Applications of Quasiconformal Geometry and Partial Differential Equations
National Science Foundation, DMS 1955992
Capogna, L
Start Date: 7/1/2020 (3 years). Amount: $187,987
Partial Differential Equations (PDE) are used to model real life systems, and in particular, subelliptic PDE are helpful in settings where there is a constrained dynamics. Examples of such systems include the motion of robot arms, structural functions of the first layer of the mammalian visual cortex, the Black-Scholes model for financial markets and quantum computing. Geometric and analytic properties of such spaces are captured in a quantitative fashion by studying the behavior of certain families of transformations of the space into itself. This project aims at studying fine properties of such transformations. In terms of broader impacts, the PI will involve graduate and undergraduate students in several aspects of the research and design outreach activities to attract K12 students to mathematics.
The technical focus of the proposed research addresses a curve shrinking flow in Carnot groups, the study of harmonic extensions of quasiconformal mappings between boundaries of certain Gromov hyperbolic spaces, and regularity of certain nonlinear, degenerate parabolic PDE.
For more information on this grant, please visit https://www.nsf.gov/awardsearch/showAward?AWD_ID=1955992&HistoricalAward...
Vladimir Druskin receives $200,000 award from Air Force Research Laboratories for “Active Array Imaging in Complex Media in the Presence of Losses and Dispersion Effects”
In Vladimir’s own words: This project is focusing on extension to the air force imaging applications of the novel data-driven model reduction approach introduced earlier by the PI and his collaborators. The scope includes algorithmic development and mathematical justification of this approach for radar imaging in strongly inhomogeneous media with multiple scattering (echoes) effects, losses and dispersion.
Learn more about Vladimir Druskin
Total award period: July 2020 - June 2023 Award amount: $200,000
Christopher Larsen receives $250,000 award from the Division of Mathematical Sciences (DMS) at the National Science Foundation "New Mathematical Methods for Dynamic Fracture Evolution"
In Professor Larsen's own words: "For a large range of applications, from civil infrastructure to national defense, understanding the failure of materials is critical. Yet, our ability to predict this failure is limited by both modeling, which is somewhat ad hoc, and the mathematics available to formulate and analyze models, as well as to justify numerical methods. These issues are most severe in dynamic problems, such as impacts, when loading changes quickly. The main goal of this project is the development of new mathematical methods for dynamic fracture evolution. In particular, the principal investigator (PI) will extend methods for regular crack paths to more realistic paths, with kinking and branching. A second goal is to address fundamental mathematical issues that are necessary for further progress in completely general settings. Finally, the PI will study phase-field approximations of fracture, which have become very popular tools in the engineering community but remain poorly understood.
The ability to accurately predict failure depends on the quality of the underlying mathematical models of defects as well as on understanding fundamental properties of solutions. When crack paths are regular, mathematical methods are available to study these evolutions. However, when they are not, the only methods so far involve considering the paths to be limits of more regular paths. The main technical issue here is that strong convergence of the corresponding elastodynamics is necessary for energy balance, as well as for other properties of solutions, but this convergence remains open in many situations. Another fundamental issue is uniqueness of elastodynamic solutions for a given crack path. The investigator will show uniqueness in certain settings, and explore general consequences, such as bounds on crack speed. The final goal of the project is to analyze phase-field models for fracture. While very popular in the engineering community, a number of properties, including whether they approximate the correct surface energy, or satisfy a maximal dissipation condition, remain open questions."
