African Diaspora Joint Mathematics

2023 African Diaspora Joint Mathematics Workshop

June 19, 2023 to June 30, 2023

Description

The African Diaspora Joint Mathematics Workshop (ADJOINT) will take place at the Mathematical Sciences Research Institute in Berkeley, CA from June 19 to June 30, 2023. ADJOINT is a two-week summer activity designed for researchers with a Ph.D. degree in the mathematical sciences who are interested in conducting research in a collegial environment. The main objective of ADJOINT is to provide opportunities for in-person research collaboration to U.S. mathematicians, especially those from the African Diaspora, who will work in small groups with research leaders on various research projects. Through this effort, MSRI aims to establish and promote research communities that will foster and strengthen research productivity and career development among its participants. The ADJOINT workshops are designed to catalyze research collaborations, provide support for conferences to increase the visibility of the researchers, and to develop a sense of community among the mathematicians who attend. The end goal of this program is to enhance the mathematical sciences and its community by positively affecting the research and careers of African-American mathematicians and supporting their efforts to achieve full access and engagement in the broader research community. Each summer, three to five research leaders will each propose a research topic to be studied during a two-week workshop. During the workshop, each participant will: conduct research at MSRI within a group of four to five mathematicians under the direction of one of the research leaders participate in professional enhancement activities provided by the onsite ADJOINT Director receive funding for two weeks of lodging, meals and incidentals, and one round-trip travel to Berkeley, CA After the two-week workshop, each participant will: have the opportunity to further their research project with the team members including the research leader have access to funding to attend conference(s) or to meet with other team members to pursue the research project, or to present results become part of a network of research and career mentors * The 2023 on-site director is Dr. Anisah Nu'Man (Spelman College). Eligibility Applicants must be a U.S. citizen or permanent resident, possess a Ph.D. in the mathematical sciences, and be employed at a U.S. institution. Selection Process The guiding principle in selecting participants and establishing the groups is the creation of diverse teams whose members come from a variety of institutional types and career stages. The degree of potential positive impact on the careers of African-Americans in the mathematical sciences will be an important factor in the final decisions. 2023 Research Leaders and Topics Folashade Agusto (University of Kansas) Ticks, Fire and Control: Implementing prescribed fire and control measures at ticks invasion front In recent times, ticks have been expanding their habitat ranges due to climate change, among other factors, leading to increased tick-borne illness risks in the United States. Thus, it is important to find a practical and cost-efficient way of controlling and managing ticks. Prescribed burns are intentional fires used for land and forest management; they are time and cost efficient, since they can be applied across large amounts of land. They can also be effective in controlling tick populations. Prerequisites: Knowledge of PDEs and their numerical simulation. Knowledge of the SIR model and R0 will be appreciated. Pre-readings: The following three resources are meant to familiarize applicants with the project: Centers for Disease Control and Prevention (CDC): Preventing ticks in the yard Emily Guo and Folashade B. Agusto: Baptism of Fire: Modeling the Effects of Prescribed Fire on Lyme Disease Alexander Fulk, Weizhang Huang, Folashade Agusto: Exploring the Effects of Prescribed Fire on Tick Spread and Propagation in a Spatial Setting. (Code used in this study can be found at Github.) Jonathan Esole (Northeastern University) Elliptic fibrations, flops, and collisions of singularities Elliptic fibrations are beautiful and elegant geometries at the interface of algebraic geometry,number theory, and string theory. While ubiquitous, they still have many secrets to teach us. In particular, their uses by physicists indicate deep connections of their birational geometry with representation theory and hyperplane arrangements of weight systems. The geometric singular fibers over generic points of the discriminant locus of a smooth elliptic fibration are famously classified by Kodaira and Neron. They have dual graphs corresponding to twisted affine Dynkin diagrams. Under mild conditions, an elliptic fibration is birationally equivalent to a (singular) Weierstrass model. Two different crepant resolutions are connected by a sequence of flops and have some matching topological invariants, such as their Euler characteristics. We are interested in the structure of crepant resolutions of Weierstrass models of elliptic fibrations corresponding to given types of Kodaira fibers. We will explore their flops and compute their topological invariants. Prerequisites: Familiarity with basic algebraic geometry. Background in intersection theory is a plus but not needed. Someone familiar with elliptic curves could also quickly digest the required algebraic geometry. Donald Eugene Kemp Martin(North Carolina State University) Inference and applications of sparse Markov model Analysis of a categorical time series is facilitated by a model that captures the statistical properties of the sequence, while being simple enough so that statistical analysis is feasible. Markov models of an order of dependence m serve this purpose. However, the number of parameters in a Markov model is exponential in m, leading to variance considerations because many parameters need to be estimated from data. Thus first-order Markov chains (m = 1) are more frequently used. Yet using a lower-order model when higher-order dependence is called for leads to bias. Sparse Markov models help with this bias-variance trade-off. A sparse Markov model (SMM) is a higher-order Markov model for which conditioning m-tuple histories are grouped into classes such that the conditional probability distribution is constant over m-tuples in the same class. The clustering reduces the number of parameters. Prerequisites: Basic understanding of probability and statistics, including knowledge of Markov chains, model fitting and statistical inference. Bibliography Nsoki Mavinga (Swarthmore College) Nonlinear elliptic partial differential equations with nonlinear boundary conditions Since Bernstein's pioneering work in nonlinear elliptic equations in 1906 and Leray's work on hydrodynamical problems in 1933, there has been a significant amount of interest in the study of nonlinear elliptic partial differential equations. Henceforth, there has been a tremendous emphasis in studying partial differential equations that enable understanding and modeling of nonlinear processes such as chemical, biological or ecological processes. This area has become central in modern-day mathematical research. The study of nonlinear elliptic partial differential equations with linear boundary conditions is relatively well studied and the literature is extensive. However, problems with nonlinear boundary conditions have recently gained attention due to their importance in the study of several branches of pure and applied mathematics such as the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary, non-Newtonian fluids, nonlinear elasticity and glaciology, among others. In this research project, we will focus on the qualitative (analytical) study of nonlinear elliptic equations subject to nonlinear boundary conditions. In particular, we consider problems with nonlinear reactions in the interior as well as on the boundary of the domain in consideration. We will investigate the existence, nonexistence, multiplicity, and properties of solutions. In the treatment of the proposed problems, new analytical tools will be developed as well as utilize the methods of nonlinear functional analysis such as sub-supersolution method, degree theory, bifurcation theory and fixed points theorems. Prerequisites: Participants need to have some knowledge of (graduate) partial differential equations, and real and functional analysis. Bibliography ADJOINT Program Directors Dr. Edray Goins, Pomona College Dr. Caleb Ashley, University of Michigan Dr. Naiomi Cameron, Spelman College Dr. Jacqueline Hughes-Oliver, North Carolina State University Dr. Anisah Nu’Man, Spelman College (2023 site director)

