MATHEMATICS

Speaker: Kira Adaricheva (Hofstra University) and Eric Rowland (Hofstra University)
Title: Summer Research in NYC
Abstract: This summer, Kira Adaricheva and Eric Rowland were faculty mentors in the New York Discrete Math REU program, which ran at Baruch College in Manhattan.

Kira Adaricheva mentored two students, who chose to work on questions related to convex geometries (CG), discrete structures with the anti-exchange axiom. In one project it was possible to generalize the property for a representation of CG by circles on the plane into higher dimensions, and in another, the so-called Weak Carousel Property holding for specific convex shapes on the plane was extended to more shapes, depending on the number of common supporting lines of any pair of these shapes.

Eric Rowland mentored two students in research projects. The first project concerned two-dimensional arrays of 0s and 1s that avoid a given contiguous subarray. For example, how many m x n arrays don’t contain a 2 x 2 block of 0s? Fixing m and varying n, we get a sequence that satisfies a recurrence. The size of this recurrence seems to behave nicely as m varies, but it’s not clear why. The second project was motivated by Chebyshev’s 1850 precursor to the prime number theorem. Certain ratios of factorials always produce integers. For example, n!(30n)!/((6n)!(10n)!(15n)!) is an integer for each positive integer n. A characterization of such ratios is known. Moreover, they all have the property that their generating series satisfies a polynomial equation. What is the degree of this equation? We have a conjectural bound for one family.

Speaker: Quinn Murphy (Hofstra University) 
Title: Predicting Post-Discharge Survival Probability of Brain Bleed Patients: A Machine Learning Approach
Abstract: This summer, Quinn Murphy was a researcher at the Big Data Summer Institute (BDSI) REU program at the University of Michigan. Throughout the course of this 6-week program, Quinn, along with Hanrui Yu (Peking University), Eliana Dietrich (University of Washington), and Joshua Brown (Johns Hopkins University) worked on “Predicting Post-Discharge Survival Probability of Brain Bleed Patients: A Machine Learning Approach” under the mentorship of Dr. Rahul Ladhania and Dr. Katherine Brumberg.

Brain bleeds are severe medical conditions which can result in long-lasting complications or death.  Intracerebral Hemorrhages, a type of brain bleed, account for 10-20% of all strokes. Overall, brain bleeds present a mortality rate of 40-50% in the first days following the event. However, even for patients discharged, the risk of mortality still remains. Patients discharged, on average, face a 22% mortality rate for 90-day survival. The Beth Israel Deaconess Medical Center in Boston, MA created the dataset MIMIC III, which contains the electronic health records of ICU patients. In this dataset, there were a total of 1330 brain bleed patients with only 756 discharged. This study aims to predict the survival probabilities across different time intervals of discharged brain bleed patients in order to better inform post-discharge care and long-term outcomes. Through use of machine learning, this project seeks to aid clinicians by creating models with both predictive accuracy and interpretability.

Speaker: Athar Abdul-Quader (Purchase College, SUNY)
Title: Nonstandard Models, Part II
Abstract: When we spoke (last semester) about nonstandard models, we talked about formalization of the basic axioms of arithmetic, the birth of first-order logic, and nonstandard models of Peano Arithmetic. These are mathematical structures where the axioms of arithmetic hold, but contain numbers larger than any standard natural number (infinite numbers?). We discussed what these models look like, and some basic results of number theory that must be true of these structures. In this talks, I will do a quick refresher on nonstandardness, and then launch into a proof of a well-known result in combinatorics, Ramsey’s Theorem for pairs, using nonstandard models of arithmetic. If time allows, I hope to explore some other peculiarities of nonstandard models of arithmetic, including Gödel’s incompleteness results.

Speaker: Richard Gustavson (Farmingdale State College)
Title: The Algebra of Integral Equations
Abstract: An integral equation is one involving an unknown function inside an integral operator. In order to study integral equations algebraically, the governing identities of the integral operators involved must be known. For the standard integral operator, this is the integration by parts identity. Most integral equations seen in applications, however, contain a nontrivial kernel as part of the integrand, in which case integration by parts no longer holds. In this talk, we will discuss various modifications that can be made to the integration by parts identity for separable Volterra integral operators, and the interesting algebraic theories that arise as a consequence.

Speaker: Dan Turetsky (Victoria University of Wellington)
Title: Isomorphisms and games
Abstract: What does it mean to say that two mathematical objects are the same?  The usual answer is “isomorphism”, which you might have seen in algebra: groups and rings.  We’ll focus on linear orders instead (and I’ll explain what it means, so don’t worry if you haven’t seen this before).

