Browsing by Department "Mathematics"

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Adaptive Multiscale Simulation Framework for Reduced-Order Modeling in Perforated Domains
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/682; National Science Foundation
Processes in perforated domains occur in many important applications. These include complex processes in soil, membranes, and filters. With current imaging techniques, detailed microscale geometries of these perforated materials can be constructed. However, it is prohibitively expensive to solve complex processes in these perforated domains due to a rich hierarchy of scales. For this reason, some types of reduced-order computational techniques are needed. The goal of this project is to develop and analyze novel computational techniques for solving challenging multiscale problems in perforated domains. The new approaches will bring the information from the detailed geometries to large-scale simulations and will improve the predictions in the simulations. This will further allow deigning new materials and optimize processes. Many current approaches for multiscale methods for problems in perforated domains have been restricted to homogenization, which is applicable when the media has scale separation. However, in many realistic perforated media, there is no scale separation, i.e., pore sizes can have a wide variety of scales. The multiscale methods of this project develop a general framework that allows rigorous and systematic reduction. The PI's goals are: (1) to develop systematic local model reduction tools for computing multiscale basis functions; (2) to develop and analyze new finite element techniques using these basis functions; (3) to study the interplay between localization of the basis functions and the global coupling mechanism; (4) to apply the developed methods to a wide variety of flows with nonlinearities and multiphysics in 3D; (5) to test and demonstrate their capabilities for solving problems in engineering and geosciences.
Advanced numerical methods for multiphysics Magnetohydrodynamics
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/339; National Science Foundation
The objective of this project is to develop innovative numerical methods capable of solving energy-related problems in the context of renewable and alternative energies. The numerical techniques developed in this project will help design grid-scale liquid metal batteries capable of storing large quantities of renewable energies. This research will also help improve the performance of large power electric transformers cooled by environment-friendly vegetable-based oils containing ferromagnetic particles. Finally, by facilitating the understanding of magneto-hydrodynamic instabilities in liquid metals, this project will help to ascertain the integrity and the efficiency of the electromagnetic pumps that will be used to extract energy from the next generation of Liquid-Metal Fast-Breeder Reactors and Tokamaks. This project will be done in collaboration with an European team; the project will foster diversity, international exchanges, and multidisciplinarity. The educational component of the project will contribute to increase the competitiveness of the STEM workforce in the US in computational magnetohydrodynamics. The research program will be organized into four areas: (1) Development of new efficient semi-implicit algorithms to solve partial differential equations with variable material properties (density, electric conductivity, magnetic permeability) using spectral or very high-order methods; (2) Modeling of ferromagnetic fluids and development of new numerical techniques to solve the magneto-static equations in the context of liquid metals and ferromagnetic fluids; (3) Development of level set techniques to account for more than two phases, and development of new high-order level set techniques to guarantee mass conservation and maximum principle; (4) Integration of the mathematical models and numerical techniques developed in (1)-(2)-(3) into a massively parallel open source code to test the proposed methods on realistic applications (liquid metal batteries, thermo-convection of ferromagnetic oil in high-voltage transformers, liquid metal dynamos). This project will involve the Principal Investigator, one post-doctoral collaborator, one graduate student, and European collaborators.
