Meaningful Mathematics: Social Justice in College Mathematics 

Mathematics is often viewed as the bastion of neutrality and logic, the "universal language", but it is also a tool used to describe and validate the values of individuals and society.  Mathematics is not necessarily a value-free subject. There are two major conversations surrounding college-level mathematics and social justice:  social justice pedagogy and a social justice curriculum.  This talk will briefly touch on how we teach and assess mathematics through a social justice lens with the main focus on intertwining social justice into curriculum from introductory quantitative reasoning courses, through calculus, and into graduate study to inform STEM students on the active role mathematics has in a democratic society.

The Role of Identity Development in STEM Student Retention

Statistical modeling has been used to explore social phenomena since the early 1990's. The lack of inclusivity in classes and social inequality, particularly in STEM disciplines, are known to impact retention in the science disciplines. This talk will explore students' social behavior, based on identity and other factors, that impacts their academic success. Microsimulation studies will be implemented to determine which behaviors are most favorable, and the influence of identity development on student's experiences.

Populations, Individuals, and What the Doctor Tells the Patient

We are awash in medical advice. Our doctors quote the latest studies on the virtues or hazards of exercise, alcohol, fruits, coffee, breads, statins, proton-pump inhibitors, antidepressants, sleep, broccoli, and almost every other matter of medication and lifestyle. Much of this comes from well-designed randomly controlled trials (RCT, “the gold standard”), often published in the top journals. Doubts have been raised about the validity of many of these studies, including a built-in “publication bias.” But even if we accept the validity of the trial, the correctness of the analysis, and the wisdom of the editorial board, how are we to interpret population statistics in terms of individual patients? The response of a person to a treatment depends on many things, most of which are not known to the investigator, and in any case can not be controlled by either the investigator or the individual. 

I will suggest a simple thought experiment as a way to bound the probabilities of significant benefit and significant harm resulting from the treatment of an individual. I will apply these bounds to treatments shown to produce positive outcomes at the population level, and reported on in our best medical journals. I will argue for new metrics to help doctors and patients make rational choices.

The Mathematician as a Public Intellectual

We commonly think of mathematicians primarily as researchers and teachers. This is natural, as these have historically been the aspects of our job that are most prominent. However, previously reclusive mathematicians are starting to develop public personae with recent widespread use of social media (tweets, blogs, facebook posts, op-eds, etc.,) and gaining both notoriety and admiration. In this talk, I will highlight some of the social benefits of making public the scholarship of mathematicians, the boundaries that some have pushed, the conversations that have been sparked by controversy, and the backlash that some people have had to endure. In particular, we will explore the question:  What are the rights and responsibilities of mathematicians as public intellectuals?

A Combinatorial Proof of an Euler Type Identity Due to Andrews

The most famous identity in the theory of partitions is likely Euler’s identity which states that the number of partitions of n into distinct parts equals the number of partitions of n into odd parts. It is natural to ask what happens if one relaxes the conditions on the parts of the partitions counted in Euler’s identity. Let a(n) be the number of partitions of n such that the set of even parts has exactly one element, b(n) be the difference between the number of parts in all partitions of n into odd parts and the number of parts in all partitions of n into distinct parts, and c(n) be the number of partitions of n in which exactly one part is repeated. Beck conjectured that a(n) = b(n) and Andrews, using generating functions, proved that a(n) = b(n) = c(n). We give a combinatorial proof of Andrews’ result. Our proof relies on bijections between a set of partitions and a multiset of partitions, where the partitions in the multiset are decorated with bit strings.