How can you tell if two objects are isomorphic?  If you can write down an isomorphism, then they are.  But how do you show that two objects are not isomorphic?  We’ll talk about how you can answer this question with a specific kind of game played between two players.

Speaker: Hamza Virk
Title: When Can Second-Order Rational Recurrences Produce Integers?
Abstract: When do integers appear in second-order linear recurrence sequences with rational coefficients? We explore this question through extensive computational searches that reveal surprising patterns. By carefully selecting initial conditions, we can construct sequences that produce remarkably long consecutive runs of integers, with some examples yielding over thirty integer terms before all subsequent values become non-integers. Under certain coprimality conditions, we establish restrictions that prevent consecutive integers beyond the initial terms. We also develop a recursive method that combines multiple sequences to extend these integer runs even further.

Speaker: Quinn Murphy
Title: Experimental Design using multi-reader multi-case statistical analysis for two-alternative forced choice and forced localization task evaluation of undersampling in MRI
Abstract: Observer studies are used to evaluate medical imaging systems. In an observer study, the observer typically performs a task such as finding a small tumor in an MRI image. In this study, we evaluated the Two-Alternative Forced Choice (2-AFC) and Forced Localization (FL) tasks in estimating human observer signal detection performance in undersampled MRI images. In a 2-AFC study, the observer determines whether the lesion is at one of two locations. In an FL study, the observer needs to find the signal in a region. We used Multi-Reader Multi-Case analysis (MRMC) to estimate the variance of human observer performance which incorporates reader variability, case variability and the correlation between the two. Undersampling refers to the process of collecting less data to speed up the MRI scan. We undersampled in the k-space by a factor of 5x, collecting varying percentages of fully collected low frequencies: 0% (pure aliasing) to 20% (pure blurring). The choice between a fully crossed (where every reader sees every case) and split-plot design (where every case is only seen by one reader) in experimental design is dependent on the availability of reader-time and ease of image acquisition. We chose a fully crossed design. The observer studies were made up of the 2-AFC and FL tasks in each condition, each with 200 trials in which a signal was placed. The FL task average human performance improved between the 0 and 1.25% conditions and remained relatively constant for the following conditions. Through hypothesis testing, we determined that there is a statistical difference in the performance between these conditions, suggesting that there is a decrease in performance solely under the full aliasing condition. This difference is not observed in 2-AFC performance. We found that the 2-AFC task is too simple to capture the effects of pure aliasing on image quality. The FL task is able to capture the loss in image quality in purely aliased images through a decrease in human observer performance.

Speaker: Justine Prasad
Title: Modeling human observer performance with neural network observers in a forced localization task using undersampled MRI images
Abstract: In a forced localization (FL) task, people must identify the location of a pathology in MRI images. This is difficult to model because there are many possible locations, accounting for human behavior, and the anatomical backgrounds are complex. FL tasks are used to optimize medical imaging, but face a challenge that the images vary drastically for different imaging conditions being considered for example, the percentage of low frequencies fully collected in MRI. In this study, we examined human FL performance using undersampled MRI images in which different amounts of low-frequency data were preserved: from 0% (aliasing) up to 20% (blurring). We developed and tested two neural networks. The first, FLNet, is a modified EfficientNet-B1 trained directly to predict the target coordinates of the lesion. The second, V1FLNet, added a biologically inspired “V1Block” which uses Gabor filters to mimic early stages of human vision before passing the data to FLNet. To better match human variability, we added Gaussian noise to FLNet and Poisson noise to V1FLNet. When trained on a single type of image, both networks outperformed average human accuracy but did not generalize like humans to new types of images. Training on both extreme conditions (0% and 20%) allowed for interpolation, which produced more human-like generalization. An adaptive approach that trained across all conditions but was calibrated to human performance at 20% provided the closest match to human behavior overall. Previous linear models inspired by the human visual system were unable to reach human performance in the forced localization task. Our findings show that neural network-based observers can surpass human accuracy and, when trained adaptively, can approximate human-like generalization to challenging out-of-distribution images. More work is needed to validate the effectiveness of V1FLNet.

Speaker: Angel Pineda, Hofstra University
Title: Do you want to get a grant? Obtaining external funding for mathematics students and faculty
Abstract: Are you a student who is interested in getting funding to go present your research at a conference, getting a scholarship or getting a fellowship to attend graduate school? Are you a faculty member who is interested in external support for your teaching, research or service?
In this talk, I will share my experience in seeking and sometimes getting external funds both for students and faculty. I will describe the process of identifying funding opportunities which match one’s interests and what that funding can do. I will also share some strategies for increasing your success rate and the importance of resilience. The goal is for everyone to have an opportunity for external funding they would like to explore in the future and for us to support each other in the application process.

