YOU CAN DO ANYTHING YOU SET YOUR MIND TO

MIDGE COZZENS, RUTGERS UNIVERSITYUNIVERSITY OF NEBRASKA CONFERENCE ON

WOMEN IN THE MATHEMATICAL SCIENCESFEBRUARY 1, 2020

LIFE TAKES YOU WHERE IT WANTS YOU TO GO!

23rd grade

High school1st teaching job

College

1st faculty position

BELIEVE IN YOURSELF

• Graduate school seemed out of reach, but it wasn’t

• Rutgers, among other universities, offered me a teaching assistantship in 1965, then brutalized me

• I met my husband and only one of us could continue to a Ph.D; he continued

• Two children and 10 years later, I went back to graduate school

YOU WILL LEARN WHAT YOU NEED TO KNOW

• Encouraged by my husband, the goal now was to get a Ph.D as quickly as possible

• I met Fred Roberts, who showed me some interesting, unsolved applied discrete math problems, mostly in graph theory

• My second passion was discovered• Ecology, psychology, computer science all

had interesting mathematics problems

1977

5

One of the key goals of conservation biologists is to determine the critical factors of habitat formation.

HABITAT FORMATION

MODELING PREDATOR-PREY RELATIONSHIPS WITH FOOD WEBS

Food webs, through both direct and indirect interactions, describe the flow of energy through an ecosystem, moving from one organism to another.

Let the vertices of a directed graph be species in an ecosystem. Include an arc from y to x if x preys on y. There are no cycles.

SIMPLE FOOD WEB

FOOD WEB OF WOLVES IN YELLOWSTONE NATIONAL PARK

COMPETITION GRAPHS

• We will try to derive the “dimensions” of habitat formation starting from properties of ecosystems, in particular normal, healthy competition between species.

• Using food webs and competition graphs• Arose from a problem of ecology• Joel Cohen 1968

• Key idea: Two species compete if they have a common prey.

CONSTRUCT COMPETITION GRAPHS

• For a digraph D = (V, E), its corresponding competition graph: G is an undirected graph with vertex set V and there is an edge between two vertices if and only if they share a common prey.

• Simple food web:

FOOD WEB OF WOLVES IN YELLOWSTONE NATIONAL PARK

COMPETITION GRAPH FOR WOLF FOOD WEB

INTERVAL GRAPH

• A key idea in the study of competition graphs is the notion of interval graph. It arose from a problem in genetics posed by Seymour Benzer.

• Benzer’s Problem (1959): How can you understand the “fine structure” inside the gene without being able to see inside?

We need to find intervals on the line that have the same overlap properties:

Given a graph, is it an interval graph?

a b

c de

c

a b d e

The following is not an interval graph. Once we give intervals for a, b, c, y and z there is no

room for x without overlapping b.

ya b c

a

b c

x

y zz

Consider the graph C4.

It is not an interval graph.

. a

bc

a b

cd

G = C4

HARD WORK DRIVES SUCCESS

• What do we wish we knew and how can we come to know it?

• Look to others for guidance and to articulate open problems

• You don’t need to solve problems by yourself –often the image of mathematicians

• Helpers come from unusual places

OPEN QUESTIONS

• How do researchers think about open questions?• Who can think about open questions?• One big example: Is the competition graph of a

food web always an interval graph?• If the answer is no, what are the conditions on a

food web so its competition graph is an interval graph?

BRAINSTORM IDEAS

• Characteristics of the community• Predator – prey relationships• Implications of a forbidden subgraph in ecology – can

such predator-prey relationships exist to cause it?• Graph theory equivalents – order maximal cliques - very

few cliques generated in competition graphs – easier to order

• Does transitivity play a part: when A competes with B and B competes with C, does A compete with C?

1980 TO 2016

• Passion for teaching and for solving applied problems continue in parallel

• Northeastern University – tenure, promotion, and chair of a large department – solved a satellite placement problem

• National Science Foundation – program officer for less than a year, then K-12 Division Director

• Research and teaching become one and I am named Distinguished Research Professor

ADMINISTRATION CALLS AGAIN

• Provost and Vice Chancellor for Academic and Student Affairs at CU-Denver/Health Sciences 1998

• President of the Colorado Institute of Technology 2002

• Teaching continued – Graduate Game Theory class taught each year, and developed a calculus online course

WHAT DO WE NEED TO SOLVE THE EVER PRESENT FOOD WEB PROBLEM?

• More examples of food webs – real examples, not artificial examples. Joel Cohen had a book of real examples – but were they real?

• Experts in ecology, including my daughter, a conservation biologist

• What has been learned since the 60s and 70s?

YOU WILL SEEK OUT OTHERS TO HELP YOU ACHIEVE

• Pratik Koirala, an REU student from Howard, working with me the summer of 2016 – he says we can solve the food web problem

• Re-enter my daughter, now a conservation biologist that worked for the Greater Yellowstone Coalition and the Nature Conservancy, and now executive director of the Kassisi Project, a Ugandan conservation NGO

OPTIONS: POSSIBLE WEIGHTINGS

• Weight the edge of a competition graph by the number of predators in common.

• Weight the arcs of a food web with the proportion of the diet consumed

• Set a threshold, for example the diet has to be more than 20% of a prey.

WEIGHT EDGES BY PROPORTION OF DIET – GO BACK TO ECOLOGISTS

• wolf eats .3 bison, .5 elk, .1 bighorn sheep and .1 beaver• bear eats .03 pronghorn, elk .05, deer mouse .02, pine .3

moths .2, berries .4, an omnivore.• coyote eats pronghorn .3, elk .4, deer mouse .2, grasses .1• big horn sheep eats grasses .9 and willow .1• beaver eats pond lily.5 and pine .5• elk eats grasses .4, pond lily .3, pine .3• All others have a single food source illustrated in the food

web.• Mark as red arcs those that are .2 or less.

REMOVE RED EDGE AND NOW AN INTERVAL GRAPH

ARE WE DONE?

• Not yet!• We need to create a theorem that says:

• If a food web indicates all prey that constitute more than 20% of a predator’s diet, then the corresponding competition graph is an interval graph. And then prove it.

• Pratik proved it using the consecutive ones property for interval graphs; I used the forbidden subgraph characterization.

"We make our world significant by the courage of our questions and by the depth of our answers.“

- Carl Sagan

THANK YOU!