Karen Smith, Ph.D. - US grants

This proposal seeks matching funds from the NSF to support modification of a 250 MHz Fourier transform nuclear magnetic resonance spectrometer for use in the undergraduate chemistry laboratories at UNM. At the present time there is no NMR instrument available for use by undergraduates in the department of chemistry. The addition of an NMR instrument will impact the educational experience of approximately 350 students per year at the sophomore, junior and senior levels. The four major laboratory courses targeted for immediate introduction to NMR experimentation include organic chemistry, chemistry lab III, synthesis and structure determination and instrumental analysis. The acquisition of an NMR spectrometer for these undergraduate laboratories will benefit students by improving and augmenting existing experiments in the curriculum and by allowing them hands-on experience with a modern, computer based instrument similar to research-grade instrumentation. It will also permit the addition of interesting and challenging experiments that are not implemented because their success and educational value relies heavily on the availability of an NMR.

With this renewal award, the Chemistry Division supports a Research Experiences for Undergraduates (REU) site directed by Dr. Lorraine M. Deck and four co-principal investigators in the Department of Chemistry at the University of New Mexico (UNM). The ten participants, recruited regionally and from UNM, will take part in a program that offers an intensive and comprehensive research experience. The conclusion of the research experience will include a final seminar, a scientifically written research report and a poster. The opportunity will also be afforded for students to participate in the writing of a paper for publication and/or travel to a professional meeting to make a presentation.

The main goal of the proposed project is to study the strong factorization conjecture for birational morphisms: every proper birational morphism between nonsingular complex varieties can be factored as a composition of smooth blowups followed by a composition of smooth blowdowns. The conjecture follows from the more fundamental problem of making a morphism toroidal by smooth blowups. There are several approaches to solving the toroidalization problem, for example, reducing it to a problem about resolution of singularities of a differential form with logarithmic poles, or using a composition of several constructions in lower dimensions to achieve toroidalization in higher dimension.

The main theme of the proposed research is to study the classification of algebraic varieties. More precisely, given two varieties that are the same generically (they are the same after removing some small subsets), can one transform one variety to another by some elementary operations? The simplest elementary operations are blowups and blowdowns of subvarieties: replacing a subvariety by another subvariety of larger (resp. smaller) dimension. The blowup-blowdown conjecture states that one can always get from the first variety to the second by blowing up several times and then blowing down.

This proposal seeks funds to implement and evaluate a model of instructional strategies for
prekindergarten teachers in order to enhance their ability to support young childre's
learning in language, early literacy, and math. The model will be experimentally tested
across Head Start and public school educational settings and is guided by a strong body of
recent research. As the two settings have different goals for children, funding of this
proposal will provide important information on the conditions under which the model is
effective. The outcomes of this research will determine critical elements for models of
instructional approaches such as intensity and duration, teacher background characteristics,
role of mentor teachers, and the importance of including a home literacy component. The
model is based on research that has clearly established a group of key elements for
prekindergarten language/literacy programs (e.g., expressive language, vocabulary, early
reading skills). Results will answer a critical remaining question regarding how a model
of instructional strategies can support teachers to implement a comprehensive set of
teaching practices supporting language/literacy development in ways that are sensitive to
the broadest range of social, emotional, and cognitive needs of young children.

The "Next Generation of Manufacturing Engineers for the Automotive Sector" project provides scholarships for students interested in all aspects of manufacturing for the automotive sector. All College of Engineering programs, as well as computer science and mathematics benefit from the project. The project strengthens existing alliances and resources, as well as partnerships with community colleges and industry sponsors throughout the Mid-South region.

The intellectual merit of the project is to supplement existing resources within the University to facilitate a life-altering, environmental change for financially disadvantaged, upper-division undergraduate CSEM students with academic potential. Scholarships are provided supplemented with summer industry or research internships between the junior and senior years. CSEMS students become an integral part of the academic culture and take more advantage of tutoring, peer mentoring, career development, research, student collaboration, as well as other academic and personal advantages gained by being fully engaged in the University community.

