Richard Durrett - US grants

This group of researchers will work on various aspects of Markov processes and infinite particle systems. Specific problems include first passage and bond percolation, particle systems in random environments and functionals of Markov processes. Percolation and particle systems in random environment provide many useful models to problems in physics and biology. The tools used involve partial differential equations, random graph theory and Markov processes.

This research involves applications of interacting particle systems to biology. In particular, the investigator will construct models to explain biological phenomena such as the temporal oscillations of measles and the patchiness of plankton in the sea. Another area of interest is the connection with nonlinear partial differential equations. By mixing the particles at a fast rate and scaling space, the system converges to a nonlinear partial differential equation. This makes it possible to derive partial differential equations that more accurately model the biological systems and also to prove results about the particle systems with fast mixing. Interacting particle systems have a structure that makes them appropriate for use in a variety of contexts in physics, chemistry, and biology. There is a grid of sites, each of which can have a fixed set of values. For example, a city of individuals can be susceptible to, infected with, or immune to measles. Each site changes its value at a rate that depends on the values of its neighbors, and this research is interested in determining what happens to the system over time. These models should eventually provide a better understanding of global ecological problems.

In recent years the stochastic point of view has achieved great prominence, both as an active area of probability theory and as a powerful tool in problems of analysis, geometry, and mathematical physics. For example, the classical Ito theory of stochastic differential equations, which defines a finite- dimensional diffusion process, already leads to many interesting connections with analysis in Euclidean space and finite-dimensional differentiable manifolds. The same stochastic differential equations can also be used to define a stochastic flow, which is a diffusion process on the group of diffeomorphisms, an infinite- dimensional space. This naturally leads to the study of other infinite-dimensional processes, especially those defined by stochastic partial differential equations. This project will support the 1993 AMS Summer Research Institute on Stochastic Analysis to be held July 11-30, 1993 at Cornell University in Ithaca, NY. This Institute is the 41st in a series designed with the purpose of bringing together a group of mathematicians interested in a particular field of mathematical research. Emphasis is placed on instruction at a very high level with seminars and lectures by distinguished mathematicians in related fields, in order to promote interaction between participants while broadening their mathematical perspectives. The goal of the Institute will be to highlight the main directions of the field through principal lectures by leaders in the following general areas: stochastic (ordinary) differential equations, applications to analysis, applications to geometry, stochastic flows, infinite dimensional problems, and stochastic partial differential equations.

9321087 Durrett This award will support the development of software for simulating stochastic spatial models, such as those found in ecology. Interacting particle systems, a vigorous area in probability theory, have a structure that makes them appropriate to the development of ecosystems. Their use as models is impeded by the fact that the mathematical theory is not easy to read and understand. To bridge the gap between theory and applications the project will develop software to allow biologist interested in using these models to learn about existing results and to simulate and analyze their own models. ***

Many biological systems can be adequately treated by assuming there is homogeneous mixing and reducing the system to a differential equation. In some systems, however, the spatial arrangement of individuals and stochastic effects are important. The principal investigators will use stochastic spatial models to explore four phenomena in which these effects play an important role: the unexpectedly low fraction of heterozygotes for t haplotypes in the common house mouse, the dynamics of hybrid zones which are maintained by heterozygote unfitness, attempts at using a virus to control the spread of chestnut blight, and competition of various strains of potato virus Y. This award is being jointly funded by the Computational Biology Activity, the Statistics and Probability, and the Ecological Studies programs.

Durrett will continue to apply the theory of stochastic spatial models to problems in biology, specifically studying: (a) abundance distributions generated by the voter model with mutation that are an alternative to the traditional lognormal, (b) hybrid zones and their scaling limits -- motion by mean curvature, (c) a generalization of the biased voter model used by Silvertown to model the competition of grass species, and (d) a nonlinear predator prey model that displays oscillatory behavior. In an interacting particle system, space is represented as a grid of sites each one of which can be in one of a finite set of states and changes its state at a rate that depends on the states of finitely many neighbors. These models have a structure that is well suited to many problems in biology and have recently seen many applications there. Durrett's investigations will continue previous work seeking to understand when the spatial distribution of individuals has an important effect on the outcome of competition and when it can be ignored. Earlier work has already identified a number of situations in which non-spatial approaches give incorrect answers to questions about the persistence or coexistence of competing species.