Joint seminar with the Zucker School of Medicine

Speaker: Allison Siegel, Hofstra University 
Title: Rifampin Resistance Among Tuberculosis Patients in Ukraine
Abstract: Tuberculosis (TB) remains a global public health challenge despite advancements in medical research and interventions.

This study focuses on the emergence of drug-resistant TB strains, particularly multidrug-resistant tuberculosis (MDR-TB) and extensively drug-resistant tuberculosis (XDR-TB), with a specific emphasis on rifampin-resistant TB (RR-TB).

The data was collected in Ukraine from 2015-2018, and it was analyzed by a group of UG students under the supervision of Dr. Helen Jenkins (Associate Professor in Biostatistics at Boston University’s School of Public Health), in the framework of Summer Institute in Biostatistics and Data Science.

This study aims to highlight the need for comprehensive surveillance and targeted interventions to control the spread of resistant TB strains, particularly in light of the alarming rise in RR-TB cases.

Speaker: Zoran Sunic, Hofstra University 
Title: Twin Towers of Hanoi
Abstract: We consider a version of the well-known Tower(s) of Hanoi game in which we work with two sets of pegs and disks. The pegs in the two sets are paired up, and whenever we move a disk between, say, peg 1 and peg 2 in the first set, we also move a disk between peg 1 and peg 2 in the second set (whether we like this move or not). This way, we are forced to solve two problems (one in each set of pegs/disks) simultaneously by using the exact same sequence of moves in both sets. Unlike the classical case of a single set of three pegs and any number of disks, which is fairly well understood, the twin version seems to offer a large set of more interesting/challenging problems. We will consider several such problems, provide answers or partial answers to some of them, while pointing out open questions that range from “most likely easy” to “looks difficult”. The presentation itself will be very accessible — it is a game after all.

Speaker: Anil Ventakesh, Adelphi University 
Title: What makes music sound good: Mathematical models of dissonance perception
Abstract: The concept of dissonance in music perception has been variously associated with the physical concepts of roughness, instability, and tension by appealing to a subjective cross-sensory analogy. For all the subjectivity around the term, it is noteworthy that non-musical test subjects in clinical experiments have produced remarkably consistent rank-orderings of musical sounds, according to the perceived dissonance of those sounds. Further studies in psychology, neuroscience, and mathematics show that consonance perception appears to be influenced not only by convention and culture but by the psychoacoustics of sound perception, likely by some combination of the Helmholtz theory of roughness and the theory of harmonicity originally contemplated by Galilei. In this talk, we lay out the competing mathematical theories of consonance perception and discuss the difficulties involved in applying these theories to real audio recordings (as opposed to perfectly sinusoidal waves). We show that the theory of harmonicity extends successfully to many familiar musical chords, yet fails to explain the empirical dissonance of others.

Speaker: Heidi Goodson, Brooklyn College (CUNY)
Title: An exploration of curves over finite fields
Abstract: Questions of how to find, count, and characterize points on curves have been studied since the days of Diophantus. Nearly 2000 years later, number theorists and arithmetic geometers are still fascinated by curves and are asking ever bigger and bolder questions. In this talk, I will define what it means to be a point on a curve over a finite field and attempt to answer the question “How many?” I’ll discuss estimates, exact values, and surprising trends as the order of the field varies.

Speaker: Michael Cole, Hofstra University
Title: A nonrelativistic approach to quantum spin and spin current density
Abstract: The topic of my talk is the mathematics that characterizes angular momentum in quantum physics. Besides the orbital angular momentum that a moving particle has with respect to an origin, each type of fundamental particle has an intrinsic angular momentum called spin.

By rough analogy, one may imagine a particle such as an electron as having an extension in space and rotating about an axis rather the way planet Earth rotates about its axis with a 24 hour period. For a charged particle such as an electron, the spin gives rise to a magnetic moment that causes the particle to experience a torque when in the presence of an external magnetic field. In the 1920s (the early days of quantum mechanics) physicists were quite puzzled by the experimentally proven fact that the intrinsic magnetic moment of electrons has twice the value that would be predicted by classical electromagnetic theory. The anomalous magnetic moment of electrons, and certain other particles, was first explained by the physicist Paul Dirac in the context of relativistic quantum physics (the union of quantum theory with Einstein’s special relativity).

