The Modern Math Workshop (MMW) is a two-day workshop which will take place in conjunction with the 2023 NDiSTEM Conference of the Society for the Advancement of Chicanos and Native Americans in Science (SACNAS). It will showcase contemporary research happening at NSF-funded mathematical sciences institutes around the country. It grew out of a similarly named yearly event originally organized by MSRI and first held at Howard University in 2002. It became a collaboration with SACNAS in 2006 and has been jointly organized by the Mathematical Sciences Institutes since 2008. Since 2011 this event has been funded by the NSF through the Mathematical Sciences Institute Diversity Initiative. This year, IMSI is the lead organizer for the MMW. The workshop is a mix of activities including research expositions aimed at graduate students and researchers, mini-courses aimed at undergraduates, a keynote lecture by a distinguished scientist, and a reception where participants can learn more information about the Mathematical Sciences Institutes. This page will be updated as specifics about workshop activities become available.
The Modern Math Workshop encourages undergraduates from underrepresented groups to pursue careers in the mathematical sciences, and builds research and networking opportunities among undergraduates, graduate students and recent PhDs. Funding is available to cover travel and lodging and SACNAS NDiSTEM registration for participants in the MMW, as well as to fund lodging for MMW participants who also wish to attend the NDiSTEM conference.
Undergraduate Mini Course I: An Introduction to Data Analytics (Part 1)
Speaker: Marco Martinez & Alicia Prieto Langarica (North Central College & Youngstown State University)
We will offer an introduction to Data Analytics, which refers to a set of tools designed to extract valuable information from large datasets. Our course will cover the general theoretical concepts that enable the selection of various techniques. Additionally, we will provide a hands-on introduction to the specialized software R. Our primary focus will be on regression methods, highlighting their significance and practical applications.
12:50-14:20 PDT
Undergraduate Mini Course II: Polynomials, Topology, and Us (Part 1)
This minicourse will be divided into three parts, with related but distinct topics and prerequisites, and focuses broadly on the geometry and topology of solving equations. The first sub-session, accessible to a general audience, begins with a historical survey of an ancient question (“How do you solve a polynomial?”) and some of the beautiful mathematics that has developed through its study. Next we explore spaces of polynomials from a more topological perspective—you are guaranteed to learn something new about the quadratic formula!—in an open-ended way that hints at venues of ongoing research. We conclude by contextualizing the work of solving algebraic equations in the miracles of classical geometry, pushing beyond the solvable-unsolvable dichotomy, and contemplating math as a rich human endeavor that we find ourselves at the heart of.
12:50-13:30 PDT
Research Talk – AIM: Electrical Power Networks, Combinatorial Optimization, & Collaborative Research
Speaker: Daniela Ferrero (Texas State University)
Electric power networks need continuous monitoring in order to prevent blackouts and power surges. This is usually accomplished by placing Phasor Measurement Units (PMUs) at strategically selected network locations. Each PMU measures magnitude and phase angle of the electromagnetic wave at the location where it is placed. The synchronized PMU readings are then combined to determine the magnitude and phase angle of the electromagnetic wave at network locations without a PMU. This method requires a PMU placement that provides sufficient information to monitor the entire network, while due to their cost, the number of PMUs should be minimized. When an electrical power network is modeled by a graph, a PMU placement using the minimum number of PMUs corresponds to a minimum power dominating set for the graph.
In 2002, soon after PMU systems were proposed in electrical engineering, power domination was introduced to graph theory. However, the large scale deployment of wide area measurement systems of PMUs started a decade later. Since then, the cost of a PMU and its installation have been reduced, and PMU readings have been proven useful for many other purposes besides monitoring, which helps to offset system costs. In recent years, the analysis of existing large scale PMU systems has shown that minimizing the number of PMUs alone yields unsatisfactory results, since the lack of redundancy results in lost, or incorrect, PMU readings. While having redundancy implies increasing the number of PMUs, the advantages of having a few additional PMUs outweigh the cost increase. As a consequence, new challenges have appeared in both, electrical engineering and graph theory.