The Center for Applied Mathematics at Cornell University will purchase computer equipment which will be dedicated to the support of research in the mathematical sciences. The equipment will be used for several research projects in mathematics, including in particular: Stochastic spatial models in the biological sciences Computational analysis of dynamical systems Computational research in coding theory Computational Algebraic Geometry Computation of Eigenvalues and Applications

9705658 Durrett This award provides funds to partially support a research conference in probability to be held at Cornell University, June 28 - July 1, 1998. The purpose of the conference is to look at exciting areas of current research in probability and to identify important problems for the next ten years. One hour expository talks will be given by some of the leaders in the study of probability.

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Durrett
There are many interesting mathematical problems that are motivated by a desire to understand the workings of various biological systems. In order to facilitate and stimulate interdisciplinary research, a workshop on "Mathematical Problems in Biology" will be held at the Fields Institute for Research in Mathematical Sciences, Toronto, Canada from June 14 to 24, 1999, as part of a special year long program on Probability and its Applications. Eighteen prominent biologists, mathematical biologists, and mathematicians, all of whom are talented expositors, will give two one-hour lectures to an audience of mostly young (postdoctoral) researchers. This award will allow fifteen US postdoctoral researchers to participate and train themselves for research in these exciting and important areas.

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This investigator will study a number of questions that arise from ecology. An important direction of the research will be to prove general results in support of the picture advocated by Durrett and Levin (1994): the behavior of stochastic spatial models can be predicted from that of the mean field ODE obtained by pretending all sites are always independent. He will also continue work on hybrid zones, progressing now to the derivation of PDE limits for these systems as selection tends to 0, and the study of spatially inhomogeneous models to understand the structure of mosaic hybrid zones that have been seen in field work on the cricket species Gryllus firmus and G. pennsylvanicus. Other projects include understanding limits to diversity by analyzing Tilman's competition model on a finite grid, studying the asymptotic behavior of an evolutionary arms race, and demonstrating the existence of chaotic oscillations in a simple discrete time model of gypsy moth populations.

The basic question that drives this research is: when does the spatial distribution of individuals change the behavior of a biological system? The comparison is made to a homogeneously mixing system in which each individual interacts equally with all the others. In the past five years the investigator and his collaborators have developed some rules for predicting the answer to this question and have applied them to a variety of biological systems. In the next three years they will apply this theory to some new examples. One of the most important questions they will study is: What are the limits to diversity? They will address this question mathematically by investigating the number of species that can coexist in a stochastic competition model. Other questions will concern the fluctuations of the size of gypsy moth populations and the structure of hybrid zones that result from spatially varying selection.

Research supported by this grant will focus on mathematical and statistical analyses of genome rearrangements and the interaction of natural selection and spatial population structure in shaping genome diversity. Statistical methods will be developed to study the evolution of DNA sequences due to large scale processes: inversions within chromosomes, reciprocal translocations between chromosomes, transpositions that move genes, chromosome fissions and fusions, and duplication of whole genomes. Major goals are to obtain statistical estimates of the number of events with associated measurements of uncertainty, test hypotheses concerning for example, constancy of mutation rates among lineages, and to compare the likelihood of various genome duplication scenarios in Arabidopsis thaliana, maize, and other organisms. Mathematical results and statistical techniques will also be developed to distinguish the causes of observed levels and patterns of DNA sequence variation across genomes, concentrating on the effects of the fixation of advantageous mutations ("selective sweeps") and of spatial population structure. Statistical work will include the development of composite likelihood methods for data from large genomic regions for which full likelihood or Bayesian methods are not tractable.