For more information please see https://www.nsf.gov/awardsearch/showAward?AWD_ID=1909991&HistoricalAwards=false
Total award period: 2019-2022 Award Amount: $250,000
Further Advancing the Northeast Combinatorics Network
National Science Foundation
Martin, W., (Co-PI), Ellis-Monaghan, J., (PI)
Start Date: 3/1/2019 (3 years). Amount: $39,000
The Northeast Combinatorics Network (NCN) will host several events including Spring and Fall Discrete Math Day (DMD) conferences and a Summer Combo conference as well as a Virtual Combinatorics Colloquium series in each year of the three-year project. The Spring 2018 DMD will be hosted at UMass Amherst, https://sites.google.com/view/umass-dms/dmdspring19, and universities in the region are being given the opportunity to host the remaining five DMD meetings. This project aims to further cultivate and enhance a network of students, post-doctoral researchers, faculty and industry mathematicians spread across the institutions in New England and New York working in the area of combinatorics (discrete mathematics), which is a thriving and rapidly advancing mathematical discipline.
Further information on this grant can be found at https://www.nsf.gov/awardsearch/showAward?AWD_ID=1853455&HistoricalAward...
Andrea Arnold receives 2019 Women's Impact Network (WIN Impact) Grant
Assistant Professor Andrea Arnold has received the 2019 Women’s Impact Network (WIN Impact) Grant “Sonia Kovalevsky Day: Math Outreach Initiative for Female Middle-School Students”. The grant will support Andrea’s initiative in organizing SK days at WPI. In her words:
Sonia Kovalevsky Days (SK Days) are nationally-held, locally-organized outreach events with the aim of supporting and encouraging young women to continue their study of mathematics (https://awm-math.org/programs/math-outreach/). The recently-formed WPI Chapter of the Association of Women in Mathematics (AWM) is planning to host an SK Day outreach event on campus this spring, specifically targeting the participation of local female middle school students. The day's events will include math activities and problems for the students, as well as a keynote talk from Dr. Misha Kilmer (Tufts), with volunteers from the WPI community. This year's inaugural event is scheduled for Saturday, March 23, 2019, with plans to host subsequent events annually each spring. This award will support the funding of SK Days at WPI for the next two years, to be held in Spring 2019 and Spring 2020
Total award period: 07/01/2019 - 06/30/21 Award Amount: $10,500
Professor Suzanne Weekes named winner of prestigious Humphreys Award
The Executive Committee of the Association for Women in Mathematics has established this prize in memory of M. Gweneth Humphreys to recognize outstanding mentorship activities. The prize is awarded annually to a mathematics teacher (female or male) who has encouraged female undergraduate students to pursue mathematical careers and/or the study of mathematics at the graduate level
This award is named for M. Gweneth Humphreys (1911–2006). Professor Humphreys earned her master’s degree from Smith College and her PhD at age 23 from the University of Chicago in 1935. She taught mathematics to women for her entire career, at Mount St. Scholastica College, Sophie Newcomb College, and for over thirty years at Randolph-Macon Woman's College. This award, funded by contributions from her former students and colleagues at Randolph-Macon Woman's College, recognizes her commitment to and her profound influence on undergraduate students of mathematics.
Professor Suzanne Weekes exemplifies fully the tradition of Gweneth Humphreys. Through her research, her project advising, her REU leadership, her CIMS activities and more, she has contributed significantly to our mission, as few others have. Dr. Weekes’ track record in mentoring students (and faculty!) is superb and extends well beyond WPI, to her work with PIC math, and includes her impactful service in MSRI-UP.
