remainingnodes and edges still form a singlenetwork.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
1
2
3 4
5
My Interest
Graph connectedness is a measure ofrobustness.
Example
This graph is not robust againstlosing nodes.Lose node #5, and the remainingnodes and edges form two separatenetworks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Connectedness
1
2
3 4
5
My Interest
Graph connectedness is a measure ofrobustness.
Example
This graph is not robust againstlosing nodes.Lose node #5, and the remainingnodes and edges form two separatenetworks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
A network structure can be encoded into a matrix using nodeadjacency.
Definition (Adjacency Matrix)
The ijth entry of the n × n adjacency matrix A of a graph G is
Aij =
1 if i 6= j and the ith and jth nodes areconnected with an edge
0 otherwise
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
2
3 4
5A =
0 1 1 1 11 0 1 1 11 1 0 1 11 1 1 0 11 1 1 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
2
3 4
5A =
0 1 1 1 11 0 1 1 11 1 0 1 11 1 1 0 11 1 1 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
3
2
4
5A =
0 1 1 1 11 0 1 1 11 1 0 1 11 1 1 0 11 1 1 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
4
2
3
5A =
0 1 1 1 11 0 1 1 11 1 0 1 11 1 1 0 11 1 1 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix of the Complete graph)
1
52
3 4
A =
0 1 1 1 11 0 1 1 11 1 0 1 11 1 1 0 11 1 1 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix)
2
3 4
5
1
A =
0 1 0 0 11 0 0 0 10 0 0 1 00 0 1 0 11 1 0 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix)
2
3 4
5
1
A =
0 1 0 0 11 0 0 0 10 0 0 1 00 0 1 0 11 1 0 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix)
1
3 4
2 5A =
0 1 0 0 11 0 0 0 10 0 0 1 00 0 1 0 11 1 0 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Example (Adjacency Matrix)
1
2
3
5
4
A =
0 1 0 0 11 0 0 0 10 0 0 1 00 0 1 0 11 1 0 1 0
(Note: i 7→ column, j 7→ row)
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of thegraph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly intoanother matrix call a Laplacian matrix whose eigenvalues andeigenvectors give structural information about the graph. We hopeto exploit this information to describe the robustness of vascularnetworks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of thegraph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly intoanother matrix call a Laplacian matrix whose eigenvalues andeigenvectors give structural information about the graph. We hopeto exploit this information to describe the robustness of vascularnetworks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of thegraph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly intoanother matrix call a Laplacian matrix whose eigenvalues andeigenvectors give structural information about the graph. We hopeto exploit this information to describe the robustness of vascularnetworks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of thegraph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly intoanother matrix call a Laplacian matrix whose eigenvalues andeigenvectors give structural information about the graph. We hopeto exploit this information to describe the robustness of vascularnetworks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Theorems on Connectedness
Adjacency
From the matrix, we can deduce much about the structure of thegraph G . For example,
the number of edges that meet at each node (degree)
whether the graph is a single connected object (connectivity)
Spectral Graph Theory
An adjacency matrix for a graph can be tweaked slightly intoanother matrix call a Laplacian matrix whose eigenvalues andeigenvectors give structural information about the graph. We hopeto exploit this information to describe the robustness of vascularnetworks.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients throughdiffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vesselsnear to or inside the tumor. (Some attract host vessel, otherscreate their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? Howcan they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients throughdiffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vesselsnear to or inside the tumor. (Some attract host vessel, otherscreate their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? Howcan they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients throughdiffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vesselsnear to or inside the tumor. (Some attract host vessel, otherscreate their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? Howcan they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients throughdiffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vesselsnear to or inside the tumor. (Some attract host vessel, otherscreate their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? Howcan they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients throughdiffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vesselsnear to or inside the tumor. (Some attract host vessel, otherscreate their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? Howcan they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Vasculogenesis
A tumor, an abnormal growth of tissue, is bad for you.Cancerous tumors are really bad for you.
For cancerous tissue to grow, it need nutrients.
Growth of tumorous tissue that acquire nutrients throughdiffusion is limited; dead inside.