AF: Small: Geometry and Complexity Theory
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/371; National Science Foundation
Linear algebra, which includes computing the solutions to a system of linear equations, is at the heart of all scientific computation. The core computation of linear algebra is matrix multiplication. In 1968 V. Strassen discovered that the widely used and assumed best algorithm for matrix multiplication is not optimal. Since then there has been intense research in both developing better algorithms and determining the limits of how much the current algorithms can be improved. There are three parts to the project. The first two are: practical construction of algorithms for matrix multiplication, and determining the above-mentioned limits. The third addresses a fundamental question of L. Valiant which is an algebraic analog of the famous P versus NP problem. Valiant asked if a polynomial that can be written down efficiently also must admit an efficient algorithm to compute it. All three parts will be approached using theoretical mathematics not traditionally utilized in the study of these questions (representation theory and algebraic geometry). The practical construction of algorithms in this project could potentially have impact across all scientific computation. The novel use of modern mathematical techniques previously not used in theoretical computer science will enrich both fields, opening new questions in mathematics and providing new techniques to computer science. The exponent of matrix multiplication, denoted omega, is the fundamental constant that governs the complexity of basic operations in linear algebra. It is currently known that omega is somewhere between 2 and 2.38. Independent of the exponent, practical matrix multiplication (of matrices of size that actually arise in practice) is only around 2.79. For example, matrices of size 1000x1000 may be effectively multiplied by performing (1000)^{2.79} arithmetic operations. If algorithms to achieve an omega of 2.38 were known, the same matrix operation can be performed using 220 million fewer arithmetic operations. By exploiting representation theory, this project will develop practical algorithms with the goal of lowering this practical exponent. It will also address the exponent by analyzing the feasibility of Strassen's asymptotic rank conjecture and its variants, which are proposed paths towards proving upper bounds on the exponent. The project will also address two aspects of Valiant's conjecture on permanent versus determinant. First, commutative algebra will be used to improve the current lower bound for the conjecture, which has not advanced since 2005. The investigator and a co-author have proven that Valiant's conjecture is true under the restricted model of equivariance. The second aspect will investigate loosening this restriction to weaker hypotheses under which the conjecture is still provable. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Analysis and Computation for Inverse Problems in Differential Equations
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/526; National Science Foundation
Many objects of physical interest cannot be studied directly. Examples include the following: imaging the interior of the body, the determination of cracks within solid objects, and material parameters such as the conductivity of inaccessible objects. When these problems are translated into mathematical terms they take the form of partial differential equations, the lingua franca of the mathematical sciences. However, since one may have additional unknowns in the model, these introduce unknown parameters in the equations that have to be resolved by means of further measurements. Specific problems addressed in this project include the recovery of the location and shape of interior objects from surface measurements or the determination of obstacles from acoustic or electromagnetic scattering data. In this project the PI deals with the practical aspects of such "inverse problems" from a mathematical and computational perspective. The main challenge is when a unique determination can be made from a given amount of data, but as these inverse problems are characterized by often severe "ill-conditioning", meaning that even when there is only one solution to the problem, two very different objects may produce data sets that are infinitesimally close. This lack of stability aspect makes designing and analyzing algorithms for the efficient numerical recovery of the unknowns extremely challenging. The PI will concentrate on developing extremely fast algorithms designed to detect significant features utilizing only minimal data. The PI also looks at inverse spectral problems, and a classic example of which is to be given the vibrational frequencies of a body and seek to determine its internal construction. Here the body can be a metal beam or a star such as the sun. A central theme of this proposal is the investigation of inverse problems for so-called anomalous diffusion models. Classical diffusion is based on Brownian motion and has its roots in 19th century physics together with Einstein's 1905 random walk model. Here a very localised disturbance spreads with the characteristic shape of a Gaussian and, further, the process is Markovian; at a given time step the state depends only on that at the previous time step. While this serves well for a wide range of models, it fails for those that exhibit a "history" or "memory" effect. This includes many materials that been developed over the last twenty years as well as economic forecasting such as stock and commodity market modeling. It turns out that degree of ill-conditioning in anomolous diffusion inverse problems can be very different from those of the classical case suggesting that indeed fundamental new physics is involved. From a mathematical and computational standpoint this comes at a price; the resulting analysis is considerably more complex and challenging. The project also has a significant educational component in the training of graduate and undergraduate students.