The African Diaspora Joint Mathematics Workshop (ADJOINT) will take place at the Mathematical Sciences Research Institute in Berkeley, CA from June 19 to June 30, 2023.

ADJOINT is a two-week summer activity designed for researchers with a Ph.D. degree in the mathematical sciences who are interested in conducting research in a collegial environment.

The main objective of ADJOINT is to provide opportunities for in-person research collaboration to U.S. mathematicians, especially those from the African Diaspora, who will work in small groups with research leaders on various research projects.

Through this effort, MSRI aims to establish and promote research communities that will foster and strengthen research productivity and career development among its participants. The ADJOINT workshops are designed to catalyze research collaborations, provide support for conferences to increase the visibility of the researchers, and to develop a sense of community among the mathematicians who attend.

The end goal of this program is to enhance the mathematical sciences and its community by positively affecting the research and careers of African-American mathematicians and supporting their efforts to achieve full access and engagement in the broader research community.

Each summer, three to five research leaders will each propose a research topic to be studied during a two-week workshop.

During the workshop, each participant will:

conduct research at MSRI within a group of four to five mathematicians under the direction of one of the research leaders
participate in professional enhancement activities provided by the onsite ADJOINT Director
receive funding for two weeks of lodging, meals and incidentals, and one round-trip travel to Berkeley, CA

After the two-week workshop, each participant will:

have the opportunity to further their research project with the team members including the research leader
have access to funding to attend conference(s) or to meet with other team members to pursue the research project, or to present results
become part of a network of research and career mentors

* The 2023 on-site director is Dr. Anisah Nu'Man (Spelman College).

Eligibility
Applicants must be a U.S. citizen or permanent resident, possess a Ph.D. in the mathematical sciences, and be employed at a U.S. institution.

Selection Process
The guiding principle in selecting participants and establishing the groups is the creation of diverse teams whose members come from a variety of institutional types and career stages. The degree of potential positive impact on the careers of African-Americans in the mathematical sciences will be an important factor in the final decisions.

2023 Research Leaders and Topics

Folashade Agusto (University of Kansas)
Ticks, Fire and Control: Implementing prescribed fire and control measures at ticks invasion front

In recent times, ticks have been expanding their habitat ranges due to climate change, among other factors, leading to increased tick-borne illness risks in the United States. Thus, it is important to find a practical and cost-efficient way of controlling and managing ticks. Prescribed burns are intentional fires used for land and forest management; they are time and cost efficient, since they can be applied across large amounts of land. They can also be effective in controlling tick populations.