Research supported by this grant will focus on quantitative studies of variation in genomes within and between organisms. Two scales are considered. The first is the evolution of genomes due to large-scale processes which rearrange gene order on chromosomes or exchange genetic material between them. An understanding of the rates and sizes of these changes will be important for improving comparative maps between species. These maps which, for example, identify the mouse homologues of human genes or give the relative order of genes in cattle and sheep are important in medicine and agriculture. A second scale is the level of DNA variation observed within species. Mathematical and statistical studies will focus on distinguishing the effects of adaptation (natural selection) and spatial (geographical) population structure on shaping patterns observed in DNA sequence data from humans and other organisms. These results will provide insights into how organisms respond genetically to novel environments and challenges (including pathogens and environmental stress), and can help identify those genes important in that adaptation. These approaches will be important for understanding the enormous genome diversity detected in humans and most other organisms. An important component of the proposed activities is the training of researchers at the interface between biology and mathematics. This grant is made under the Joint DMS/NIGMS Initiative to Support Research Grants in the Area of Mathematical Biology. This is a joint competition sponsored by the Division of Mathematical Sciences (DMS) at the National Science Foundation and the National Institute of General Medical Sciences (NIGMS) at the National Institutes of Health.

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Durrett
Durrett proposes to study a variety of problems in probability theory that arise from questions in ecology and genetics. In the area of stochastic spatial models, he will study the asymptotic behavior of multitype voter models where voters have several opinions and are more likely to imitate neighbors with similar opinions. With Ted Cox he will work to extend their conclusions for the stepping stone model to samples of size k > 2, and will study equilibrium properties of a hybrid zone which results when each of two alleles is favored in one half of the space. With Ed Perkins he will study measure valued limits of particle systems to produce limit processes with interactions between different types of particles. Not all of the systems Durrett studies will be spatial however: he will work with Jason Schweinsberg to develop approximations of genetic hitchhiking that involve the coalescent with multiple collisions, will study the phase transition in a quasi-species model which models properties of viruses with high mutation rates, and will continue his work with Vlada Limic on Kauffmann's NK model, which is a prototypical example of a fitness landscape with a large number of local maxima.
Durrett's research is motivated by a variety of applications. Durrett will work with Ted Cox to extend their work on spatial genetics models to investigate explanations for the surprisingly large amount of linkage disequilibrium (genetic correlation) in the human genome. Understanding the possible sources of patterns in DNA sequences is important for using association mapping to locate genes and to identify footprints of positive selection. In relation to the latter question, Durrett will work with Jason Schweinsberg to develop an approximation to the genetic hitchhiking that occurs with the fixation of favorable mutation, which will allow for the derivation of analytical results for quantities that can currently be understood only by simulation. A major theme of Durrett's research is the influence of a spatial distribution of competitors on equilibrium properties of a system. With Simon Levin, he will study a collection of stochastic spatial models for the evolution of social norms. With Ted Cox, he will investigate hybrid zones where closely related but genetically differentiated populations coexist in close proximity. With Ed Perkins he will investigate scaling limits of stochastic spatial models, in order to construct new interesting examples of interacting measure-valued diffusions and to obtain approximations for particle systems with large range.

In recent years, there has been a tremendous expansion in the use of probability models in finance, geosciences, neuroscience, artificial intelligence and communication networks in addition to an increase in its
use in traditional application areas in engineering disciplines, physics, ecology, genetics, and various fields of mathematics. This has created a strong demand for researchers trained in probability to develop new
methodologies and to work in an interdisciplinary context. We propose a variety of activities to meet these needs. Graduate fellowships will enhance the training of new researchers at Cornell, where a cohort of more
than 20 Ph.D. students exists and 4-6 students receive their Ph.D.'s in this subject each year. On a national scale, two yearly 2.5 day hot topics conferences will bring to Cornell a small group of established researchers and young investigators (in equal numbers). The conference will feature talks describing recent developments, and the young researchers will have a unique opportunity to discuss their research and open problems with the leaders in the field. A two week summer school featuring 4-6 lectures by three prominent researchers, two series of 1-3 on interdisciplinary opportunities, and a limited number of short talks by selected participants, will benefit a large number of graduate students and researchers (young and old) throughout the country.

This project will be under the direction of six probabilists from Math and Operations Research at Cornell (Durrett, Lawler, Protter, Resnick, Saloff-Coste, and Samorodnitsky). Their combined research covers a wide variety of topics in probability and its applications. However, to ensure that this is truly a national resource and covers all aspects of modern probability, they will receive advice from a nationwide committee of prominent researchers that represent a wide variety of specialties and many of the major probability groups throughout the country: David Aldous (U.C. Berkeley), Thomas Kurtz (U. of Wisconsin, Madison), Claudia Neuhauser (U. of Minnesota), Charles Newman (Courant Institute), Yuval Peres (U. C. Berkeley), Simon Tavare' (U. of Southern California).