For this reason it was wrongly thought for decades that quantum spin in general, and the anomalous magnetic moment of electrons in particular, must be understood as purely relativistic phenomena. This erroneous idea was corrected in the 1960s by the French physicist Jean-Marc Levy-Leblonde who demonstrated an adequate nonrelativistic way to account for the spin and magnetic moments of electrons. It was later noticed that Levy-Leblonde’s analysis of spin also gives a nice clarification for the existence of a spin contribution to the probability current density that is easily understood in Dirac’s relativistic context, but was hard to account for in the nonrelativistic setting without introducing unnatural ad hoc assumptions.

In my talk I will give a mathematically breezy introduction to the Schrodinger equation, develop the basic mathematics of particle spin, and argue the case that Levy-Leblonde’s approach to these issues is the most mathematically natural. The physics meaning of this mathematics is, of course, quite deep and notoriously hard to conceptually grasp even for the world’s most expert physicists. That is simply the nature of quantum physics. But much of the relevant math is surprisingly and pleasantly simple and quite interesting and deserves to be better known.

Speakers: Sandro Amaglobeli (CS/Math, Hofstra University), Allison Siegel (Math, Hofstra University), Skylar Homan (CS/Math, Hofstra University), Anoop Krishnadas (CS/Math, Hofstra University)
Title: (Amaglobeli): Modeling human observer performance in a forced localization task using transfer learning with neural networks
Abstract: To reduce the number of human trials needed to evaluate medical imaging systems, models are used to predict human observer performance. We implemented transfer learning with neural networks to predict human performance in a forced localization task with undersampled magnetic resonance imaging (MRI). This approach surpassed the performance of the previous state-of-the-art methods using physiologically derived models of the human visual system.

Title: (Siegel): Estimating the variance of human observer performance including covariance between observers and cases in forced localization tasks
Abstract: Our research addresses statistical challenges in evaluating medical imaging systems by quantifying image quality through human observer signal detection. We employ a multiple-reader multiple-case (MRMC) approach to estimate variances in observer performance considering correlations between human readers (observers) and images (cases) in a task where a human localizes a signal in an image.

Title: (Homan): Space Management in the Optimized D-basis Algorithm
Abstract: The D-basis algorithm uses ideas from lattice theory to find attribute relationships in binary tables for data analysis. The main problem with this algorithm is the storage space required: even a relatively small data set requires exponentially more space to store every implication. The goal of this project is to improve the implementation of the code used for this purpose, which reduces the space requirements of the D-basis from double exponential to linear on the input size.

Title: (Krishnadas): Identifying factors influencing STEM student retention at Long Island community colleges
Abstract: This study continues the work by the STEM2 Network to address retention
challenges among STEM students at Long Island institutions of higher education, this time focusing on Community Colleges. The research aims to identify key factors affecting students’ persistence in pursuing Bachelor of Science degrees in STEM fields. Current progress regarding data pre-processing and binning to produce binary tables for the D-basis processing will be reported.

Speaker: Rehana Patel, Wesleyan University
Title: Symmetric random constructions via model theory
Abstract: The Erdös-Rényi “coin-flip” construction is a random process that produces the Rado graph (also known as the countable infinite random graph) almost surely.  A special symmetry property exhibited by this construction is that its distribution does not depend on the order in which edges are decided.  A natural question to ask is: Which other countable infinite objects admit such symmetric random constructions? In this talk, I will provide a complete answer to this question, obtained using model-theoretic methods.  No prior knowledge of model theory will be assumed, and all definitions will be explained. This is joint work with Nathanael Ackerman and Cameron Freer.

This talk will take place over Zoom. E-mail Dr. Johanna Franklin for the link.

Speaker: Kimberly Hadaway, Iowa State University
Title: Parkordle = Parking + Wordle: Counting Lucky Spots and Lucky Cars in Parking Functions 
Abstract: Parking functions correspond with preferences of n cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. When a car parks in its preferred spot then the corresponding car and corresponding spot are deemed “lucky.” In this talk, we look briefly at lucky cars which have been studied previously and in simple cases can be understood by a generalization of a result due to Pollak. We also consider lucky spots where the situation is more complex and not previously studied. Probabilities and asymptotics for lucky spots are given for the first few spots on the one-way street. We close with an exploration of the special cases when cars enter the one-way street in either weakly-increasing or weakly-decreasing order of their preferences.

TLDR: If you’ve seen a car park before or if you’ve played Wordle before, you should come to this talk!