Some tumors can “arrange for” the formation of blood vesselsnear to or inside the tumor. (Some attract host vessel, otherscreate their own vasculature.)
Big Question
What are some of the mechanisms at work that allow this? Howcan they be inhibited?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Angiogenesis
Vessel formation can be good, too.
Example
Wounds heal.
Example
Blood flow reroutes when vessels are blocked (e.g., stroke).
Big Question
What are some of the mechanisms at work that allow this? Howcan they be promoted?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Angiogenesis
Vessel formation can be good, too.
Example
Wounds heal.
Example
Blood flow reroutes when vessels are blocked (e.g., stroke).
Big Question
What are some of the mechanisms at work that allow this? Howcan they be promoted?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Angiogenesis
Vessel formation can be good, too.
Example
Wounds heal.
Example
Blood flow reroutes when vessels are blocked (e.g., stroke).
Big Question
What are some of the mechanisms at work that allow this? Howcan they be promoted?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Background: Angiogenesis
Vessel formation can be good, too.
Example
Wounds heal.
Example
Blood flow reroutes when vessels are blocked (e.g., stroke).
Big Question
What are some of the mechanisms at work that allow this? Howcan they be promoted?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Research Project
Question
How can we effectively measure the effects of promoting orinhibiting vasculogenic or angiogenic processes?
This is a question posed to a group of faculty andundergraduates in 2004 by Robert Baer.
Example (Model system)
Human umbilical vein endothelial cells (HUVEC) self organize intonetworks of vessels.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Research Project
Question
How can we effectively measure the effects of promoting orinhibiting vasculogenic or angiogenic processes?
This is a question posed to a group of faculty andundergraduates in 2004 by Robert Baer.
Example (Model system)
Human umbilical vein endothelial cells (HUVEC) self organize intonetworks of vessels.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Research Project
Question
How can we effectively measure the effects of promoting orinhibiting vasculogenic or angiogenic processes?
This is a question posed to a group of faculty andundergraduates in 2004 by Robert Baer.
Example (Model system)
Human umbilical vein endothelial cells (HUVEC) self organize intonetworks of vessels.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Mathematical Biology Initiative, summer 2004
An NSF training grant in mathematical biology allowed this groupto take an image analytic approach to this question.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
number of junctions
network length
network area
number of meshes
size of meshes
Computer Aided Analysis
How can we get a computer to make these measurementseffectively with a minimum of human direction?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
number of junctions
network length
network area
number of meshes
size of meshes
Computer Aided Analysis
How can we get a computer to make these measurementseffectively with a minimum of human direction?
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature(view 1)
medial axis
meshes
segmented vasculature(view 2)
medial information,nodes
medial graph
newtworkrepresentation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature(view 1)
medial axis
meshes
segmented vasculature(view 2)
medial information,nodes
medial graph
newtworkrepresentation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature(view 1)
medial axis
meshes
segmented vasculature(view 2)
medial information,nodes
medial graph
newtworkrepresentation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature(view 1)
medial axis
meshes
segmented vasculature(view 2)
medial information,nodes
medial graph
newtworkrepresentation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature(view 1)
medial axis
meshes
segmented vasculature(view 2)
medial information,nodes
medial graph
newtworkrepresentation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature(view 1)
medial axis
meshes
segmented vasculature(view 2)
medial information,nodes
medial graph
newtworkrepresentation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature(view 1)
medial axis
meshes
segmented vasculature(view 2)
medial information,nodes
medial graph
newtworkrepresentation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Vascular Networks
Product: Vascular Network Toolkit
raw image
segmented vasculature(view 1)
medial axis
meshes
segmented vasculature(view 2)
medial information,nodes
medial graph
newtworkrepresentation
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Research Groups
Mathematical Biology Initiative, summer 2004
At the same time in 2004, another research group was supportedby the same NSF training grant - statistical habitat suitabilitymodel for Lesquerella filiformis (the MO Bladder-pod).
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved intosomething bigger.
Biweekly Mathematical Biology Seminar, — a life sciencefashion show
Connected more research active biology faculty with moretalented mathematics faculty
Supported the evolution of faculty scholarship in math andbiology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved intosomething bigger.