For more information about the Humphreys Award, please see https://www.wpi.edu/news/wpi-mathematician-receives-prestigious-teaching-award
Burt Tilley and Vadim Yakovlev receive $775,336 award from AFOSR titled ``Permittivity Gradients, Polarization, and Gas Dynamics in Composite Electromagnetic Heat Exchangers’
In Professor Tilley's and Professor Yakovlev's own words, the project consists of a five-year research program to quantify the fundamental heat-transfer processes in the conversion efficiency from incoming electromagnetic radiation into elevated internal energy of a compressible coolant. The idea is to use electromagnetic-radiation absorbing materials, either porous or designed with channels through which a coolant can flow, that can withstand temperature up to 2000 K, heat these materials through the application of electromagnetic waves, and then run coolant through the material to harness the desired energy. Since electrical conductivity of these materials depends on temperature, multiple steady temperatures are possible at the same input power. The research program centers on using asymptotic multiscale methods including homogenization to formulate an effective medium theory to describe the energy conservation and electric field propagation through this medium, for incompressible and compressible coolants. The spatial variation of loss factor will be examined in order to compensate for conduction losses found at higher temperatures, and as a simple model for metal-ceramic composites. These results depend on the examination of the energy transfer between the composite and the coolant, and we shall consider these modes for incompressible and compressible coolants. Three-dimensional electric field amplitude equations, developed in our current award using high-frequency homogenization, will be extended to incorporate heat transfer, viscous fluid flow spatial dimensions, and comparisons with solutions using finite-difference time-domain and with finite-element methods will be performed. The goal is to develop a systematic method to better understand wave propagation, heat transfer and power delivery to the coolant for a general three-dimensional spatially-periodic microstructure.
Award period: September 2018 - August 2023 Award amount: $775,336
Luca Capogna Receives Five-Year Grant to Use Mathematics to Classify and Understand the Nature of Shapes
Simons Foundation
Luca Capogna, professor and head of the Department of Mathematical Sciences, has received a five-year, $42,000 grant from the Simons Foundation, a 24-year-old organization that funds research into mathematics and the sciences. The project being funded is titled “Topics in Nonlinear PDE and in Quasiconformal Mappings.”
The goal of Capogna’s research is to classify all possible shapes—whether the curls of a lettuce leaf shape or the space swept up by a robot arm as it moves through all of its possible configurations—using a mix of geometry and analysis methods. By better classifying and understanding shapes, scientists will better understand their nature and how they behave.
The research should enable Capogna to determine whether or not two spaces are similar. For instance, two shapes may look very different but have hidden similarities, from a mathematical perspective.
Capogna’s project, which is a continuation of research he’s been conducting for the past 10 years, could ultimately be applied to the study of cloud computing or even social networks. For instance, he is using mathematical models that relate to curved spaces, such as a curling lettuce leaf, to understand spaces that expand very quickly, a technique that connects a leaf that takes up more space as it uncurls to a cloud of data or a social network that rapidly expand in size.
Award Date: July 31, 2018 Total Award Amount: $42,000
Research Experiences for Undergraduates in Industrial Mathematics and Statistics
National Science Foundation
Tilley, B., (PI) Sturm S., (Co-PI)
The Worcester Polytechnic Institute (WPI) REU in Industrial Mathematics and Statistics provides a distinctive educational experience for students in the mathematical sciences by introducing them to the ways in which mathematicians and statisticians use math in the "real world". Students work in teams on research problems of industrial and mathematical significance that come directly from industry and that are of immediate interest to the companies involved in the program. Students work closely with company representatives to define the problem and to develop solutions. They work closely with faculty advisors to maintain a clear focus on the mathematics and statistics at the core of the project. This summer research provides challenges not faced in traditional undergraduate programs and helps to develop skills usually not addressed in academic programs. It also provides a glimpse of some of the many career possibilities that are open to students with a strong mathematical background. For further information on this grant, please visit https://www.nsf.gov/awardsearch/showAward?AWD_ID=1757685
Total Award Period: February, 2018 to January 31, 2022 (estimated) Total Award Amount: $299,606
William Martin Receives $150,000 NSF Grant to Further Our Understanding of Association Schemes
William Martin, professor of mathematical sciences, has received a three-year, $150,000 grant from the National Science Foundation for a research project titled, “Association Schemes and Configurations in Real and Complex Space.”
The project is aimed at taking on questions that have stumped the best mathematical minds in the world for decades. Martin is looking to advance mathematicians’ understanding of association schemes, which are finite combinatorial structures that can be viewed algebraically, geometrically, or as highly symmetric networks.