The broader impact of the project is achieved via the following objectives, each with specific, measurable outcomes, which are analyzed to continually improve the program: i) to decrease the financial debt of CSEM graduates via scholarships and counseling, ii) to decrease the number of CSEM students working in non-intellectually engaging jobs by increasing opportunities for students to work on a faculty member's funded research or through approved co-op or summer internships with industry partners of the project, iii) to improve employment placement in highly sought-after positions via the institution's relationship with industry partners who support seamless transition of internships into full-time positions, iv) to increase the graduation rate of CSEM students, particularly minority and female students, v) to decrease the average time to degree completion for CSEM students and, vi) to increase the number of CSEM graduates who enroll in graduate school.

These objectives are achieved through a unique combination of student environment modifications, one-on-one counseling and tutoring services, scholarships, retention and intervention, professional development and research activities, and industrial collaborations while leveraging and not replacing existing resources. The project strives to maintain the underrepresented participation from our baseline CSEMS awards, which consisted of 52% minorities and 40% females. A committee of faculty, staff and industry partners from the Tennessee Valley Authority (TVA), Federal Express Corporation (FedEx), The Solae Company, Wright Medical Technology, Inc., and the Army Corps of Engineers will select and mentor the recipients.

This project focus on problems at the interface of commutative algebra and
algebraic geometry. Specifically, several commutatative algebraic
approaches to problems in higher dimensional birational algebraic geometry
are proposed. These include building on the PI's prior work on ideal cores
to prove a conjecture of Kawamata's on the existence of a global section
for any ample line bundle on a smooth projective variety which is adjoint
to an ample bundle. It also includes the development of the theory of
multiplier ideals in the singular setting, and a study of the commutative
algebraic properties of jet schemes, especially for combinatorially rich
cases such as monomial schemes.

The work proposed here is part of the broader landscape of research in
"pure" algebraic geometry and commutative algebra, the fields that study
the geometric objects (called algebraic varieties) that can be described
by polynomial equations. These fields provide the underpinnings of several
important areas of applications of mathematics to problems in government
and industry. Examples include error-correcting codes based on algebraic
varieties, and computer-aided design (CAD) which "draws" geometric objects
on the computer screen by plotting pieces of algebraic varieties. While
this project does not address specific current problems in applications,
it does contribute to the healthy thriving infrastructure of the subject
necessary to support applications in the future. Furthermore, it is
expected that several of trainees in this project will continue later with
more applied aspects of the subject by moving to positions in industry and
government (such as the National Security Agency).

The project establishes a research training program in algebraic geometry
and its boundary areas at the University of Michigan. Target trainees are
recent PhDs and advanced graduate students. The project is run by five
senior faculty members representing a wide swath of current research
trends in algebraic geometry, in consultation with an additional 14 senior
faculty at Michigan in related areas. Activities include innovations in
seminar structure offering trainees broad exposure to a wide variety of
basic and research level topics in algebraic geometry (broadly construed);
an annual school (open also to young researchers outside Michigan)
covering a research topic of current importance taught by a prominent
researcher from outside Michigan; opportunities for post-doctoral
researchers to design and lead REU projects under the mentorship of a
senior faculty member; opportunities for advanced graduate students and
post-docs to develop their technical lecturing and writing skills; and
numerous opportunities for post-docs and advanced graduate students to
receive mentoring on all aspects of a research career in mathematics.


Algebraic geometry is one of the most central and active branches of
mathematics today, with increasingly important connections to other
branches of mathematics and science. Algebraic geometry also underlies
many applications of mathematics to technology and government, including
coding theory (bringing us, for example, the compact disc) and spline
theory (bringing us computer aided design and the computer graphics
essential to certain medical applications and Hollywood movies), as well
as important issues of national security pioneered by the National
Security Agency. The explosive growth of algebraic geometry at the end of
the twentieth century has made this a very exciting time to begin research
in the field, but it has also made it difficult for young researchers to
get started. This project will increase the flow of broadly trained
researchers in algebraic geometry and its boundary areas, therefore
enhancing the training infrastructure and the research workforce in these
vital areas of mathematics in the twenty-first century.

The symposium focuses the nation's attention on a critical policy issue -- the severity of teacher turnover in STEM K-12 mathematics and science -- and identifies evidence-based strategies for addressing this issue. The project proposes a symposium that brings together researchers, policymakers, practitioners and opinion leaders. The meeting is designed to: (1) develop a comprehensive picture of the current knowledge base around teacher turnover; and (2) identify key strategies to improve the retention, quality and diversity of the STEM teacher workforce. The Intellectual merit of this proposal lies in building on the established record of the National Commission on Teaching and America's Future (NCTAF) who have produced several far-reaching reports. The broader impact will be in the identification of policies that can address teacher turnover and retention, disseminate them widely and begin to identify a future research agenda.