Arithmetic of Function Fields and Diophantine Geometry
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/460; National Science Foundation
This award supports mathematicians from US institutions to participate at the conference Arithmetic of Function Fields and Diophantine Geometry, to be held on May 20-24, 2019, at the National Center for Theoretical Sciences (NCTS) on the campus of National Taiwan University in Taipei, Taiwan. The purpose of the conference will be to bring together both established and junior researchers in Number Theory from across the globe, and especially researchers in Function Field Arithmetic, Diophantine Geometry, and related fields. Primary goals will be to discuss recent advances in these areas and to encourage further research and future collaboration, and one important aspect of the conference will be to increase participation of junior researchers. In addition to twenty invited speakers ranging from established mathematicians to recent Ph.D.'s, the schedule will include 6-8 contributed talks by graduate students and post-doctoral researchers. The conference website is http://www.ncts.ntu.edu.tw/events_2_detail.php?nid=216. Since early work of Artin and Weil in the last century, it has been known that the study of function fields of algebraic curves over finite fields in positive characteristic runs in close analogy with the study of number fields. At the same time Diophantine geometry also applies techniques from algebraic geometry to problems in number theory, and so the two subjects are closely intertwined in their efforts to resolve deep problems in number theory using geometric methods. The conference will focus on aspects of function field arithmetic and Diophantine geometry that strengthen common points of view and present opportunities for cross-pollination of ideas, including modular forms and Galois representations, special values of L-functions, arithmetic dynamics, and rational points on varieties. These topics are on the forefront of research in Number Theory, and there have been several major advances in these areas in recent years. The conference will afford participants the opportunities to communicate their research with a broader mathematical community and to initiate new research projects that advance these important areas of Number Theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Automorphic Forms and L-Functions
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/450; National Science Foundation
Automorphic forms and L-functions are special kinds of mathematical functions that are useful for studying a broad array of questions in number theory. For instance, the Riemann zeta function, which has proven to be indispensable in studying the distribution of the prime numbers, is the simplest example of an L-function. Other types of L-functions are crucial for understanding whether certain polynomial equations have solutions, or more generally, how many solutions there are. Often these questions are related to how large the value of an L-function is at a special point. This project concerns the development of new tools for studying the values of L-functions. The investigator will study moments of L-functions that are beyond the convexity range, and deduce new bounds for central values of L-functions. A related theme is to develop foundational tools to study new families of L-functions, including a Petersson formula for newforms of arbitrary level. Such tools are necessary to advance the analytic method for studying the representation problem for ternary quadratic forms. In addition, the investigator will study different aspects of the equidistribution of automorphic forms, such as restricted quantum unique ergodicity (QUE) as well as QUE in higher rank settings. Finally, the investigator will study the zeros of automorphic functions of various types. The methods employed will be techniques from analytic number theory such as summation formulas, exponential sums and integrals, and the spectral theory of automorphic forms.
Banach Space and Metric Geometry-
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/594; National Science Foundation
Metric spaces, especially those so-called Banach spaces, form the conceptual framework in which mathematicians, scientists, and engineers work when investigating problems that involve estimation or approximation. Discrete metric geometry, including dimension reduction that has been established by the PI and a collaborator, is important in the design of algorithms and in compressed sensing. Work on the classification of operators that are commuters was originally motivated by the uncertainty principle in quantum mechanics (which, from a mathematical point of view, arises because multiplication of operators, unlike multiplication of numbers, is not commutative). Nonlinear phenomenon often occurs in nature but is difficult to deal with. This makes it important to understand when non linearity actually conceals underlying linear structure, and this is central to the nonlinear study of Banach spaces. Parts of this research project are coordinated with the Workshop in Analysis and Probability Theory at Texas A&M University, of which the PI is Associate Director. The Workshop encourages interactions among researchers and apprentices in different mathematical fields by bringing together graduate students and junior and senior postdoctoral participants in several areas of analysis. Activities of the Workshop include seminars, Concentration Weeks, introductory lectures, and an annual conference. The efforts of the principal investigators and other participants in the Workshop are helping to break down barriers between different areas of mathematics and also promote the outreach of pure mathematics to other sciences, especially to computer science. The problems in Banach space and metric geometry to be considered fall into several subcategories: commutators of operators on Banach spaces, approximation properties of Banach spaces, the structure of finite and infinite dimensional spaces of p-integrable functions, the nonlinear classification of Banach spaces, discrete metric geometry, quantitative linear algebra, and cluster value problems in an infinite dimensional setting. These topics are at the heart of the geometries of Banach spaces and of metric spaces and make contact with many other areas within mathematics, including operator theory, group theory, geometric analysis, and linear algebra, as well as with other areas of science, including the design of algorithms and compressed sensing.
