James Gray, (Hon) LLD, (Hon) DSc - US grants

"Access in Mathematics for All" (AMA) supports an emerging partnership between the Community College of Aurora (CCA) and the University of Denver (DU) to develop a program in mathematics education designed specifically to improve access and opportunities for underrepresented racial and ethnic minorities and low-income students who are mathematics majors, to teach in high-need schools. The legacy of a lack of access to a high quality and rigorous education in mathematics for historically marginalized populations is that these groups are significantly and consistently underrepresented in STEM education, degrees, and careers. The primary goal of AMA is to develop capacity at DU to significantly increase the number of diverse and low-income students graduating from DU with a degree in mathematics and a master's degree and teaching license in secondary mathematics. A secondary goal is to create an academic pipeline for students at CCA who have finished substantial mathematics coursework to transfer to DU as a mathematics major and eventually become mathematics teachers in secondary schools.

The project will be directed by the following five guiding principles: (a) instruction that is considerate of students' backgrounds and needs carried out by well-trained and dedicated faculty; (b) academic support for students to develop intellectually as well as confidence in their mathematical abilities; (c) social support for students to be integrated into the DU community; (d) ongoing student access to successful (student and faculty) role models; and (e) recognition and celebration of students' achievements. This project will address the underrepresentation of diverse and low-income students in STEM by creating a mathematical pipeline between CCA and DU. Three courses will be developed in the early stages of this project. The first two courses, a "transition to college calculus" course, and a proofs/logic course in mathematics, have specifically been selected because success in these courses is critical to the future success of mathematics majors. The third course is a "capstone" course in secondary mathematics. A goal in all of these courses will be to foster students' mathematical reasoning as they solve problems and share and develop mathematical ideas with peers. A working hypothesis of the project is that through engaging students in instruction in which the development of mathematical reasoning through discourse is modeled, students will learn how to implement such instruction and this will benefit them when they start teaching. Each of these courses will include instruction in best practices for teaching mathematics to English language learners (ELLs) and will leverage the extensive social and financial support provided for students in two existing DU programs, the Volunteers in Partnership (VIP) program and the Center for Multicultural Excellence (MCE). The VIP program provides financial support for low-income students who commit to working on a volunteer basis in local high schools through the year. In addition, a Math Club at CCA will be initiated and members of this new club will carry out supplemental instruction in Aurora schools. AMA Fellows will also serve as volunteers in participating VIP high schools. The service activities that AMA Fellows will experience through VIP and the CCA Math Club will help prepare them to be effective teachers in high-need schools. This project will initiate a research study to understand the complexities associated with recruiting, retaining, and graduating diverse students and low-income students who major in mathematics in order to become secondary mathematics teachers in high-need schools. It will also initiate a longitudinal study of diverse, low-income students to determine best strategies for recruitment as well as obtaining student perspectives of what program activities are most beneficial. This will involve individual and focus group interviews with the students.

This award funds the research activities of Professor James Gray at the Virginia Polytechnic Institute and State University.

Two theories underlie our understanding of modern physics: quantum mechanics and Einstein's theory of gravity. Electrons experience gravity, however light they may be, and as such these quantum mechanical objects must fit within a gravitational theory. However, if one applies the usual rules of quantum mechanics to theories containing gravity, one gets nonsense. For example, the probability of certain events occurring, that is calculated in this way, turns out to be infinite. Many theories have been proposed to try and solve this problem. To date, by far the most successful is a theory called string theory. The idea is that instead of the fundamental building blocks of nature being particles, they are tiny, one dimensional, pieces of string. The troublesome probabilities mentioned above can be calculated to be proportional to one divided by the length of these strings. Thus one sees that, as one shrinks the length to zero and recovers a point like particle, one finds division by zero - an infinite answer. If the string length does not vanish, however, the theory can make sense. String theory comes with a very big catch. Mathematically, the theory only makes sense if the strings live in more than just three spatial dimensions. If some of the dimensions are wrapped up in some very small shape, we would not necessarily see them. Think of a piece of paper. This is clearly a two dimensional sheet. Now wrap the paper up into a cylinder and imagine furling it up more and more to make a smaller and smaller tube. If you could keep doing this (and if the paper was thin enough) then eventually the sheet would simply look like a line - a one dimensional object. The shape that the extra dimensions of string theory take determines the physics that someone living in the resulting three-dimensional universe would see. As an example, the volume of the shape determines the strength of gravity, as governed by Newton's constant. More subtle geometrical properties of the extra dimensions determine all of the rest of the physics of the resulting three-dimensional universe - for example, whether electrons exist and whether they carry electric charge. Given this, an obvious question arises. Is there a shape, which if used to hide the extra dimensions of string theory, gives rise to a three dimensional universe like our own? The answer to the question is still unknown. The proposed research is to use computers to study hundreds of millions of different possible shapes for the extra dimensions of string theory. In this way the PI will obtain a comprehensive survey of which three dimensional universes can be described by string theory. The PI will search this database to see if any of the shapes give rise to physics which is like that we see around us. The broader impact of this project will largely be through the technical training of graduate students, undergraduate researchers and a postdoctoral research assistant. Funding for six months of a postdoctoral researcher's tenure is included, with the other years and personnel involved being provided by resources from Virginia Tech. The PI plans to host an interdisciplinary workshop on computational algebraic geometry and string compactification at his home institution, to help develop links between these two different fields.