In this talk, I will introduce power domination as a combinatorial optimization problem and then I will present results of my research in this area, obtained in collaboration with a large number of co-authors. While doing so, I will adopt a personal perspective to emphasize the collaborative nature of my research and highlight the role of several NSF supported institutes to enable the collaboration.
13:35-14:15 PDT
Research Talk – ICERM: Computational Simulation of Maxwell’s Equations in Complex Materials
Speaker: Vrushali A. Bokil (Oregon State University)
Maxwell’s equations are a set of four vector partial differential equations that govern the evolution of electromagnetic waves. In most cases, due to the presence of heterogeneities and complex geometries, it is impossible to solve these equations exactly. Thus, efficient computational methods that are accurate, consistent, and stable are required to be developed to address numerous applications in materials research involving the design of engineered composites such as metamaterials; radar, environmental and medical imaging; noninvasive detection of cancerous tumors; communication and computation; global climate assessment, among others. Many applications involve multiple spatial and temporal scales, spatial and temporal heterogeneities, complex geometries, among other issues. In this talk, I will discuss some of the aspects of the development of robust computational tools for simulating Maxwell’s equations. The numerical simulation of these equations and many other complex partial differential equations is routinely addressed at workshops and semester programs organized by the Institute for Computational and Experimental Research in Mathematics (ICERM). I will discuss some of the current programs organized by ICERM that are relevant to this talk and include efforts to diversify the computational mathematics community.
14:15-14:45 PDT
Break
14:45-16:15 PDT
Undergraduate Mini Course I: An Introduction to Data Analytics (Part 2)
Speaker: Marco Martinez & Alicia Prieto Langarica (North Central College & Youngstown State University)
We will offer an introduction to Data Analytics, which refers to a set of tools designed to extract valuable information from large datasets. Our course will cover the general theoretical concepts that enable the selection of various techniques. Additionally, we will provide a hands-on introduction to the specialized software R. Our primary focus will be on regression methods, highlighting their significance and practical applications.
14:45-16:15 PDT
Undergraduate Mini Course II: Polynomials, Topology, and Us (Part 2)
This minicourse will be divided into three parts, with related but distinct topics and prerequisites, and focuses broadly on the geometry and topology of solving equations. The first sub-session, accessible to a general audience, begins with a historical survey of an ancient question (“How do you solve a polynomial?”) and some of the beautiful mathematics that has developed through its study. Next we explore spaces of polynomials from a more topological perspective—you are guaranteed to learn something new about the quadratic formula!—in an open-ended way that hints at venues of ongoing research. We conclude by contextualizing the work of solving algebraic equations in the miracles of classical geometry, pushing beyond the solvable-unsolvable dichotomy, and contemplating math as a rich human endeavor that we find ourselves at the heart of.
14:45-15:25 PDT
Research Talk – IAS: A Random Walk Through TCS
Speaker: Fernando Granha Jeronimo (Simons Institute and Institute for Advanced Study)
In this talk, we will take a random walk through the field of theoretical computer science (TCS),
which lies at the intersection of mathematics and computer science. The “laws” that govern computation
are fundamentally mathematical, and the study of their power and limitations has uncovered a vast range
of beautiful results while also raising challenging questions. We will begin with the foundations of the field
and explore some of its modern ramifications. Our main goal is to offer a taste of this fascinating field,
including some of its contemporary aspects, in order to serve as an invitation for further exploration.