This proposal requests travel support for U.S. participants attending the 31st Conference on Stochastic Processes and their Applications, an international meeting to be held July 17-21, 2006 in Paris, France. The conference on Stochastic Processes and their Applications have become the principal annual international forum for researchers studying applied and theoretical problems in stochastic processes. These conferences provide a balanced and broad overview of current research in probability and areas of application and bring together distinguished scholars from all over the world.

We propose to continue two successful local outreach activities at the high school level, improving and diversifying our collection of activities and materials. The Math Explorer's Club (MEC), meets once a week at Ithaca High School in the 2:45-3:30 time slot set aside for after school clubs. Activities are organized into six to eight week modules, each of which is devoted to a specific topic and led by a Cornell graduate student. The aim of MEC is to show high school students that there is more to mathematics than the centuries old progression from algebra to calculus, and to introduce them a wide spectrum of mathematical ideas, through games, puzzles, model building, and other explorations. The Senior Seminar is a class on advanced mathematics designed for students who have taken most of the available math classes. It meets for one period during school hours three days week at the high school and introduces students to topics such as cryptography, number theory, graph theory, game theory, combinatorics, probability, and topology that they would typically not see until their junior or senior years in college.

The intellectual merit of this proposal lies in the development of high quality materials that introduce advanced mathematical concepts to high schools in an engaging and entertaining manner. The broader impact of this proposal is to export these successful activities to a number of other communities across the country. To facilitate the development of similar activities in other locations, the faculty coordinator and some of the graduate student members will make presentations each year at the Joint Membership meetings of the MAA and AMS, and work with individuals from other universities that want to use our materials. We will create a web-based archive of activities for the MEC and of lecture notes for the Senior Seminar that can be used by students and teachers throughout the country. The availability of tested materials will make it much easier to start similar activities in other locations. The Math Explorer's Club and the Senior Seminar aim to interest more high school students in majoring in math in college, and to increase the number of young women and men who choose mathematics as a career, by increasing the flow into the pipeline.

We propose to investigate a number of problems in probability theory that arise from genetics and ecology. We will extend our analysis of the impact of the fixation of beneficial mutations, develop analytical results to help quantify the effects of Hill-Robertson interference, which occurs when a second advantageous mutation occurs before the first one reaches fixation and study diffusion approximations for the related problem of codon usage bias. We will obtain rigorous results for the sequentially Markovian coalescent, which has recently been introduced as an approximation for genealogies with recombination, investigate the nature of regulatory sequence evolution in Drosophila and other organisms with large effective population size; study the subfunctionalization explanation of the persistence of gene duplicates, and consider random Boolean networks, in order to gain insights into gene networks. Inspired by ecology, we will use spatial models to understand conditions needed for the emergence and maintenance of cooperation between individuals, and we will investigate changes that occur when the parameters governing the interactions in predator-prey and epidemic models evolve.
The last decade has seen the generation of incredible amounts of DNA sequence data. However, having the text of the instruction manual that controls life is just the beginning, we need to understand how the expression of genes is controlled and how genes interact in biological networks. These interactions cannot be directly observed so one needs to develop mathematical models to test hypotheses, and to infer how genetic systems evolved to operate as they do. In particular, our research will investigate (i) how rapidly changes can occur due to regulatory sequence evolution, which is commonly quoted explanation for many of the differences between humans and chimpanzees, (ii) whether or not the subfunctionalization hypothesis provides a satisfactory explanation of the paradox of the preservation of gene duplicates, and (iii) to improve our understanding the patterns of variations caused by adaptive evolution. These are just three examples of many at the interface between biology and mathematics. An important aspect of our proposal is to train graduate students to work in this exciting area.