Banach Spaces and Graphs: Geometric Interactions and Applications
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/321; National Science Foundation
It can arguably be said that the world one lives in is geometric in nature. Numerous practical everyday-life issues, as well as fundamental scientific mysteries, can be expressed in geometric terms. For instance, networks are ubiquitous in our modern society. From the World Wide Web and its powerful search engines to social networks, from telecommunication networks to economic systems, networks represent a wide range of real world systems. They can naturally be seen as geometric objects by considering the number of edges of the shortest path connecting two nodes as a quantity measuring their proximity. The study of physical laws has led as well to the development of a refined mathematical framework where elaborate geometric structures are able to depict and model the interactions of elementary particles and the symmetries underlying quantum physics. The notion of a metric space is a central concept that is pivotal in mathematical models of optimization problems in networks, and in a vast range of application areas, including computer vision, computational biology, machine learning, statistics, and mathematical psychology, to name a few. This extremely useful abstract concept generalizes the classical notion of an Euclidean space, where the distance from point A to point B is computed as the length of an imaginary straight line connecting them. The heart of the matter usually boils down to understanding whether a given metric space, in particular a graph equipped with its shortest path distance, can be faithfully represented in a more structured space, typically a Banach space with some desirable properties. Our ability to do so usually has tremendous applications. The problems investigated in this project are motivated by their potential applications in theoretical physics and theoretical computer science. Most of the problems considered find their origins either in practical issues (e.g. the design of efficient approximation algorithms), or in fundamental mathematical problems in topology or noncommutative geometry (e.g. the Novikov conjecture(s), the Baum-Connes conjecture(s)). Embedding problems that arise in connection to these problems have been considered independently by several groups of mathematicians (Banach space geometers, geometric group theorists, computer scientists...). An underlined aspect of this proposal is to consider these embeddings problems from a unified and global standpoint. Fundamental and long-standing open problems in quantitative metric geometry (e.g. Enflo's problem, a metric reformulation of uniform smoothability, the coarse embeddability of groups and expander graphs into super-reflexive Banach spaces...) will be tackled from a different angle with new and innovative techniques. In particular, the project will develop a certain asymptotic theory of Banach spaces and explore its connections to the geometry of infinite graphs. The approach here to solve the local problems above, is to study asymptotic counterparts of the local properties involved, in order to gain new insights and to devise new approaches. This approach is motivated by the fact that the asymptotic setting usually provides a finer picture, is on some occasion better understood, and requires completely different techniques. A general outline of the research methodology of this project is to utilize powerful tools from surrounding fields (graph theory, probability theory, Ramsey theory...), and cross-over techniques (e.g. techniques from theoretical computer science to solve geometric group theoretic problems). This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Banach Spaces: Theory and Applications
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/572; National Science Foundation
A focus of this project is the theory of Banach spaces and their geometry which provide an important conceptual framework to study problems in engineering, physics, signal processing, and the analysis of large data sets. The second main object of this investigation are "graphs". In computer science they are used to represent networks of communication, data organization, computational devices. An embedding of graphs, or more generally metric spaces, which may represent a large data set, into a Banach space, is necessary to be able to store this data set efficiently. Secondly it is necessary to provide the Banach space with the right coordinate system, it is therefore also necessary to be able to embed a Banach space, into a Banach space admitting the appropriate coordinate system. The problems that will be studied in this project revolve around the issue of embedding less structured mathematical objects like graphs or, more generally, metric spaces, into better structured objects like Banach spaces. On one hand the goal is to obtain information about the structure of the graph, from the property that it embeds in certain Banach spaces, and on the other hand deduce geometric properties of a Banach space, from the property that certain graphs embed or do not embed in it. The principal investigator will investigate possible extensions of Ribe's Program on metric characterizations of local properties of Banach spaces to characterizing asymptotic properties. An important question for example is the question whether or not the property of a Banach space to be reflexive can be metrically characterized. Other important properties to be considered are uniform asymptotic convexity and uniform asymptotic smoothness. The principal investigator will also study the problem of isomorphically embedding Banach spaces, having a certain property, into Banach spaces with a coordinates system like a Schauder basis or an unconditional basis. Finally the principal investigator will continue the study of closed sub-ideals of the spaces of operators on specific Banach spaces. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
CDS&E-MSS: Recovery of High-Dimensional Structured Functions
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/323; National Science Foundation
Many scientific problems of crucial importance for the United States involve a very large number of parameters. This high dimensionality has long been a challenge for the numerical treatment of physical, chemical, and biological processes, and now it also represents an obstacle in data science, especially for the task of extracting useful information from only a limited amount of observations. Propitiously, the high-dimensional objects prevailing in real-life problems often possess some underlying structure simplifying their manipulation. The purpose of this project is to exploit this structure in order to develop a consistent theoretical framework and to conceive novel computational methods for the exact recovery (or good approximation) of the high-dimensional objects sketchily acquired. More specifically, the objects considered in this project are functions of very many variables. Handling them via traditional numerical methods is doomed by the so-called curse of dimensionality. But modern ideas such as sparsity and variable reduction make it possible to bypass the curse and thus are revolutionizing the approach to high-dimensional problems. Building upon the fundamentals from the theory of compressive sensing, the project will consider sparse recovery and simultaneously structured recovery as part of the more general recovery of high-dimensional functions depending on few reduced variables. The research strategy starts by investigating the theoretical limitations of any recovery method within a model, then continues by refining the model through confrontation with real-life problems, and finishes by implementing the algorithms proven to perform optimally. Since the project features interactions with several applied fields (in particular, Engineering and Bioinformatics), the novel numerical methods are to be tested in these areas, which will in turn provide fresh mathematical insight. A final part of the project is devoted to the integration of emerging concepts into the culture of the next scientific generation and in particular to the training of mathematicians in computational and data-related aspects.
