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Organizing Committee
American Institute of Mathematics
American Institute of Mathematics (AIM) & Iowa State U
ICERM/Morehouse College
Abstract
This workshop, a formal collaboration between ICERM and the American Institute of Mathematics (AIM), is one in a series of annual REUF workshops. These workshops bring together leading research mathematicians and faculty based at primarily undergraduate institutions to investigate open questions in the mathematical sciences and to equip participants with tools to engage in research with undergraduate students. REUF also serves to jump-start faculty who want to re-engage in research or who are considering a change in their research area.
The workshop will be hosted at ICERM.
The goals of this workshop are to promote undergraduate research and to forge research collaborations among the participating faculty. The majority of the workshop will be spent working on problems in small research groups, reporting on progress, and formulating plans for future work. Note that there are opportunities for participants to continue research activities beyond the workshop week, which will be discussed during the workshop.
Preference will be given to faculty who teach or mentor substantial numbers of minority students underrepresented in mathematics, students with disabilities, or first-generation students.
Workshop Leaders:
The workshop will be hosted at ICERM.
The goals of this workshop are to promote undergraduate research and to forge research collaborations among the participating faculty. The majority of the workshop will be spent working on problems in small research groups, reporting on progress, and formulating plans for future work. Note that there are opportunities for participants to continue research activities beyond the workshop week, which will be discussed during the workshop.
Preference will be given to faculty who teach or mentor substantial numbers of minority students underrepresented in mathematics, students with disabilities, or first-generation students.
Workshop Leaders:
- Elizabeth Gross (University of Hawaii at Manoa)
- Helen Grundman (Bryn Mawr College)
- Victor Moll (Tulane University)
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Workshop Schedule
Monday, August 07, 2023
9:00 - 9:20 AM EDT
Check In
11th Floor Collaborative Space
9:20 - 9:30 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:30 - 10:00 AM EDT
Welcome to REUF / Introduction
Program Overview - 11th Floor Lecture Hall
Leslie Hogben, American Institute of Mathematics (AIM) & Iowa State U
Ulrica Wilson, ICERM/Morehouse College
10:00 - 10:30 AM EDT
Project 1 Introduction - Maximum likelihood degree of stochastic block models
11th Floor Lecture Hall
Project Leader
Elizabeth Gross, University of Hawai'i at Mānoa
Abstract
A popular class of statistical models studied in algebraic statistics are log-linear models. Log-linear models are popular since they have a monomial parameterization and correspond to toric ideals. Toric ideals are polynomial ideals generated by binomials with lots of combinatorial structure, making them great objects for exploration at varying levels of depth. For our project, we will focus on log-linear models that appear in network analysis, in particular, stochastic block models, which are used for clustering tasks in networks. We will study the geometry of maximum likelihood estimation for these models, specifically the maximum likelihood degree. The maximum likelihood degree is the number of complex solutions to the system of likelihood equations, and thus, for part of our project we will explore different computational software to solve such systems and different techniques to bound the number of solutions.
10:30 - 11:00 AM EDT
Coffee Break
11th Floor Collaborative Space
11:00 - 11:30 AM EDT
Project 2 Introduction - A New Variation on Happy Numbers
11th Floor Lecture Hall
Project Leader
Helen Grundman, Bryn Mawr College
Abstract
The happy function maps a positive integer (expressed in base 10) to the sum of the squares of its digits. A happy number is a positive integer that, under repeated applications of the happy function, maps to 1. The iterative behavior of the happy function and its many variations has been the subject of study for over 75 years. In this project, we'll study a new variation on the classic happy function, working to establish results paralleling those known for the classic
happy function and/or its other variations.
11:30 AM - 12:00 PM EDT
Project 3 Introduction - BRACKETS: A NEW METHOD FOR INTEGRATION
11th Floor Lecture Hall
Project Leader
Victor Moll, Tulane University
Abstract
The evaluation of integrals is one of the subjects taught in Calculus courses. Even tough this appears to be an elementary subject, there is no complete algorithm to compute every integral. The project will be based on a new method to evaluate definite integrals. This is called the method of brackets, invented by Ivan Gonzalez at his Physics Ph.D. thesis at Universidad Santa Maria, Valparaiso, Chile in the context of Feynman diagrams. These diagrams explain the interaction of elementary particles and their parametrization produces very complicated integrals. The evaluation of this integrals give all the fundamental properties of particles. This is a heuristic method, based on a small number of rules, some of which have established rigorously.
The project will involve a combinations of symbolic computation and also some theory needed to prove the rules of the method.
This is a remarkable method: it reduces the evaluation of definite integrals to the solution of a system of linear equations, usually of very small size.
12:00 - 2:00 PM EDT
Lunch/Free Time
2:00 - 2:15 PM EDT
Selection of Project Groups
Group Work - 11th Floor Lecture Hall
2:15 - 3:30 PM EDT
Group Work
Assigned Group Work Space
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 5:00 PM EDT
Group Work
Assigned Group Work Space
Projects
Maximum likelihood degree of stochastic block models
Elizabeth Gross (University of Hawai'i at Mānoa)
A popular class of statistical models studied in algebraic statistics are log-linear models. Log-linear models are popular since they have a monomial parameterization and correspond to toric ideals. Toric ideals are polynomial ideals generated by binomials with lots of combinatorial structure, making them great objects for exploration at varying levels of depth. For our project, we will focus on log-linear models that appear in network analysis, in particular, stochastic block models, which are used for clustering tasks in networks. We will study the geometry of maximum likelihood estimation for these models, specifically the maximum likelihood degree. The maximum likelihood degree is the number of complex solutions to the system of likelihood equations, and thus, for part of our project we will explore different computational software to solve such systems and different techniques to bound the number of solutions.
A New Variation on Happy Numbers
Helen Grundman (Bryn Mawr College)
The happy function maps a positive integer (expressed in base 10) to the sum of the squares of its digits. A happy number is a positive integer that, under repeated applications of the happy function, maps to 1. The iterative behavior of the happy function and its many variations has been the subject of study for over 75 years. In this project, we'll study a new variation on the classic happy function, working to establish results paralleling those known for the classic happy function and/or its other variations.
Brackets: A New Method for Integration
Victor Moll (Tulane University)
The evaluation of integrals is one of the subjects taught in Calculus courses. Even tough this appears to be an elementary subject, there is no complete algorithm to compute every integral. The project will be based on a new method to evaluate definite integrals. This is called the method of brackets, invented by Ivan Gonzalez at his Physics Ph.D. thesis at Universidad Santa Maria, Valparaiso, Chile in the context of Feynman diagrams. These diagrams explain the interaction of elementary particles and their parametrization produces very complicated integrals. The evaluation of this integrals give all the fundamental properties of particles. This is a heuristic method, based on a small number of rules, some of which have established rigorously. The project will involve a combinations of symbolic computation and also some theory needed to prove the rules of the method. This is a remarkable method: it reduces the evaluation of definite integrals to the solution of a system of linear equations, usually of very small size.