Abstract
DMS 0635449
Principal Investigator: James O. Berger

SAMSI is a national institute that will deeply impact the future
of the statistical and mathematical sciences and, through them,
science in general, by catalyzing creation of the theory and
methodology necessary to confront the central data- and
model-driven scientific challenges of our time. SAMSI will focus
on new syntheses of the statistical sciences, applied mathematics
and disciplinary science. To illustrate the vision, consider an
activity central to modern science and technology, and with which
SAMSI will be heavily engaged: numerical modeling of complex
physical processes. Developing numerical models and evaluating
their fidelity to reality requires merging knowledge from
multiple disciplines. Applied mathematics builds on disciplinary
understanding to construct a fine-detail numerical representation
of the deterministic aspects of a process; probability provides
concepts and insight to characterize stochastic elements of the
process; and statistics provides the mechanisms to relate these
constructs to observational data on the real-world process. But,
despite a multiplicity of context-specific advances, there is
currently no general framework for combining these disciplines,
much less a formal system for simultaneously applying them. The
SAMSI efforts in this direction will focus on filling this gap,
by bringing together statisticians, mathematicians and modelers
from across the country (and beyond) to establish frameworks for
model development and validation, at a high level that spans
multiple application contexts.

To carry out this synthesis of the statistical sciences, applied
mathematical sciences and disciplinary sciences, SAMSI engages
established researchers - from academia, industry, national
laboratories and government - as well as young researchers
(postdoctoral fellows and graduate students) at the formative
stages of their careers. Each research program lasts from 6
months to one year, and involves a host of activities ranging
from research by intensive interdisciplinary working groups to
broad, energizing workshops. Outreach to undergraduate students,
high-school teachers and faculty from teaching institutions
extends SAMSI's impact on the essential development of the
national human resource base for science. To enable activities of
the breadth and depth necessary for the success of SAMSI, the
institute is a partnership between the National Science
Foundation and the consortium of Duke University, North Carolina
State University, the University of North Carolina at Chapel Hill
and the National Institute of Statistical Sciences. These
partners provide a uniquely strong base for SAMSI's national
scope. Scientific input to SAMSI comes, in part, from a National
Advisory Council composed of eminent statistical and mathematical
scientists. Most important, SAMSI will engage the entire
nationwide statistical and mathematical sciences communities, by
means of widely publicized opportunities to bring problems to
SAMSI, or direct SAMSI's attention to them.

This project will support research on probability problems that arise from a wide variety of fields. Topics from genetics include regulatory sequence evolution, gene duplication, and the analysis of data generated by cancer genome projects. In finance, understanding bubbles in markets, swing options, and credit risk associated with subprime mortgages require sophisticated mathematical ideas from control theory, backward stochastic differential equations, and enlargement of filtration. The study of financial markets, insurance risk, and communication and computer networks lead to stochastic models that exhibit long range dependence and heavy tails. The analysis of these processes and the new phenomena they present, involves techniques much different from the classical theory of independent random variables, making use of ideas from a wide variety of sources including ergodic theory. In addition, there is a fertile interaction between probability and a variety of topics that arise from algebra, geometry, and analysis, including the use of analytic techniques to study convergence rates of Markov chains.

Probability theory, born in the analysis of gambling games, now plays an important role in biology, physics, economics, finance, insurance, communication networks, and in many topics within mathematics. The main aim of this proposal is to train more researchers in probability to tackle the many important problems that arise from its applications. Cornell is an ideal place for these activities because of the strong traditions of research in probability and interdisciplinary work in applied mathematics. Support for summer research projects will show undergraduates that probability is an interesting research area with many important applications. At the graduate level, increased fellowship support will enhance the training of our Ph.D. students. Postdoctoral positions will help new graduates expand their horizons, while hot topics conferences and annual summer schools will be a national resource for broadening the education of researchers (young and old). In all of these activities, the focus will be on developing theory to treat problems that arise from applications.

Travel support is requested for 20 U.S. participants (young researchers, women, and members of underrepresented minority groups) to attend the World Congress in Probability and Statistics, an international meeting to be held July 14-19, 2008, in Singapore. The conferences on Stochastic Processes and their Applications have become the principal annual international forum for researchers studying applied and theoretical problems in stochastic processes. Every four years that sequence of meetings is interrupted by a World Congress in Probability and Statistics, a much larger conference that bring together distinguished scholars from all over the world and provides a broad overview of current research in probability and statistics, and their applications.