Cohomology of Noncommutative Rings: Structure and Applications
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/545; National Science Foundation
Representation theory is a branch of mathematics that studies symmetry and motion algebraically, for example, by encoding such information as arrays of numbers, or matrices. It arises in many scientific inquiries, for example, in questions about the shape of the universe, symmetry of chemical structures, and in quantum computing. Cohomology is a tool that pulls apart representation-theoretic information into smaller, more easily understandable components. The PI's research in cohomology and in representation theory aims at answering hard questions about fundamental structures arising in many mathematical and scientific settings. Her research program impacts that of many other mathematicians, particularly the students and postdocs that she mentors. She leads research teams both at her university and internationally, she co-organizes conferences in her research area, and she is writing a book at an advanced graduate level. This project concerns several inter-related problems on the structure of Hochschild cohomology and Hopf algebra cohomology, and on applications in representation theory and algebraic deformation theory. Hochschild cohomology of an associative ring has a Lie structure that is an important tool and yet is difficult to manage. Some of this difficulty was recently overcome through work of the PI and others in developing new techniques for understanding the Lie structure in terms of arbitrary resolutions. This opens the door to much more potential progress in understanding the structure of Hochschild cohomology, for example, to making connections to other descriptions such as by coderivations and loops on extension spaces. The PI will also work on related questions in deformation theory. Another part of this project concerns a finite generation conjecture in Hopf algebra cohomology. The PI will prove the conjecture for some important classes of Hopf algebras using a variety of techniques, including resolutions for twisted tensor products and the Anick resolution. The PI will work on related support varieties for understanding questions about representations of these Hopf algebras and related categories, using techniques she is developing with a postdoc for handling module categories over tensor categories.
CombinaTexas 2020: A Combinatorics Conference for the South-Central U.S.
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/350; National Science Foundation
The CombinaTexas 2020 conference will be held at Texas A&M University, College Station TX on April 10-11, 2020. The conference will feature six fifty-minute plenary lectures and a number of contributed talks in various areas of Combinatorics and Graph Theory. The aim of the CombinaTexas conference series is to enhance the educational and research atmosphere of combinatorialists in Texas and the surrounding states, increase communication between mathematicians of the region, and provide a forum for presentation and discussion of the most recent developments in the field of Combinatorics. CombinaTexas was established in 2000 and rotated among different institutions in the South Central United States until 2014. Since then it has been hosted at Texas A&M University. CombinaTexas 2020 is the nineteenth conference in this series. The topics of the CombinaTexas Series include all branches of Combinatorics, Graph Theory, and their connections to Algebra, Geometry, Probability Theory, and Computer Science. In 2020 the confirmed plenary speakers are Miklos Bona (University of Florida), Tri Lai (University of Nebraska- Lincoln), Chun-Hung Liu (Texas A&M University), Nathan Reading (North Carolina State University), Stephanie van Willigenburg (University of British Columbia), Josephine Yu (Georgia Institute of Technology). They will present research in enumerative combinatorics, combinatorial representation theory, cluster algebras, tiling theory, topological graph theory, and tropical geometry. About 70 participants are anticipated, with an estimated 20 contributed talks in parallel sessions. More information about the conference is available at the webpage https://www.math.tamu.edu/conferences/combinatexas/ This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Concentration, Convexity and Structure
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/335; National Science Foundation
