The Mathematics of Data
MATH 4513
The ways in which humans want to understand, manipulate, and learn from data are growing every day. As our computing abilities grow, we discover new ways to collect more and more data in higher and higher dimensions. Mathematical theory leads us to discover how to process and understand data in more detailed, higher-order, and/or faster ways.
In The Mathematics of Data, we will introduce students to the mathematical theory of various aspects of data and image processing. The main themes will include categorization and clustering; sparsity and compressed sensing; machine learning; and topological data analysis. This class will be a mix of theory and application. Students will learn to use Python packages that perform some algorithms in each section. Students will also learn the mathematical underpinnings of the algorithms, using what they have learned in linear algebra, differential equations, and – if they have taken these courses - analysis and topology. Participants will gain experience with some of the most rapidly-growing and important applications of mathematics - the applications to data and image processing.
Although this is a Mathematics Capstone course, students from other majors who have the required mathematical background are encouraged to enroll. No prior programming experience is required. The guest speakers will be internationally-renowned experts in the areas related to the course. They will be able to talk to the students directly about applications they have worked on and the specific techniques they use. Each visiting scholar will give a public lecture showcasing an application of their work.
Public Lecture Series
Information for this Lecture Series will be posted here as it comes in.
Transport transforms for data analysis and machine learning
March 22, 2022
5:30 PM
Physical Sciences, Room 201
Open to the public
Dr. Akram Aldroubi
Professor of Mathematics
Transforms are mathematical machines that reorganize data, signals, or images in new configurations that allow a better understanding and manipulation of the data. For example, to store an image on a computer, the image is first sampled on a grid and then transformed into a set of binary numbers (zeros and ones) that are then stored on a chip. There are many types of transforms, so they can be selected based on the type of data and what one wants to do with it. Recently, new transforms have been developed that are based on optimal transport theory. These transforms are well-suited for many applications and have outperformed state-of-the-art methods. In this talk, we will present several applications including cell morphology in drug discovery, nuclear structure in cancer, and MRI for knee cartilage in the prognosis of osteoarthritis.
Akram Aldroubi received a diploma (M.S. equivalent) of Electrical Engineering from Ecole Polytechnique Federale de Lausanne in 1982, and a Ph.D. in Mathematics from Carnegie Mellon University in 1987. His interest in pure and applied mathematics include functional analysis, modern harmonic analysis, sampling and reconstruction, and their applications to image processing and data analysis.
Before taking a position at Vanderbilt University in 1998, he was at the Biomedical Engineering and Instrumentation Program (BEIP) at the National Institute of Health (NIH) where he developed new techniques for image and signal processing, including contributions to the development of wavelet theory. He developed the theory of sampling and reconstruction in shift-invariant spaces, spline signal processing, and DT-MRI image reconstructions. He received the Special Achievement Award in 1996 from the NIH. He was a visiting professor at several universities and institutions including, CNRS and Ecole Normal Superieure de Cachan, Ecole Normale Superieure de Lyon, The Hausdorff Institute in Bonn, and the Institute of Mathematics at the Technical University of Berlin. He was awarded a Fulbright award in 2009, and was a visiting professor at the University of Buenos Aires. He was inducted Fellow of the AMS 2014 for his "contributions to modern harmonic analysis and its applications, and for building bridges between mathematics and other areas of science and engineering." He has written over 100 research articles.
Aldroubi is the Co-editor-in-Chief of “Sampling Theory, Signal Processing, and Data Analysis’’ (Birkhauser), and he is on the editorial board of several other mathematical journals including the Journal of Fourier Analysis and Applications, and the Journal of Numerical Functional Analysis and Optimization. He has organized and co-organized over 15 national and international conferences, including several conferences on Applied and Computational Harmonic Analysis and recently a one-month Focus Program on Data Science, Approximation Theory, and Harmonic Analysis, at the Fields Institute, May 9 -- June 10, 2022.
We have Data and Computers, why do we need Math?
April 5, 2022
5:30 PM
Physical Sciences, Room 201
Open to the public
Dr. Konstantin Mischaikow
Rutgers University
With today's technology, we can collect massive high-dimensional sets of data from experiments and generate massive high-dimensional sets of data numerically, but at the end of the day, these are still finite sets of points. Assume that we are trying to understand a continuous process that generated the data. One such example could be a process that can be modeled by a differential equation. In this case we probably want to be able to extract a continuous function, which raises the question, "How can one go from finite data to continuum objects with some sense of certainty?"
This talk will focus on how to go from finite data to the identification of a periodic orbit using algebraic topology.
Konstantin Mischaikow is a Distinguished Professor of Mathematics at Rutgers University. He is also a member of the Rutgers Center for Quantitative Biology. He is on the editorial boards of several mathematical publications and served as Editor-in-Chief for the Journal of Differential Equations from 2000-2011.
Data, Algorithms, Justice and Fairness
April 21, 2022
5:30 PM
Physical Sciences Center, Room 201
Open to the Public
Dr. Cristopher Moore
Sante Fe Institute
Algorithms are being used to decide whether defendants will show up for court, whether they should be released on bail, and whether they will be good citizens if they are given parole. How accurate are these algorithms? What data are they based on? And how fair are they to different subgroups of the population? Over the past few years, a controversy has erupted over the issue of algorithmic fairness – whether these algorithms treat some people differently than others. We will go through how these algorithms work, what they are based on, and how "fairness" and "accuracy" are slippery terms. Can decisions made by AI be explained to the humans affected by them? What recourse do we have if we disagree with them? Will algorithms help us move forward to a better future, or will they encode and enshrine the biases of the past?
Cristopher Moore received his B.A. in Physics, Mathematics, and Integrated Science from Northwestern University, and his Ph.D. in Physics from Cornell University. From 2000 to 2012 he was a professor at the University of New Mexico, with joint appointments in Computer Science and Physics. Since 2012, Moore has been a resident professor at the Santa Fe Institute. He has also held visiting positions at the Niels Bohr Institute, École Normale Superieure, École Polytechnique, Université Paris 7, Northeastern University, the University of Michigan, and Microsoft Research.
Moore has written over 160 papers at the boundary between mathematics, physics, and computer science, ranging from quantum computing, to phase transitions in Bayesian inference and NP-complete problems, to the theory of social networks. He is an elected Fellow of the American Physical Society, the American Mathematical Society, and the American Association for the Advancement of Science. With Stephan Mertens, he is the author of The Nature of Computation from Oxford University Press.
Moore collaborated with the Santa Fe Symphony on an award-winning PBS documentary, The Majesty of Music and Mathematics, which has been broadcast widely. His non-technical writing on mathematics has appeared in Nautilus and The American Scholar. Most recently, he and his colleagues in the Interdisciplinary Working Group on Algorithmic Justice have analyzed risk assessment algorithms for accuracy and fairness, and argued against the use of proprietary algorithms in housing and pretrial supervision.