An important part of the intellectual development of young researchers is to attend international conferences where they have an opportunity to listen to stimulating lectures and interact with probabilists and statisticians who work on a wide variety of topics. The World Congress in Probability and Statistics, which meets every four years, and will be held in Singapore in 2008 will feature one hour talks by about a dozen internationally known researchers and special sessions on 33 topics. Travel support is requested for 20 U.S. participants (young researchers, women, and members of underrepresented minority groups) to attend this important meeting.

Research will be carried out on three topics: stochastic spatial models, processes taking place on random graphs, and questions related to the evolution of biological systems. Three of the proposed questions in the first topic concern "When can species coexist", while a fourth concerns the possibility that the quadratic contact process in two dimensions can have two phase transitions one for the existence of stationary distributions and a larger one for survival from a finite set. In the second topic one interesting mathematical problem concerns "explosive percolation" conjectured to have a discontinuous transition, while the more biologically important question concerns how the outcomes of epidemics and ecological competitions change when they take place on random graphs, which arguably provide better models of the real social networks. The third topic concern situations in which the characteristics of individuals in ecological competitions are also not static but evolve in response to their environment. My first steps in the area "adaptive dynamics" were taken in a study of predator-prey systems with John Mayberry. Here, we propose to study more complex examples that lead to evolutionary cycling and a second problem on the evolution of virulence, which leads to consideration of the role of spatial structure in increasing the virulence of diseases.

Many interesting mathematical questions arise from biology. Here we address some questions that arise from ecology and evolution. Three examples should illustrate the nature of our work. (1) At the turn of the century, observations of social networks revealed that we live in a small world in which everyone on the planet is separated by six degrees of separation. Now we need to understand how this geometry of social networks effects the spread of epidemics and the other biological and social processes. (2) The world shows much more biodiversity than mathematical models predict, so it is important to understand the mechanisms which allow for species coexistence. More generally, we will also be interested in how spatial structure changes the outcome of ecological competition. (3) In most situations the characteristics of individuals involved in competition with other species or with infectious agents are not static but evolve in time.
For example, in most cases diseases evolve to be less virulent, but in a spatially structured population the opposite may occur. Co-evolution of hosts and parasites can lead to interesting evolutionary cycling, sometimes called ?Red Queen Dynamics after the character in Alice in Wonderland who has to keep running to stay in the same place.

This award will provide partial support for 25-30 participants of the third Workshop for
Women in Probability to be held October 14-16, 2012 at Duke University. All of those receiving support will be young female researchers (graduate students or recent PhD's) and will be recruited from a wide variety of institutions. This conference features talks on current exciting research areas by ten women probabilists, who are leading researchers in their field. Topics will include mathematical biology, stochastic partial differential equations, fractional Brownian motion, random matrices, dynamical systems, interacting particle systems, stochastic networks, Markov chain Monte Carlo, and randomized algorithms. Participants will have an opportunity to interact with the speakers in a way that is difficult at a large national or international meeting, and to give poster presentations on their own research. This conference will foster the formation of networks of a support structure for female mathematicians at the beginning of their careers. The young researchers will have the chance to network with their peers as well as more senior women in their field, and to discuss the challenges they face in graduate school and early professional life. The interpersonal contacts resulting from this meeting will help guide them through early career transitions, which will ultimately improve the retention of talented women in the mathematics work force.

Research will be carried out on various aspects of stochastic spatial models taking place on regular lattices and on complex networks. (i) With Ted Cox and Ed Perkins, the author has studied voter model perturbations. This work will be the starting point for studying evolutionary games with weak selection in order to try to resolve some recent conjectures of Martin Nowak. (ii) The proposer has studied various processes on random regular graphs in joint work with Shirhsendu Chatterjee. The main thrust of the new work will consider process on graphs with more general degree distributions, but we will also consider models in which a threshold of more than one neighbor with a different opinion is needed for an individual to change. (iii) A third theme is to study networks in which the state of the system and the connections in the network coevolve. This work leads to difficult questions about stochastic evolutions on the space of all labeled graphs.