15:30-16:10 PDT
Research Talk – IPAM: Integrability and Combinatorics of Skew Formulas for Skew Tableaux
Speaker: Alejandro Morales (Université du Québec à Montréal)
Algebraic combinatorics and integrable probability are two areas of mathematics with recent interactions. In one direction, we have seen many applications of tools and emergence of objects from algebraic combinatorics in integrable models in statistical mechanics. In the opposite direction, tools and ideas from probability and statistical physics have seen application in problems from Algebraic Combinatorics. One example of this is enumerative and asymptotic formulas for important structure constants like standard tableaux of skew shape. No product formula is known to count such tableaux. In 2014, Naruse presented a formula as a positive sum over excited diagrams of products of hook-lengths. These diagrams can be viewed as lozenge tilings of certain regions. Shortly after, Morales, Pak, and Panova gave a q-analogue of Naruse’s formula for semi-standard tableaux of skew shapes in terms of restricted excited arrays. They also showed, partly algebraically, that the Hillman-Grassl map restricted to skew shapes is the bijection between skew SSYTs and excited arrays. We study the problem of making this argument completely bijective. For a skew shape, we define a new set of semi-standard Young tableaux, called the “minimal SSYT”, that are equinumerous with excited diagrams via a new description of the Hillman–Grassl bijection. The minimal skew SSYT are the natural objects to compare with the terms of another formula for counting skew tableaux by Okounkov–Olshanski. This is joint work with Greta Panova (USC) and GaYee Park (UQAM).
16:25-17:25 PDT
Panel Discussion: Personal Stories from the Path to Academia
Moderator: Selenne Bañuelos (IPAM)
Panelists: Jose Israel Rodriguez (University of Wisconsin – Madison), Johnny Guzman (Brown University), Juanita Pinzón Caicedo (University of Notre Dame), Daniela Ferrero (Texas State University), Vrushali Bokil (Oregon State University), Alejandro Morales (University of Massachusetts Amherst), Fernando Granha Jeronimo (Simons Institute and Institute for Advanced Study)
17:30-19:30 PDT
Modern Math Workshop Reception
Thursday, October 26, 2023
8:30-9:00 PDT
Coffee & Pastry
9:05-9:45 PDT
Research Talk – SL Math: Toroidal Integer Homology Spheres have Irreducible SU(2)-Representations
Speaker: Juanita Pinzón Caicedo (University of Notre Dame)
The fundamental group is one of the most powerful invariants to distinguish closed three-manifolds. One measure of the non-triviality of a three-manifold is the existence of non-trivial SU(2)-representations. In this talk I will show that if an integer homology three-sphere contains an embedded incompressible torus, then its fundamental group admits irreducible SU(2)-representations. This is joint work with Tye Lidman and Raphael Zentner.
9:50-10:30 PDT
Research Talk – IMSI: 4×4 Matrices in Algebraic Statistics: From Independence to Mixing the Tree of Life
Speaker: Jose Rodriguez (University of Wisconsin, Madison)
A mantra of Algebraic Statistics is that a statistical model can be realized as the restriction of an algebraic variety to a semi-algebraic set. For discrete models this semi-algebraic set is the probability simplex, while for Gaussian models it is the positive definite cone. With this perspective, the statistical inference method maximum likelihood estimation (MLE) can be viewed as solving a system of polynomial equations. For discrete models, the likelihood equations and methods to solve them, were introduced by Hosten, Khetan, and Sturmfels. These equations are with respect to observed data and a model. The number of solutions for generic data is called the maximum likelihood (ML) degree of the model. Thus, the ML degree can bound the algebraic complexity of solving MLE. In this talk I will show how the ML degree connects to statistics, topology and phylogenetics with examples stemming from 4×4 matrices.
10:30-11:00 PDT
Break
11:00-12:00 PDT
Plenary Talk: Connections Between Numerical PDEs and Algebraic Topology
Speaker: Johnny Guzman (Brown University)
The finite element method (FEM) is an important tool in approximating solutions to partial differential equations (PDEs). For a few decades now connections between algebraic topology and FEM have been exploited, most extensively by the work of Arnold Falk and Winther in the finite element exterior calculus. In this talk, we will discuss a few of these connections.
The deadline to apply for funding was August 11, 2023.
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