In various scientific disciplines such as mathematics, statistical mechanics, quantum information, and others, high-dimensional structures play a central role. It has been observed that these distinct areas share the common feature that basic probabilistic principles govern the underlying high-dimensional behavior. In most cases, efficient approximation and study is facilitated by (non-asymptotic) high-dimensional probability. The investigator intends to work on several questions related to the most widely applied principle in high-dimensional probability: the concentration of measure phenomenon. This principle is commonly the main reason behind the frequently-observed tendency of high-dimensional systems to congregate around typical forms. To quantify this phenomenon, one needs precise inequalities for high-dimensional objects (for instance, measures or random vectors), where independence properties can be lacking. The questions under study have a strong geometric component. Results of the study will have implications in disciplines that depend vitally on high-dimensional objects, including asymptotic geometric analysis, geometric probability, machine learning, sparse recovery, random matrices, and random polynomial theory. The main goal of the project is to find the quantities or to isolate characteristics of a function that govern its concentration (say with respect to the Gaussian measure); in particular, to determine the quantities that control small fluctuations (variance) and small ball probabilities. The project undertakes a systematic study of this problem and initiates some new methods to compute deviation inequalities (especially in the small ball regime). It is planned to test these methods on more general measures such as log-concave probability measures. The project will also investigate limit theorems for geometric quantities that complement concentration inequalities at the asymptotic level. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Conference: International Workshop on Operator Theory and its Applications 2018
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/232; National Science Foundation
This award provides funding to help defray the expenses of participants in the meeting "International Workshop in Operator Theory and its Applications" (IWOTA) a workshop that will take place during July 23-27, 2018, on the campus of the East China Normal University in Shanghai, China. Additional information about the conference can be found on the website http://iwota2018.fudan.edu.cn The 2018 IWOTA meeting will continue the tradition of past meetings in this series going back to 1981. The meeting will be focused on the latest developments in functional analysis, specifically, operator theory and and related fields. This includes applications in engineering and mathematical physics from areas such as differential and integral equations, interpolation theory, system and control theory, signal processing, and scattering theory. The IWOTA meeting will run the week before the international symposium "Mathematical Theory of Networks and Systems"(MTNS) and this pairing provide opportunities for analysts to get exposed to engineering problems. Priority for funding will be given to early career mathematicians, women, members of underrepresented groups and those without other means of support. This is an important conference in operator theory and its applications that will offer participants the opportunity to learn of state-of-the-art research in operator theory with applications to engineering and other sciences. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Efficient and adaptive methods for simulating multiscale effects in optical metamaterials -
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/427; National Science Foundation
The project centers on the development of novel computational approaches to simulate the optical properties of metamaterials. Metamaterials are small optical devices that manipulate light on a microscopic scale. An important aspect of the project is the dissemination of the developed numerical algorithms in form of publicly available open source software. As such the project will enable and foster research directions in optics that require a strong computational component. In addition to the dissemination of the research to the scientific community, results will be presented to students through graduate courses, mentoring of students, and university-internal research and student seminars. In particular, the PI plans to develop a new graduate-level course about advanced finite element methods for optical problems. Efforts will be made to attract female and minority students and stimulate their interest by presenting and incorporating exciting new research topics in numerical methods courses on upper-division undergraduate level. Metamaterials are specifically engineered, periodically aligned microstructures that exhibit unusual optical properties. A major challenge that manifests in the simulation of scattering processes involving metamaterials is that they are of pronounced two-scale character, meaning that relevant optical processes act on very different length scales. This is complicated by the fact that realistic experimental geometries contain 1D discontinuities at boundaries of the 2D material sheets. Such discontinuities cause edge effects that are challenging to simulate due to their dominant and singular behavior. The project centers around the development and analysis of novel computational approaches for the simulation of scattering processes in complex optical devices, that are able to cope with the two-scale character and the occurrence of edge effects: (1) a parallel and adaptive finite element method for 3D device simulations will be developed and implemented that combines goal-oriented local mesh refinement and domain decomposition for MPI parallelization and preconditioning; (2) a heterogeneous multiscale method will be developed and analyzed that incorporates model-adaptive strategies for an efficient sampling of effective metamaterial parameters; (3) special emphasis will be given in both research directions to connect the algorithmic and numerical development to interdisciplinary applications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Finite Element Approximations of Bending Actuated Devices