The world around us is spatially structured. In biological systems, organisms and infections only spread over short distances. Social systems have a network structure that is described by a graph with vertices being individuals and edges between those that interact. Connections between individuals may extend over large physical distances, but diseases, opinions, and new technologies spreading due to the interactions between an individual and their neighbors in the graph. Most mathematical analyses ignore spatial or network structure because its inclusion greatly complicates the analysis. However, as our past work with Simon Levin and others has shown, incorporating spatial structure changes the outcome of ecological competition. Our new work will focus on how spatial structure changes the impact of frequency dependent selection, and how the network structure of a social system changes the behavior of processes that occur in it. The models are idealized but the results give valuable insights about real systems.

This award supports junior U.S. researchers, allowing graduate students and those in early stages of their career to participate in the thematic semester on Biodiversity and Evolution that will be held at the Centre de Recherches Mathematiques (CRM) in Montreal during the second half of 2013. This is part of the more general program of the thematic year on Mathematics of Planet Earth (MPE2013). Five workshops are devoted to (i) random trees, which arise in a wide variety of biological application, (ii) modeling the complex evolutionary dynamics that have shaped the structure of contemporary biodiversity, (iii) models of DNA sequence evolution, (iv) coalescent theory and other genealogical processes, and (v) applications of evolutionary game theory. This grant will allow 25-30 young U.S. researchers to participate in this exciting program.

Understanding the rise and decline of species in interaction with one another and with the environment, not to mention the astonishing variability within species, requires a combination of approaches from distinct scientific disciplines (genomics, ecology, economics, computational biology, mathematical modeling, statistical genetics and bioinformatics). The thematic semester on Biodiversity and Evolution that will be held at the Centre de Recherches Mathématiques (CRM) in Montreal August-November 2013 will have conferences featuring research on a wide variety of mathematical, statistical, and computational approaches. Participating in conferences will allow 25-30 young researchers to learn about fertile and important research directions, and to interact with the leading researchers in their field.

This project concerns spatial models for ecological and social interactions motivated by various applications. The theme of this research is to see how predictions change when systems previously studied under the assumption that each individual interacts with all the others are made more realistic by incorporating space. The four main examples are the following: (i) the Staver-Levin forest model, which predicts that forest and savannah (grassland with isolated trees) are alternative stable states; (ii) evolutionary games, which have long been used in ecology to explain phenomena such as the persistence of altruistic behavior; (iii) Axelrod's model, which studies the spread of opinions when individuals interact with a probability based on the number of the number of opinions they share; (iv) the latent voter model, which studies the spread of technology in a social network when consumers who have just acquired a new product will wait some time before they are willing to purchase a new one. The general goal of studying these idealized models is to understand how properties of the equilibrium of the system depend on the details of the interactions. When each individual interacts with all the others, the system is an ordinary differential equation and is easily studied. However, when space is explicitly taken into account the problems become very difficult.

This project has the following specific goals: (i) show that in the Staver-Levin model, the direction of movement of a boundary between forest and savannah indicates the one state that is the true equilibrium in the spatial model; (ii) show that evolutionary games have three separate weak selection regimes that can lead to a PDE, ODE, or a regime in which Tarnita's formulas are valid; (ii) complete Junchi Li's thesis work studying Axelrod's model in the situation in which there are a large number of issues about which there are a large number of opinions (this would provide the first rigorous result for that model in two dimensions); (iv) show that even if latent period is brief, it changes the dynamics so that there is only one nontrivial stationary distribution, in contrast to the one parameter family in the voter model without latency.

Successful prevention, early detection, and treatment of cancer requires an understanding of the underlying mechanisms by which cancer evolves spatially and temporally. The occurrence of different cell types in different parts of a tumor, the spatial heterogeneity of the tumor, can lead to failure of therapy that is directed only against one type of cancer cell. Furthermore, spatial heterogeneity is associated with a worse prognosis in many cancer types, and the spatial distribution of nutrients and oxygen can favor the development of more aggressive tumor cell populations. By analyzing spatial cancer models, this research will provide new insights into the mechanistic underpinnings of intra-tumor heterogeneity in solid tumors, and into the role of cellular mobility in shaping tumor phenotypes with poor prognosis. To guarantee that the results are relevant to treatment, the project will be carried out in collaboration with physicians and a cancer biologist. The involvement of graduate students in this research will enhance their ability to work at the interface between mathematics and biology.