This project concerns the theory and application of ``noncommutative projective geometry'' or the interaction of projective algebraic geometry with noncommutative algebra. Roughly speaking and by analogy with the commutative situation, the category of graded modules modulo torsion over a noncommutative graded ring of quadratic, respectively cubic growth should be thought of as the noncommutative analogue of a projective curve, respectively surface. This intuition has lead to a remarkable number of nontrivial insights and results in noncommutative algebra. Indeed, the problem of classifying noncommutative curves (and noncommutative graded rings of quadratic growth) can be regarded as settled and the motivating theme behind much of this proposal will be to understand noncommutative surfaces. A large class of these algebras can be classified in terms of the ``naive'' blow-ups developed in collaboration with Keeler and Rogalski. Although these blow-ups are constructed in a manner reminiscent of commutative blowups, and depend upon geometric data, their structure is quite unlike the classical objects. A major portion of the project will be to further understand these objects and to extend their applications. The other major theme of the project will be in applications of this general theory to specific classes of algebras. A particularly useful technique, here, is to ``complete'' the category of modules over a noncommutative algebra to those over a graded algebra and then to apply noncommutative projective geometry. This has, for example, been used to relate rational Cherednik algebras in type A to Hilbert schemes, and to Haiman's work on the n! conjecture. The project will continue this research to gain a deeper understanding of these important algebras and their relation to other areas of mathematics, for example to integrable systems and to the study of invariant eigendistributions on symmetric spaces.

Algebraic geometry, which is one of the oldest areas of modern mathematics, has its origins in the study of polynomial equations; for example a plane curve is the set of solutions of a polynomial equation in two variables. This leads to a rich interplay between that geometric object and the (commutative) algebra of the associated polynomials. Noncommutative algebra, which is a much younger subject, also has its origins in the theory of equations, in this case matrix equations, and in recent years has become increasingly important in many areas of mathematics (for example the theory of differential equations) and physics (Heisenberg's uncertainty principle is a classic illustration, but more subtle non-commutativity occurs, for example, in string theory). It has become apparent in recent years that there are definite, though often rather subtle, geometric structures hidden in these noncommutative objects and the interplay between the two has led to a rich theory, actually several theories, in their own right. These are collectively called noncommutative geometry.

This conference proposal follows up on a prior Wingspread meeting conducted by the National Commission on Teaching and America's Future. The earlier meeting focused on the research base around attracting and retaining K-12 STEM teachers. This second conference will build on that prior work and bring together education leaders, STEM researchers and policymakers to explore evidence based approaches to reducing STEM teacher turnover. The outcome of the conference will be a call to action that draws on the best evidence available to identify strategies that increase the retention, quality and diversity of the STEM teacher workforce.

Scholarships are being awarded to academically talented, financially needy upperclass students majoring in biology, chemistry, the geological sciences, or physics. The project is building on an existing successful CSEMS project for students majoring computer science. The following objectives are being met 1. decreasing the financial debt of graduates via scholarships and counseling; 2. decreasing the number of students working in non-intellectually engaging jobs by increasing opportunities for students to work on a faculty member's funded research or through approved co-op or summer internships with industry partners; 3. improving employment placement with industry partners; 4. increasing the graduation rate of students, particularly minority and female students; 5. decreasing the average time to degree completion for students; and, 6. increasing the number of graduates who enroll in graduate school.