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/630; National Science Foundation
The ability to generate complex deformations from relatively small energies has tremendous applications in micro-engineering and biomedical science. This research focuses on developing and implementing mathematical algorithms able to predict and optimize the deformation of elastic films, chosen for their potential in the production of robust and light-weight micro-scale devices. The deformations considered are triggered by exposing to external stimuli either polymers with different expansion characteristics or manufactured gels with residual stresses. Devices based on these technologies are, for instance, employed as drug delivery vesicles, cell encapsulation devices, sensors, bio-muscles and as proxies for tissue growth. In addition to these applications in biomedical science, the development of autonomous foldable structures such as self-deployable sun sails in spacecraft or deployable aircrafts, photovoltaic devices, actuators, micromotors, microgrippers, microvalves, microswimmers are very popular interests in the engineering community this research is likely to have impact on. The proposed study focuses on thin devices where bending is the principal mechanism for, possibly large, deformations. The mathematical models are derived as the two dimensional limit of thin three dimensional hyper-elastic solids. They are characterized by energy densities dominated by bending, expressed geometrically as the film's second fundamental form. In addition to the difficulties inherent in the non-divergence form of this fourth order system, fully non-linear geometrical constraints must be taken into account in the context of large deformations. The aim of this research is to derive mathematical models when not available for the targeted application, and to design, analyze and implement finite element based algorithms for their approximations. The entire process from the mathematical analysis to the actual highly parallel implementation of finite element algorithms is covered. Hence, analytical tools borrowed from differential geometry and calculus of variation are blended with numerical analysis and delicate computational efforts to achieve efficient and practical algorithms. The ability to generate complex deformations from relatively small energies has tremendous application in micro-engineering and biomedical science. The algorithms resulting from the proposed research - and in particular their relatively effortless implementations - are likely to have impact in these areas. To mention a few applications, devices based on bilayers of polymers or prestrained films are employed as drug delivery vesicles, cell encapsulation devices, sensors, bio-muscles and as proxies for tissue growth. In addition to these applications in biomedical science, the development of autonomous foldable structures such as self-deployable sun sails in spacecraft or deployable aircrafts, photovoltaic devices, engineered scaffolds, actuators, micromotors, microgrippers, microvalves, microswimmers are very popular research interests in the engineering community. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
FRG: Collaborative Research: Noncommutative Dimension Theories
Mathematics; TAMU; National Science Foundation
Approximation defines our world. For example, the letters on this screen have smooth curves and bends. But zoom in and all you see are squares: pixels. The smaller the pixels (i.e., the finer the approximation), the smoother a curve looks. Add three colors and suddenly we can approximate a rainbow. In a sense, science is all about refining and improving approximations to reality. Take Newtonian physics, for example. It works great at medium scales, but breaks down when things are too big or too small. Einstein's relativity and quantum mechanics work much better at those scales. And these theories require sophisticated mathematics. This focused research group project addresses several outstanding questions in operator algebras and their analogies in other areas of mathematics. Operator algebras arose as a framework for quantum mechanics. Over the years many classical theories were extended to this noncommutative context: geometry, topology, probability and more. The PIs will spearhead an international effort to capitalize on recent connections between operator algebras and other areas such as dynamics, measure theory, coarse geometry and K-theory. Specifically, the PIs shall push analogies between nuclear dimension and asymptotic dimension, two notions defined via approximation and encompassing a huge swath of examples, to address K-theoretic questions such as the Universal Coefficient Theorem and the Baum-Connes and Farrell-Jones conjectures.