Speaker: Abigail Raz, Cooper Union
Title: The Path Variant of the Explorer-Director Game on Graphs 
Abstract:The Explorer Director game, first introduced by Nedev and Muthukrishnan (2008), simulates a Mobile Agent exploring a ring network with an inconsistent global sense of direction. The two players, the Explorer and the Director, jointly control the movement of a token on the graph. During each turn, the Explorer calls any valid distance, d, with the aim of maximizing the number of vertices the token visits, and the Director moves the token to any vertex distance d away with the aim of minimizing the number of visited vertices. The game, on graph G with starting vertex v, ends when no new vertices could be visited assuming both players are playing optimally, and we denote the total number of visited vertices by fd(G,v). Since 2008, many authors have explored fd(G,v) for various graph families as well as analyses of complexity. In this talk, we will focus on a variation of this game focused on path lengths rather than distances. In this variant, if the token is on vertex u, the Explorer is now allowed to select any valid path length, l,  and the Director can now move the token to any vertex v, such that G contains a uv path of length l. The corresponding parameter is denoted by fp(G,v). We will discuss how far apart fd(G,v) and fp(G,v) can be for various graph families. All necessary preliminaries will be discussed – no background in graph theory is assumed.

Speaker: Athar Abdul-Quader, Purchase College, SUNY
Title: Nonstandard numbers
Abstract: The natural numbers (0, 1, 2, …) are, in some ways, the simplest mathematical objects we can study. Traditionally they are defined as the “smallest set containing 0 and closed under successors” (adding 1). This is not the most formal definition, but in the 19th and 20th centuries, mathematicians such as Dedekind, Peano, Tarski and Gödel worked on a new branch of mathematics, mathematical logic, which intended to formalize mathematics. For example, the Peano Axioms for number theory state everything you learned about (natural) numbers in elementary school, like the commutative properties (for addition and multiplication) and the distributive property, along with the induction property: if an assertion P(x) is true when x = 0 and whenever P(x) is true, it implies that P(x+1) is also true, then P(x) is true for every natural number x. P(x) here could refer to any assertion you could make about a number x, in the language of first order logic, for example: “There is an n such that either x = 2n or x= 2n + 1.”

The story didn’t end with formalization, however. Results from first-order logic and model theory give us a window into the hidden complexities behind the formal approach, including Gödel’s famous completeness and incompleteness results. Importantly, this led to the birth of nonstandard models of the Peano Axioms. These are mathematical structures where the axioms of arithmetic hold, but contain numbers larger than any standard natural number (infinite numbers?). We will explore enough of first-order logic and model theory to study what these kinds of models look like and use them to prove results from other areas of mathematics, like Ramsey’s Theorem.

Speaker: Lauren Rose, Bard College
Title: Quads, a SET-like game with a twist
Abstract: Quads is a card game similar to SET, with 81 cards, each containing 1-4 objects in one of 4 colors and one of 4 shapes. The goal of the game is to find “quads”, which are sets of 4 cards that satisfy a particular pattern. We will explore several mathematical properties of this game, including the question of how many cards you must lay down to guarantee a quad.

Speaker: Allison R. Siegel
Title: Statistical experimental design of binary outcome observer studies
Abstract: Understanding the uncertainty of results from observer studies is important in evaluating imaging systems. Multiple-reader multiple-case (MRMC) analysis estimates variance in observer studies considering the correlations between readers and cases. In this work, we will give an overview of the derivation of the components of variance and use them to decide between experimental designs. Our preliminary results suggest that adding more images reduces the variance than adding more cases and that a study where every observer sees every image has a smaller variance than one where observers see different images.

Speaker: Sandro Amaglobeli
Title: Matching human observer performance in forced localization tasks using neural networks
Abstract: To reduce the number of human trials needed to evaluate medical imaging systems, computational models are used to predict human observer performance. In this work, we implemented transfer learning with neural networks to predict human performance in a forced localization task with under-sampled magnetic resonance imaging (MRI). We matched human performance by early stopping during training, Gaussian noise injection and fine-tuning on human observer data. Our preliminary results show successful human performance matching for under-sampling conditions similar to training data.

Speaker: Justine P. Prasad
Title: Introducing students to research: neural networks for forced localization studies
Abstract: This talk addresses the problem of onboarding students for research using neural networks to model forced localization. In forced localization, participants estimate the position of a lesion in an image. We present a Jupyter notebook that generates sample signal datasets and introduces convolutional neural networks. The notebook also outlines the steps needed to train a model to predict lesion locations from input images. This resource is designed to bridge theory and practice, giving students a practical starting point for research.