[unreadable] DESCRIPTION (provided by applicant): Chromatin structure plays a critical role in maintaining proper gene expression. Epigenetic changes in the form of covalent modifications on histones including acetylation help regulate chromatin structure and gene expression. One group of enzymes, the histone deacetylases (HDACs) catalyze the removal of acetyl groups from histones which generally leads to gene repression. Since cancer can be caused by aberrant changes in the expression of genes that control cell growth, HDACs have become important targets of chemotherapeutics. Histone deacetylase inhibitors are currently being tested in clinical trials to treat several types of cancers, including breast cancer. These drugs are effective at preferentially halting growth of cancer cells over normal cells. However, the HDAC inhibitors currently in use target the catalytic sites of HDACs and do not discriminate among the several structurally similar HDACs in humans. Thus these HDAC inhibitors exhibit undesirable side effects due to their lack of specificity for individual HDACs. Individual HDACs reside in multi-subunit protein complexes, and a few known examples show that some of these HDAC-associated proteins are associated with and required for the catalytic activity of a particular HDAC. Therefore, as an alternative to inhibiting the HDAC enzymes themselves, inhibiting HDAC- interacting proteins should provide just as effective, but more specific treatment. This proposal will identify and functionally characterize proteins specifically associated with either HDAC1 or HDAC3 that can control breast cancer proliferation through modulation of their specific associated HDAC's activity. [unreadable] [unreadable] Specific Aims: 1. Perform proteomics analysis to identify proteins associated with HDAC1 and HDACS in human cells. Biochemically identify the subunit composition and the number of distinct complex(es) the HDACs and their differentially associated proteins reside in. 2. Identify which HDAC-interacting proteins are important for HDAC activity and determine how HDAC1- and HDACS-associated proteins affect the activity and integrity of their associated HDAC complex. 3. Target HDAC1- and 3-containing complexes in invasive breast cancer cells using siRNAs and identify which proteins, when abrogated, halt cancer cell growth. Relevance to Public Health: Many current anti-cancer drugs show unwanted toxic side effects in patients due to poor specificity of these drugs for their targets. This proposal seeks to find more specific protein targets for chemotherapeutics which should decrease the toxicity of these drugs. [unreadable] [unreadable] [unreadable]

ABSTRACT

Principal Investigator: Smith, Karen E.
Proposal Number: DMS - 0810844
Institution: University of Michigan Ann Arbor
Title: Commutative Algebra and its Interactions


This proposal is for a conference that will take place at the University of Michigan, Ann Arbor, July 31--August 5, 2008. The conference will focus on recent advances in commutative algebra, centered around areas which have been greatly influenced by the contributions of Mel Hochster. These include tight closure, singularities and multiplier ideals, the homological conjectures, invariant theory, local cohomology, the structure of free resolutions, and combinatorial commutative algebra. Hochster's research has had a transforming impact on the field, and there are several highly active areas which rely on his past and present work. We hope to bring together these areas for fruitful interaction. Commutative algebra is the study of equations: to paraphrase Hochster, "...algebra, after all, has to do with solving equations. Abstract algebra is the daughter of the theory of equations (in the broadest sense) and perhaps its best theorems still deal with that subject." Striking developments in the last few decades in commutative algebra have reinforced this view of commutative algebra. These developments include solutions to classical problems, as well as strengthened connections with diverse areas including algebraic topology, group cohomology, algebraic combinatorics, representation theory, invariant theory, and algebraic and arithmetic geometry. The invited speakers include experts in different areas, as well as several young researchers. The conference will provide an opportunity to present key results and recent advances. A special effort will be made to support the expenses of young researchers, women and minorities, graduate students, and those without alternative means of support.