FRG: cQIS: Collaborative Research: Mathematical Foundations of Topological Quantum Computation and Its Applications
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/524; National Science Foundation
A second quantum revolution in and around the construction of a useful quantum computer has been advancing dramatically in the last few years. Topological phases of matter, the importance of which has been recognized by scientific awards that include the 2016 Nobel prize in physics, exhibit many-body quantum entanglement. This makes such materials prime candidates for use in a quantum computer. Topological quantum computation is maturing at the forefront of the second quantum revolution as a primary application of topological phases of matter. The theoretical foundation for the second quantum revolution remains under development, but it appears clear that algebras and their representations will play a role analogous to that played by group theory in the first quantum revolution. This focused research group aims to formulate the theoretical foundations of topological quantum computation, leading to an eventual theoretical foundation for the second quantum revolution. It is anticipated that the results of the research will guide and accelerate the construction of a topological quantum computer. A working topological quantum computer will fundamentally transform the landscape of information science and technology. The project includes participation by graduate students and postdoctoral associates in the interdisciplinary research. The goal of topological quantum computation is the construction of a useful quantum computer based on braiding anyons. The hardware of an anyonic quantum computer will be a topological phase of matter that harbors non-abelian anyons. A physical system is in a topological phase if at low energies some physical quantities are topologically invariant. Topological properties are non-local, yet can manifest themselves through local geometric properties. The success of topological quantum computation hinges on controlling topological phases and understanding their computational power. This research addresses the mathematical, physical, and computational aspects of topological quantum computation. The projects include classification of super-modular categories, vector-valued modular forms for modular categories, extension of modular categories to three dimensions, simulation of conformal field theories, topological quantum computation with gapped boundaries and symmetry defects, and universality of topological computing models. The research has potential impacts ranging from new understanding of vertex operator algebras to the development of useful quantum computers. One specific goal is a structure theory of modular categories analogous to that of finite groups. Such a theory would lead to a structure theory of two-dimensional topological phases of matter.
Fundamental Decomposition Results in Finite Von Neumann Algebras
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/397; National Science Foundation
The study of operators on Hilbert space became important with the advent of Quantum Mechanics, but in addition, understanding of these operators has proven to be vital to progress in many areas of mathematics. Historically, a method of studying and understanding such operators is to break them down into simpler components, based on spectral decomposition. This consists of describing parts of the operator that behave like multiplication by certain numbers, and to explain how these parts assemble into the whole. One major goal of this project is to advance such understanding of large classes of operators. Another major goal is to study families of operators that arise in various quantum mechanical models, in light of certain deep mathematical conjectures regarding finite dimensional approximations of infinite dimensional objects. More specifically, the principal investigator, together with collaborators, has made advances in recent years on spectral decomposition results for non-selfadjoint elements of finite von Neumann algebras. These are centered around upper triangular forms, analogous to the classical results of Issai Schur for matrices, and both utilize and extend results about hyperinvariant subspaces found recently by Haagerup and Schultz. Particular proposed projects include (a) studying norm convergence properties of bounded operators and (b) extending spectral distribution and upper-triangular form results to unbounded affiliated operators. In related directions, the principal investigator will work on the hyperinvariant subspace problem for elements of tracial von Neumann algebras and to investigate the Murray--von Neumann puzzle, which is akin to the Heisenberg relations. A second area of proposed research concerns the notion of bi-freeness and bi-free products. A third area of proposed research concerns quantum correlations. Recent results of the principal investigator showing non-closedness of the set of quantum correlations for five inputs and two outputs open the door to new understanding of these small cases, which the principal investigator proposes to pursue. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
Geometry of Physical Models Governed by Diffusion
Mathematics; TAMU; https://hdl.handle.net/20.500.14641/685; National Science Foundation
Over the past decades probability theory has had tremendous success in elucidating scaling limits and large-scale phenomena. However, systems that have no apparent scaling limit are abundant in both the physical and life sciences, and there is practical need to better understand such systems. This research project aims to advance understanding of such systems by studying models augmented with additional symmetries that share the local behavior of the classical disordered systems under study. Graduate students and postdoctoral associates will receive training through involvement in the research. The topics under study in this project divide into three subareas: (1) stationary diffusion limited aggregation (DLA) and Hastings-Levitov models, (2) DLA in a wedge, and (3) chemical distance in random interlacements. These models are of interest for statistical mechanics and mathematical physics. They share the features of being generated by diffusion or the harmonic measure, and of attempting to grasp the nature of a physical phenomenon that is not amenable more classical models in the field, such as DLA or Bernoulli percolation. The methods to be employed borrow ideas and tools from various mathematical disciplines, including complex analysis, harmonic analysis, differential equations, and ergodic theory. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.