Prerequisites: Knowledge of PDEs and their numerical simulation. Knowledge of the SIR model and R0 will be appreciated.

Pre-readings: The following three resources are meant to familiarize applicants with the project:

Centers for Disease Control and Prevention (CDC): Preventing ticks in the yard
Emily Guo and Folashade B. Agusto: Baptism of Fire: Modeling the Effects of Prescribed Fire on Lyme Disease
Alexander Fulk, Weizhang Huang, Folashade Agusto: Exploring the Effects of Prescribed Fire on Tick Spread and Propagation in a Spatial Setting. (Code used in this study can be found at Github.)

Jonathan Esole (Northeastern University)
Elliptic fibrations, flops, and collisions of singularities

Elliptic fibrations are beautiful and elegant geometries at the interface of algebraic geometry,number theory, and string theory. While ubiquitous, they still have many secrets to teach us. In particular, their uses by physicists indicate deep connections of their birational geometry with representation theory and hyperplane arrangements of weight systems. The geometric singular fibers over generic points of the discriminant locus of a smooth elliptic fibration are famously classified by Kodaira and Neron. They have dual graphs corresponding to twisted affine Dynkin diagrams. Under mild conditions, an elliptic fibration is birationally equivalent to a (singular) Weierstrass model. Two different crepant resolutions are connected by a sequence of flops and have some matching topological invariants, such as their Euler characteristics. We are interested in the structure of crepant resolutions of Weierstrass models of elliptic fibrations corresponding to given types of Kodaira fibers. We will explore their flops and compute their topological invariants.

Prerequisites: Familiarity with basic algebraic geometry. Background in intersection theory is a plus but not needed. Someone familiar with elliptic curves could also quickly digest the required algebraic geometry.

Donald Eugene Kemp Martin(North Carolina State University)
Inference and applications of sparse Markov model

Analysis of a categorical time series is facilitated by a model that captures the statistical properties of the sequence, while being simple enough so that statistical analysis is feasible. Markov models of an order of dependence m serve this purpose. However, the number of parameters in a Markov model is exponential in m, leading to variance considerations because many parameters need to be estimated from data. Thus first-order Markov chains (m = 1) are more frequently used. Yet using a lower-order model when higher-order dependence is called for leads to bias. Sparse Markov models help with this bias-variance trade-off. A sparse Markov model (SMM) is a higher-order Markov model for which conditioning m-tuple histories are grouped into classes such that the conditional probability distribution is constant over m-tuples in the same class. The clustering reduces the number of parameters.

Prerequisites: Basic understanding of probability and statistics, including knowledge of Markov chains, model fitting and statistical inference.

Bibliography

Nsoki Mavinga (Swarthmore College)
Nonlinear elliptic partial differential equations with nonlinear boundary conditions

Since Bernstein's pioneering work in nonlinear elliptic equations in 1906 and Leray's work on hydrodynamical problems in 1933, there has been a significant amount of interest in the study of nonlinear elliptic partial differential equations. Henceforth, there has been a tremendous emphasis in studying partial differential equations that enable understanding and modeling of nonlinear processes such as chemical, biological or ecological processes. This area has become central in modern-day mathematical research. The study of nonlinear elliptic partial differential equations with linear boundary conditions is relatively well studied and the literature is extensive. However, problems with nonlinear boundary conditions have recently gained attention due to their importance in the study of several branches of pure and applied mathematics such as the theory of quasiregular and quasiconformal mappings in Riemannian manifolds with boundary, non-Newtonian fluids, nonlinear elasticity and glaciology, among others.

In this research project, we will focus on the qualitative (analytical) study of nonlinear elliptic equations subject to nonlinear boundary conditions. In particular, we consider problems with nonlinear reactions in the interior as well as on the boundary of the domain in consideration. We will investigate the existence, nonexistence, multiplicity, and properties of solutions. In the treatment of the proposed problems, new analytical tools will be developed as well as utilize the methods of nonlinear functional analysis such as sub-supersolution method, degree theory, bifurcation theory and fixed points theorems.

Prerequisites: Participants need to have some knowledge of (graduate) partial differential equations, and real and functional analysis.

Bibliography

ADJOINT Program Directors

Dr. Edray Goins, Pomona College
Dr. Caleb Ashley, University of Michigan
Dr. Naiomi Cameron, Spelman College
Dr. Jacqueline Hughes-Oliver, North Carolina State University
Dr. Anisah Nu’Man, Spelman College (2023 site director)

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