Exploratory Use of Tree-Based Methods in Biomedical Research: Modeling Metabolic Outcomes Based on Genetic Risk Factors in Patients with Schizophrenia

Schizophrenia is a psychiatric illness characterized by psychotic symptoms, cognitive impairment, and social dysfunction. Unfortunately, antipsychotic medications used to treat schizophrenia have severe metabolic side effects. It is unclear whether metabolic risk genes established in the general population are also associated with poor metabolic outcomes among antipsychotic-treated patients. Exploration of this clinical question requires a statistical approach that accommodates variable selection and allows for potentially complex interactions among covariates. Tree-based machine learning algorithms such as Classification and Regression Trees (CART) handle these data analytic challenges and provide results that are easily interpretable for physicians. We review implementation of CART using the R statistical software package and demonstrate the utility of tree-based algorithms using a motivating data set provided by the Schizophrenia Research Program at Massachusetts General Hospital.  Among a cohort of several hundred patients, genetic risk factors emerged as predictive of several metabolic outcomes, suggesting that genetic screening may prove useful for personalized decision-making and treatment management.

Spatial Point Analysis of Segregated Communities and Greenhouse Gas Sources in New York

In this research project, we hypothesized that sources of pollution are more likely to be built in highly segregated communities because segregated communities lack the social power to resist the establishment of these sources and disadvantaged racial minorities could be forced to live near existing sources. To test this, we designed a way to quantify segregation across a range of spatial scales using spatial pattern analysis tools: Ripley’s K function and the pair correlation function. We designed a coding framework to efficiently calculate these functions from 2010 census data. Previous studies typically only identify segregation at a single spatial scale, so our approach presents a novel way to understand this phenomenon. Using these results with EPA data on Greenhouse Gas (GHG) emissions, we examined the relationship between segregation and the location of pollution sources. Preliminary results show that areas with higher levels of racial segregation were more likely to contain major GHG emission sources. Our computational design also allows us to quantify the relationship between racial groups and pollution sources across a continuum of spatial scales, and preliminary results show that disadvantaged racial groups are more likely to be clustered near pollution sources. In the past month, we have been able to refine our code further to account for other aspects such as intensity of population in different areas and correct for edge-effects. These results further strengthen our hypothesis and are very promising. The complete project will provide a detailed analysis of correlation between GHG emissions and segregation that can act as a basis for policy reform.

Pairing-Based Cryptography in Theory and in Practice

Pairing-based cryptography, which takes advantage of the algebraic properties of elliptic curves, lies at the foundation of modern cryptosystems. In this talk, I will give an overview of the basic concepts and algorithms used in elliptic curve cryptography and I will discuss a new cryptosystem made for use in elections.

Analogue of Vector Space Basis in Groups

Ever wondered how linear algebra ideas can be applied to group theory? We look at invariants of groups that can take the place of the dimension of vector spaces. A natural idea is to look at generators of the group to form a basis. Then our adventure takes some magical (yet logical!) directions...

Optimizing Sparse Representations of Kinetic Distributions

The United States Air Force Research Laboratory uses kinetic simulations to reduce costs in their various research projects, including plasma simulations.  When performing these simulations, probabilistic methods are employed to reduce the computational expense of estimating the physical entropy of the system. These techniques introduce an error term in the estimation, which we seek to reduce by developing a more efficient algorithm.  We discuss the nature of kinetic simulations, relevant mathematical background, and methods for error analysis. We then present multiple algorithms to estimate the physical entropy from common sampling distributions.  Some techniques explore the use of Binary Trees and the roots of Legendre polynomials, as well as a combination of the two. Finally, we discuss the performance of these algorithms and provide suggestions for further research.

Efficient Approximations for Large Sparse Interacting Particle Systems

Stochastic dynamics arising in a variety of applications including epidemiology, statistical physics, and load balancing are modeled by so called interacting particle systems. The dynamics of interacting particle systems are governed by an underlying graph structure and are typically high dimensional systems involving a large number of particles. The high dimensionality makes the characterization of the dynamics of a typical particle challenging. Approximations for the dynamics have been mainly studied in the case where the underlying graph is dense. Building on recently developed theory, we develop novel computational algorithms for approximating local dynamics on sparse graphs. We demonstrate the effectiveness of our approximations through numerical results, and show that in many cases they outperform existing algorithms.