Biweekly Mathematical Biology Seminar, — a life sciencefashion show
Connected more research active biology faculty with moretalented mathematics faculty
Supported the evolution of faculty scholarship in math andbiology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved intosomething bigger.
Biweekly Mathematical Biology Seminar, — a life sciencefashion show
Connected more research active biology faculty with moretalented mathematics faculty
Supported the evolution of faculty scholarship in math andbiology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved intosomething bigger.
Biweekly Mathematical Biology Seminar, — a life sciencefashion show
Connected more research active biology faculty with moretalented mathematics faculty
Supported the evolution of faculty scholarship in math andbiology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
This small NSF supported pilot program quickly evolved intosomething bigger.
Biweekly Mathematical Biology Seminar, — a life sciencefashion show
Connected more research active biology faculty with moretalented mathematics faculty
Supported the evolution of faculty scholarship in math andbiology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Research-focused Learning Communities in Mathematical
Biology
The next NSF grant (2004) formalized this:
cross disciplinary teams working in 12 month intervals withintensive summer term
academic year seminar
field trips
peer reviewed product – presentations (poster & oral) atnational and international conferences
sending students to interdisciplinary graduate program
courses and a future minor in mathematical biology
Currently, over 9 biology faculty, 10 math & cs faculty, and 3 otherfaculty are actively involved in this community.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to useundergraduate research as a way to
expand the STEM talent pool through high-qualityundergraduate research experiences
bring together research faculty in all STEM areas into a singlesummer community
foster faculty scholarship
Community
Together, the Next STEP and MathBio programs havedramatically increased the connections between faculty andstudents of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to useundergraduate research as a way to
expand the STEM talent pool through high-qualityundergraduate research experiences
bring together research faculty in all STEM areas into a singlesummer community
foster faculty scholarship
Community
Together, the Next STEP and MathBio programs havedramatically increased the connections between faculty andstudents of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to useundergraduate research as a way to
expand the STEM talent pool through high-qualityundergraduate research experiences
bring together research faculty in all STEM areas into a singlesummer community
foster faculty scholarship
Community
Together, the Next STEP and MathBio programs havedramatically increased the connections between faculty andstudents of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to useundergraduate research as a way to
expand the STEM talent pool through high-qualityundergraduate research experiences
bring together research faculty in all STEM areas into a singlesummer community
foster faculty scholarship
Community
Together, the Next STEP and MathBio programs havedramatically increased the connections between faculty andstudents of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to useundergraduate research as a way to
expand the STEM talent pool through high-qualityundergraduate research experiences
bring together research faculty in all STEM areas into a singlesummer community
foster faculty scholarship
This is Truman’s “The Next STEP” program.
Community
Together, the Next STEP and MathBio programs havedramatically increased the connections between faculty andstudents of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Inter-STEM Research community
At the same time, a proposal went into the NSF to useundergraduate research as a way to
expand the STEM talent pool through high-qualityundergraduate research experiences
bring together research faculty in all STEM areas into a singlesummer community
foster faculty scholarship
This is Truman’s “The Next STEP” program.
Community
Together, the Next STEP and MathBio programs havedramatically increased the connections between faculty andstudents of different disciplines.
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Challenges
Sustainability
Conversion of student collaborations to peer reviewed work
Supporting continued faculty scholarship and research
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Challenges
Sustainability
Conversion of student collaborations to peer reviewed work
Supporting continued faculty scholarship and research
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Challenges
Sustainability
Conversion of student collaborations to peer reviewed work
Supporting continued faculty scholarship and research
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and arehelping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, andjoyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm forlearning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and arehelping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, andjoyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm forlearning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and arehelping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, andjoyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm forlearning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and arehelping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, andjoyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm forlearning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
Introduction Graph Theory Cells Community
Community
Acknowledgements
Truman administrative leaders who support this work and arehelping us look for solutions to the challenges
Truman STEM colleagues who have embraced this effort, andjoyfully made connections with others outside their disciplines
Rob Baer and Jim Rhoades
the hundreds of students whose raw talent and enthusiasm forlearning make all this work a joy
Jennifer Thompson, our Program Coodinator
J. Miller Department of Mathematics Truman State University
Connectedness As A Measure of Robustness
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