DESCRIPTION (provided by applicant): The loss or dysfunction of GABAergic inhibitory interneurons, particularly parvalbumin (PV) expressing interneurons, is implicated in severe psychiatric disorders including schizophrenia, bipolar depression, and Tourette syndrome. PV+ interneurons regulate excitatory neuron output and are important for coordinating brain rhythms so that excitatory neurons can fire in synchrony- properties deemed to be critical for their function in cognition and behavioral control. This proposal describes the career development plan and research aims that Karen M|ller Smith, Ph.D. will achieve during her mentored career development training. The immediate goal of this proposal is to prepare Dr. Smith for an independent research career by providing her with theoretical and knowledge-based training in the neurobiology of psychiatric disorders through formal course work in neuroscience, neurodevelopmental disorders and statistics, and through mentored research training in primary tissue culture, cell transplantation, behavioral analysis of transgenic mice, gene expression microarray experiments using the translating ribosome affinity purification (TRAP) method, and target validation experiments using in vivo electroporation of silencing RNA (siRNA) constructs. The long-term goal is to gain an understanding of the developmental events contributing to the maturation and survival of cortical interneurons during the critical period of their synaptic integration and cell maturation in the postnatal brain. Dr. Smith obtained her Ph.D. in genetics by performing candidate gene analysis of dopaminergic system genes in Attention Deficit Hyperactivity Disorder (ADHD). During her postdoctoral training at the Yale Child Study Center (YCSC), Dr. Smith participated in the T32 Neurobiology of Childhood Neuropsychiatric Disorders training program and has gained expertise in developmental neurobiology, anatomical analysis and behavioral characterization of transgenic mice while performing several studies to elucidate the role of fibroblast growth factor (Fgf) signaling in cortical development and behavior. Dr. Smith will continue her studies at YCSC, under the primary mentorship of Dr. Flora Vaccarino, with the assistance of a distinguished group of scientific advisors consisting of experts in psychiatry, neurobiology, and molecular genetics, who will provide guidance with her research design, performance of experiments, and data analysis. The diverse research environment at Yale offers numerous opportunities to learn from internationally recognized experts in the fields of psychiatry, neurobiology, and developmental biology and to attend seminars hosted by the various departments in neurobiology and psychiatric research. Dr. Smith will take advantage of these resources in order to attend courses, seminar series, journal clubs, and hands on training at core facilities. Dr. Smith will enhance these activities by attending scientific conferences and an intense workshop course in neuroscience methodologies. In the research proposal, Dr. Smith will utilize mice lacking the fibroblast growth factor receptor 1 (Fgfr1) gene, which exhibit a decrease in PV+ cortical inhibitory interneurons that is correlated with hyperactive behavior, in order to gain a better understanding of how the neuronal circuitry involved in behavioral inhibition is established. The loss of interneurons in Fgfr1 mutants occurs postnatally, when Fgfr1 is expressed in glial cells of the cortex, particularly astrocytes. Dr. Smith will investigate whether the astrocytes of Fgfr1 mutants are less capable of supporting the survival and maturation of cortical interneurons by in vitro co-culture, and by cell transplantation of interneurons into control and Fgfr1 mutant cerebral cortices. She will generate astrocyte specific mutations of Fgfr1 to test the hypothesis that Fgfr1 signaling in astrocytes is essential for proper maturation of inhibitory interneurons. Dr. Smith will utilize TRAP and microarray analysis to identify pathways that are disrupted by Fgfr1 mutations in astrocytes, and will test the effects of candidate genes upon interneuron maturation by in vivo electroporation of silencing RNAs. These studies will allow Dr. Smith to determine the role of Fgfr1 in establishing the proper proportion of PV+ interneurons in the cortex, and will shed light into the postnatal development of PV+ interneurons, a problem with direct relevance to schizophrenia and bipolar depression. Results from this research will inform future studies aimed at promoting the health and maturation cortical interneurons. PUBLIC HEALTH RELEVANCE: This project will address the postnatal development of an important subtype of neurons, cortical inhibitory interneurons, which are diminished in schizophrenia and bipolar depression, and in mice lacking the Fgfr1 gene. The research will focus on the role of glia, supportive cells of the brain that express Fgfr1, in maintaining the health and maturation of inhibitory interneurons at a time when interneurons are integrating into the brain circuitry. The ultimate goal of this research is to identify mechanisms that contribute to interneuron maturation and survival, possibly leading to novel therapies aimed at preventing or reversing interneuron disruption in psychiatric illnesses.

This RTG project continues a successful research training program in algebraic geometry and its boundary areas at the University of Michigan. Target trainees are recent PhDs and advanced graduate students, though undergraduates are also involved on a smaller scale. The project is run by five senior faculty members (William Fulton, Robert Lazarsfeld, Mircea Mustata, Yongbin Ruan and Karen Smith) in consultation with an additional 16 senior faculty at Michigan in related areas. Algebraic geometry is a central and highly active branch of mathematics today, with increasingly important connections to other branches of mathematics and science. These other areas include commutative algebra, number theory, combinatorics, representation theory, and complex analysis, as well as string theory and other aspects of theoretical physics and computer science. The explosive growth of algebraic geometry at the end of the twentieth century has made this a very exciting time to begin research, but it has also made it difficult for young researchers to get started. This project will increase the flow of broadly trained researchers in algebraic geometry and its boundary areas. Given the size and visibility of the research groups involved in this project, Michigan's Algebraic Geometry RTG program has had and will continue to have a substantial impact at the national level in building a thriving community of young researchers in and around algebraic geometry.