The project proposes to study compactifications of F-theory and heterotic string theory, which are currently the most promising candidates for reproducing known particle physics. Using modern formal methods of computational algebraic geometry, the PI will study large numbers of Calabi-Yau four- and three-fold compactifications of these theories in very fine detail. In addition, the PI will develop formalism to describe the moduli space and matter content of non-Kahler compactifications of heterotic theories. This work will try to answer two questions. First, can one find compactifications of string theory that reproduce not only the gauge group and particle content of the standard model, but also a realistic set of soft supersymmetry breaking terms? Second, if such compactifications can indeed be found, what are their common predictions for experimental observables that are yet to be measured? This research has the potential to advance mankind's knowledge of nature directly. It could help us understand whether string theory is simply a technical tool for studying some areas of mathematics and quantum field theory, or whether it is a fundamental physical theory of our universe. The work proposed will develop our understanding of string compactification on Calabi-Yau and non-Kahler manifolds in situations where the supergravity approximation is valid. The work on Calabi-Yau threefolds is designed to push the theory as close as possible to confrontation with experiment. The research on non-Kahler compactifications will lay the groundwork for more ambitious approaches to obtaining particle physics models from string theory in the future.

This award, under the direction of Professor Ronen Plesser of Duke University, supports a series of regional conferences in mathematical string theory.

String theory is the leading contender to unify general relativity and quantum field theory, two of the most significant developments of twentieth-century physics. This award provides support for a series of bi-annual regional meetings in North Carolina and Virginia which focus on the mathematical aspects of this subject. Research in this area not only advances our understanding of fundamental physics but has also proven to have extremely fruitful interactions with mathematics. As a result, research in this area advances the national interest by promoting the progress of science. These meetings fill a critical need by regularly bringing together isolated research groups to collaborate and hear research talks on various aspects of mathematical string theory. These meetings also provide essential training for students and postdocs, both by exposing them to a wide array of ideas as well as giving them a forum in which to give talks, thereby substantially aiding both training and research efforts.

The two day "Symposium on Challenges at the Interface of String Phenomenology and Geometry" will take place on July 6th and 7th, 2017 as part of the events of the 16th International String Phenomenology 2017 conference that will be held July 3-7, 2017 at Virginia Tech. The goal of the symposium is to promote and facilitate interdisciplinary research in mathematics and physics. The symposium will bring together talented researchers from a diverse range of institutions and backgrounds to address pressing problems at the interface of algebraic/differential geometry, commutative algebra, and string theory. The symposium will consist of one day of invited plenary talks from leading researchers in the field, along with an afternoon of parallel sessions which will invite the contributions of junior and other researchers from around the world. The broader impacts of this event will include the training of graduate students and the increase of interdisciplinary dialog between mathematicians and physicists.