Finding new association schemes, or even better understanding them, could enable researchers to use one quantum computer to simulate another. They also could be used to discover new error-correcting codes for digital communications, new spherical designs for estimating solutions to calculus problems in high-dimensional space, or new secure scrambling components for symmetric key encryption.
Martin is working on underlying algebraic theorems that will lead to a stronger mathematical theory, though he said he is focused on making advances that other researchers can then use in their own work, to solve problems in various applied areas.
At least one PhD student and multiple MQP teams will be working on the project.
Further information on this award can be found at https://www.nsf.gov/awardsearch/showAward?AWD_ID=1808376&HistoricalAward...
Total Award Period: 2018 - 2022
Zheyang Wu Receives NSF Award for Work Related to ALS
Zheyang Wu, associate professor of mathematical sciences, has received a three-year, $150,000 award from the National Science Foundation for a project titles, “Optimal and Adaptive p-Value Combination Methods with Application to ALS Exome Sequencing Study.” The project is focused on developing tests that will give scientists a better understanding of which DNA segments are related to ALS susceptibility. Using ALS exome-sequence data, researchers will develop better data analysis methodology to paint a clearer picture of how genes influence ALS.
ALS, or amyotrophic lateral sclerosis, is a progressive neurodegenerative disease that affects nerve cells in the brain and spinal cord, affecting one’s speech and ability to swallow and to control the muscles. Ultimately, it causes paralysis and death. Each year, more than 5,000 people in the United States are diagnosed with ALS, which is also known as Lou Gehrig's disease.
Genetics plays a critical role in ALS. Despite numerous advances in recent years, doctors still cannot trace the genetic cause of a significant amount of ALS cases. This research project is taking on the “missing heritability” problem, using innovative genetic data analysis algorithms and more powerful p-value combination tests to analyze large exome sequencing data for detecting novel ALS genes. The methodology being developed in this project has broad applications and could be used to better understand other diseases, as well. A WPI graduate student will work on this research with Wu. For more information on this grant please visit https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812082
Total award period: July 1, 2018 - June 30, 2022
Arnold’s research in applied mathematics focuses on inverse problems and uncertainty quantification, which involves estimating unknown system parameters using indirect observations and analyzing the changes in predicted outcomes due to changes in the inputs. Many applications in modern science involve system parameters that are estimated using little prior information. This poses a challenge in applied and computational mathematics, particularly for problems where knowledge of parameters is crucial in obtaining trustworthy model output.
The study will develop mathematically sound and computationally efficient systematic approaches for estimating time-varying parameters with unknown dynamics. This work also will involve developing models for parameter evolution that take into account any prior knowledge relating to the structure or behavior of the parameter over time without defining explicit functions to describe the dynamics. Arnold will include WPI undergraduates and graduate students on the project team starting in the summer of 2019. Learn more about this grant.
Mammalian sperm must navigate the female reproductive tract, swimming a distance greater than 1000 times their own length to reach and fertilize the egg. In order to aid in the treatment of reduced sperm motility, it is important to understand interactions of the sperm flagellum with different regions of the reproductive tract. In particular, fluid flow helps bring the egg to the uterus (in the opposite direction of sperm progression). Recent experiments have shown that a large percentage of sperm exhibit positive rheotaxis, the ability to reorient and swim against a background flow. Additionally, sperm will bind and unbind to the oviductal wall and the role of a background flow on sperm detachment is not known. The main scientific goals of this project include further analyzing existing experimental data (through image processing techniques) and developing new computational models to understand the clinical importance of migration through the female reproductive tract and sperm binding and detachment from walls in a background flow. Several new computational modeling frameworks will be developed to allow simulations of sperm in the presence of a background flow and a wall. The PI will provide interdisciplinary training for several students (undergraduate and graduate) as well as one postdoc in the areas of computational biofluids, image processing of experimental movies, and model development. In addition, the PI will work to develop image processing and modeling modules to be used in area High Schools and at summer programs for High School students at WPI. Learn more about the grant.