The project develops the research potential of its trainees, as well as their capability to nurture the next generation. Activities include a rich array of seminar and workshop activities, including the opportunity for post-doctoral trainees to propose, design and organize workshops under the mentorship of a senior faculty member; opportunities for post-doctoral researchers to design and lead undergraduate research projects under the mentorship of a senior faculty member; numerous opportunities for advanced graduate students and post-docs to develop their technical and expository lecturing skills, and to receive mentoring on all aspects of a research career in mathematics; funds for trainee travel to domestic and international conferences in order to establish connections, get exposure to research trends not represented at Michigan, and gain visibility as they lecture on their work.

The research program proposes several problems at the interface of algebraic geometry and characteristic p techniques in commutative algebra. Given a variety, for simplicity say defined by polynomials
with integer coefficients, we can ``reduce mod p" for each prime
integer p, and arrive at a family of varieties over finite fields of varying characteristic. The overall goal of this program is to understand the relationships between phenomena defined by resolution of singularities or integration for complex varieties with "purely algebraic" issues in these prime characteristic models. For example, continuing with a project started with her post-doc Karl Schwede, the PI proposes to prove that complex Log Fano varieties reduce mod p to globally F-regular varieties. She also proposes a possible attack on the conjecture that log canonical singularities reduce mod p to F- pure singularities, which involves direct computation of the "F- threshold", a prime characteristic analog of the log canonical threshold, for hypersurfaces. Her PhD student Daniel Hernandez is making excellent progress on this.

Algebraic Geometry is the study of geometric objects which are defined by polynomial equations. Just as lines are described by
equations like y = 3 x + 1 or circles by equations like x^2 + y^2 =
4, it is possible to describe many more complicated geometric objects, for example, in higher dimensional spaces, with polynomials.
This project attempts to understand the geometry of these complicated objects by looking at the algebraic features of the equations that define them. For example, we can see geometric properties of the line (such as its slope, or "how fast it rises") in the equation y = m x +
b (the slope is the coefficient of x--the number m), or geometric
properties (such as the radius) of the circle x^2 + y^2 = 4 in the algebra of its equation (the square root of the constant term, or 2, is the radius). This project proposes exactly that: study the geometry of much more complicated objects defined by polynomials by looking carefully at the polynomials themselves. There are several projects proposed with the mentorship of young mathematicians in mind.

This project provides scholarships to able and financially needy upper-division students majoring in all STEM fields. The project emphasizes supporting students from groups that are underrepresented in STEM disciplines. A primary goal is to ensure such students graduate in five years with a minimum of student debt. This is done through a combination of individual counseling, tutoring, group seminars, internships, and research experiences. All this takes place within strong cohorts of students with similar experiences and goals. Additional goals include improving student placement after graduation and increasing the number of students going to graduate school in STEM disciplines.

Intellectual Merit: The academic programs into which the students go are strong, and there are academic support activities. The project emphasizes placing students within local industry and in retaining contact with these students. These goals are pursued though a combination of industry presentations, internships, seminars on applying to graduate programs, and comprehensive, convenient exit surveys. The S-STEM project has considerable support within local industry. It also exploits the natural cohorts that have been formed under the aegis of the MemphiSTEP program.

Broader Impact: The project is increasing the number and diversity of students who complete a STEM major and go on to work in the field or to further education. It specifically targets students from populations underrepresented in science and first-generation college students. The project emphasizes employing local talent that serves to enrich both the individual students and the local community.

Polynomials are mathematical functions that describe many different kinds of behavior in mathematics, physics, and engineering. They are concrete enough to be easily manipulated by hand or computer, and easy to understand mathematically, but also flexible enough to model many different situations in nature and science. Commutative algebra and algebraic geometry are the large branches of mathematics concerned primarily with understanding polynomials and the geometric shapes they define. Their broader applications range from the computer aided design used in the automotive and entertainment industries to the error correcting codes used in many digital media. This project in commutative algebra and algebraic geometry focuses on basic research in these fields, while broadly training a diverse group of PhD and undergraduate students in commutative algebra and algebraic geometry.