Since its inception in 2002 at the University of Oxford, the annual String Phenomenology conference series has grown immensely and become a focal point for researchers working across a wide range of topics in string theory, particle phenomenology, cosmology and mathematics. It provides a forum to bring together active, young researchers from around the world to address the pressing question - Can string theory provide a model of observable physics? However, any path towards an answer to this question must lead to the cutting edge of modern mathematics. In string compactifications the unanswered questions of physics become those of geometry and lead to novel approaches to questions ranging from invariants of singular spaces to birational correspondences and the classification of algebraic varieties, from Mirror Symmetry to the Geometric Langlands program. A symposium to foster such interdsiciplinary dialog between mathematicians and physicists is timely and important. Topics covered by the symposium will include Differential Geometry, Algebraic Geometry, and Computational methods in Commutative Algebra. The conference website is available at http://www.cpe.vt.edu/stringpheno17/index.html.

This award funds the research activities of Professors Lara Anderson, James Gray, and Eric Sharpe at Virginia Tech.

In string theory --- a proposal for a fundamental theory of quantum gravity --- the roles of physics and geometry are intrinsically intertwined. While the questions that string theory attempts to answer are physical, the path to those answers frequently leads to cutting-edge challenges in modern mathematics. This award will fund a collaborative program of research to explore the physics that arises from string compactifications. The goals of this work include strengthening the links between string theory and current progress in particle physics by developing new foundational tools for the subject of string phenomenology. In addition, Professors Anderson, Gray and Sharpe aim to further bound and characterize the geometries arising in string compactifications. Experience shows that when strong physical requirements are expressed in the language of geometry, they can open the door to new and unexpected results in both physics and mathematics. As a result, research in this area advances the national interest by promoting the progress of basic science. Professors Anderson, Gray and Sharpe will involve junior scientists in this project, including a postdoctoral researcher and several graduate students who will take part in the collaborative research. Their efforts will include the organizing of conferences and workshops that will increase dialog between physicists and mathematicians on pressing problems at the boundary of both fields. In all of these aspects of student training and professional dialog, Professors Anderson, Sharpe and Gray are committed to actively encouraging the inclusion of under-represented groups into the frontline of progress in the sciences.

More specifically, the PIs will study two of the most flexible frameworks for four-dimensional compactifications of string theory: Heterotic string theory and F-theory. Within heterotic string theory, novel descriptions of the physical and geometric moduli spaces will be used to compute previously undetermined aspects of the effective theory, including the N=1 matter field Kahler potential and physically normalized Yukawa couplings. The nonperturbative contributions to Yukawa couplings will also be computed via quantum sheaf cohomology, a generalization of ordinary quantum cohomology. This work will explore new dualities including (0,2) mirror symmetry, as well as the global structure of the moduli space of SCFT's. Within F-theory, new results in the geometry of elliptic fibrations will be used to study the properties of singular Calabi-Yau manifolds and their links to Hitchin systems, as well as to study the implications of the ubiquity of multiply fibered manifolds for string dualities and effective theories. Recent progress in geometry will be used to extract new features of the effective theories describing F-theory compactifications, including the explicit four-dimensional field-dependent form of flux contributions to the superpotential.

This grant, under the direction of Professor James Gray at the Virginia Polytechnic Institute and State University, provides partial support for the conference "String Phenomenology 2017: The Sixteenth Annual Meeting on String Phenomenology", which will be held at Virginia Tech, Blacksburg, VA, USA on July 3rd-7th, 2017.

Since their inception at the University of Oxford in 2002, the annual String Phenomenology conferences have become the flagship annual meeting for the subfield of string phenomenology. String theory aims to describe the nature of the fundamental building blocks of matter and their interactions at the very highest energies, and string phenomenology aims to develop connections between the ideas of string theory and those of more traditional particle physics, astrophysics, and cosmology. As such, these conferences attract physicists and mathematicians representing a diverse international community and provide a venue in which they can discuss and present recent developments and challenges in this exciting interdisciplinary field. The sixteenth iteration of this conference series will be particularly important and timely, given recent important experimental results from the Large Hadron Collider at CERN in Geneva, from the Laser Interferometer Gravitational-Wave Observatory (LIGO), and from the Dark Energy Survey (DES), as well as a plethora of new results from the mathematical literature. As such, support for this series of meetings serves the national interest by promoting basic science in the United States. This grant is also envisioned to have significant broader impacts. These conferences bring together talented researchers from a diverse range of institutions and backgrounds, and special effort is made to incorporate the work of junior researchers and members of under-represented groups.