This project addresses critical challenges in the quest for more effective diagnostic tools and therapeutic approaches in the fight against cancer. Complex agent-based models have often been employed to investigate the impact of space on cancer evolution. However, even if a simulation successfully reproduces observed phenomena, it does not mean that the true underlying mechanism has been found. For this project the investigators plan to extend prior work on spatial Moran models by adding new features to the basic model. To further reduce the gap between analytically tractable and complex computational models, the behavior of the new systems will be analyzed mathematically. The biological questions to be addressed by the project are motivated by recent experimental findings, including studies of the impact of cancer cell migration on solid tumor growth and elucidation of the role of cancer stem cells in driving cancer evolution. Mathematical models that can include genomic data will be used to study genealogies and intra-tumor heterogeneity in colon cancer. Rigorous results for hybrid models that describe tumor-microenvironment interactions will also be derived. More generally, the rigorous analyses of the models will require the development of novel mathematical techniques that will contribute to a deeper understanding of spatial stochastic systems and the behavior of genealogies in an expanding population.

Ecologists have long emphasized climate as primary factor determining ecosystem dynamics at large scales, but the relationship between climate and vegetation is not always deterministic. Where vegetation-environment feedbacks are significant (e.g.,in savannas), predicting responses to climate can be especially difficult. Recent work suggests that feedbacks with fire may make savannas much more common than they would be were their distributions solely determined by climate; as a result, savanna and forest responses to global change may thus be drastic, sudden, and difficult to foresee. However, existing work does not explain why savanna is spatially aggregated with savanna and forest with forest -- a pattern that indicates that spatial processes may also play a role in determining ecosystem responses to climate. Here the investigators will consider the impacts of those spatial interactions between savanna and forest on their distributions and on their potential responses to climate and land-use change, both in the past and into the future.

The investigators have identified three hypotheses to explain the spatial aggregation of savanna and forest: H1) that savanna and forest are bistable, and that spatial structure in initial conditions (as a result of past climate) determines their distributions, H2) that savanna and forest are bistable, but that spatial processes within savanna (e.g., fire spread) result in spatially structured distributions, and/or H3) that nearest neighbor interactions between savanna and forest change their distributions on long time scales, impacting their long-term stability. These hypotheses are variously supported in the empirical literature, and existing work has not attempted to disentangle these processes. Results will allow the research team to generate informed theoretical and empirical predictions about the past and future distribution of savanna and forest globally. The proposed work will also generate novel mathematical results. Possible outcomes of theoretical, spatial-stochastic models include a) savanna and forest coexistence in landscapes, b) forest exclusion by savanna, c) savanna exclusion by forest, or d) alternative stable states in biome savanna/forest dominance. The last outcome would be unlikely in a homogeneous spatial stochastic model, where the winning biome is decided by the direction of movement of the biome boundary (i.e., the front), but most closely resembles real biome distributions. Spatial stochastic model results will be reconciled with observations using theoretical and simulation modeling.

This award will help support the 2020 edition of the Southeastern Probability Conference to be held at Duke University May 11-12, 2020. The award will also support the 2021-2023 conferences. The Southeast Probability Symposium is a small regional conference supported by local resources that has been held at the Duke mathematics department seven times in the last nines year. This award will be used to (i) continue this successful series of conferences that highlight exciting new directions in probability, (ii) provide support for participants and include more probabilists from nearby universities, especially those without colleagues in their area and those with limited access to federal funding.

Similar to the Midwest and Northeast Probability Conferences, our goal is to serve the probabilists in the Southeast. However, in contrast to those meetings, which attract a large number of participants from a wide area,ours is a small meeting that allows ample opportunity for speakers and other participants to interact. Graduate students and postdocs will also have a chance to learn about recent developments from experts. A poster session at the end of the first day will allow early career researchers to present their work. Slides of the main lectures will be made available after the meeting on the conference web page to more broadly disseminate their content. The Conference web site is: https://services.math.duke.edu/~rtd/SEPC2020/SEPC2020.html

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.