One specific type of algebra to be investigated are cluster algebras, which are commutative rings defined iteratively that have turned up in many branches of mathematics and physics, including the classical field of total positivity for matrices, the representation theory of lie algebras, number theory, Teichmuller theory, mirror symmetry, Poisson Geometry, discrete dynamical systems, string theory, wiring diagrams and networks, and more. The specific proposed research involves 1) developing the commutative algebra of cluster algebras, including basic notions such as localization and blowing up, as well as prime characteristic features such as test ideals; 2) reduction to prime characteristic and the iteration of the Frobenius map to understand singularities, cohomology and other features of algebraic varieties and commutative rings, studying invariants defined via Frobenius in prime characteristic, their relationship to cluster structures and their potential use in birational geometry; and 3) settling a question of Kollar on the behavior of local cohomology under base change (the main application of which is to reduce questions about maps between local Picard groups of complex varieties to characteristic p). The PI's role as a senior researcher is an integral part of the project: training the next generation of algebraists is woven into the very fabric of the research and one of the project's main objectives.

The Arctic surface climate has warmed dramatically over the past several decades. This warming, known as Arctic amplification, is associated, most notably, with extensive loss of sea ice. The Arctic stratosphere has also been experiencing change, including significant stratospheric ozone depletion. There is increasing evidence that polar stratospheric variability and change can have a significant downward impact on the surface climate. The proposed research will examine the connection between stratospheric circulation and chemistry and past and future changes in the surface climate of the Arctic region. Understanding the coupling between the stratosphere and the surface in the Arctic region has potentially important implications for sea ice and weather forecasting. In addition, the proposed research will lead to an improved understanding of the local and remote processes that will determine Arctic stratospheric and tropospheric ozone concentrations in the future. This is important because (a) stratospheric ozone protects the earth from harmful ultraviolet radiation and (b) tropospheric ozone is a toxic pollutant, which may have potential negative impacts on air quality and vegetation in the region.

This project will contribute to STEM workforce development in two ways. The principal investigator is a beginning investigator and this project will provide her support during the formative years of her career. The project will also provide support for the training of a graduate student. The anticipated science results have the potential to contribute to improve short-range forecast of sea ice and weather. The project will enhance collaboration between NCAR and Columbia University. Outreach to high school students will be enabled through leveraging of programs at the American Museum of Natural History and in the Inuit village of Arviat. Finally, research results will be shared with local K-12 teachers through an ongoing program sponsored by the principal investigator?s home institution.

The proposed research will be carried out by employing both observationally-based data products and global climate model integrations to address the question of whether stratospheric circulation and chemistry has a downward influence of the Arctic surface climate, particularly sea ice, on seasonal, interannual and multi-decadal time scales. From a modeling perspective, the approach outlined in the proposal will be to employ the Whole Atmosphere Community Climate Model (WACCM), with three different configurations that will allow examination of the interplay between dynamics and chemistry: (i) the standard WACCM with interactive stratospheric chemistry, (ii) the specified-chemistry version of WACCM (SC-WACCM) and (iii) the new fully coupled WACCM-TSMLT with tropospheric and stratospheric chemistry.

The project is in algebraic geometry, the science of understanding geometric shapes that can be described by polynomial equations. Polynomials are ubiquitous in mathematics because they can be easily manipulated by hand or by machine, yet they exhibit a large range of behavior which can model nearly any shape, from simple circles and lines to complicated images which could appear in medical imaging or an animated movie. The project seeks to understand the singularities of algebraic varieties---places where the shape is pinched or folded over on itself. We will devise tools to measure how "bad" these singularities are, and attempt to classify the "worst" ones. The project will be carried out by the PI with a team of trainees, including undergraduate students, graduate students, post-docs, and other collaborators.

The project investigates lower bounds on the F-pure threshold of polynomials over an algebraically closed field of characteristic p. The goal is to find sharp lower bounds, classify the singularities achieving that lower bound, and apply these results to open problems in the field. More specifically, a first direction is to prove general lower bounds on F-pure threshold in terms of other invariants, such as multiplicity, and then identify the varieties—“extremal singularities”— achieving those bounds. Next, the aim is to classify Frobenius forms, which are the “extremal singularities” for a certain lower bound on F-pure threshold in the homogeneous case. A third direction is to advance progress on a conjecture of Kleiman and Piene characterizing hypersurfaces whose Gauss maps are extremal in certain ways, using properties of Frobenius forms. The final direction is an investigation into whether Frobenius forms may define hypersurfaces admitting non-commutative resolutions of singularities, contrary to previous speculations about mildness of singularities for varieties with non-commutative resolutions.

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.