Wayne Raskind - US grants

Raskind 9700896 This award supports work on the theory of algebraic cycles on algebraic varieties over p-adically complete fields and over algebraic number fields. The principal investigator will define and study p-adic intermediate Jacobians for algebraic varieties over a p-adically complete field which have a p-adic uniformization. It is expected that these will be fundamental tools in the theory. He will also study higher obstructions to the Hasse principle for varieties over algebraic number fields. The theory of algebraic cycles is part of algebraic geometry, which is the study of solutions to systems of polynomial equations. This is one of the oldest fields in mathematics, but it has experienced an explosion of new ideas and fundamental theorems over the past thirty years. Also, new applications of the theory have been found to coding theory, cryptography and engineering.

Abstract for NSF 0070850--Raskind, Geisser, and Scharaschkin

The investigators will study the arithmetic and K-theory of algebraic varieties over fields of arithmetic type, such as algebraic number fields. They will introduce methods to describe the rational points on a wide class of algebraic varieties over number fields. One of the basic tools used to study these questions is motivic cohomology, which Geisser and Raskind will continue to develop.

The theory of systems of polynomial equations with rational coefficients is important for questions in cryptography and coding theory. Knowing a lot about the solvability of such a system of equations, or lack thereof, can play a big role in developing or breaking cryptosystems and codes. The proposers will use the latest techniques in number theory and algebraic geometry to study these questions. Although these are very old subjects, they have found spectacular applications in recent times, which has helped spur their theoretical development.

DMS-0300133
Geisser, Thomas H.

Abstract


Title: Motivic cohomology and arithmetic geometry


The investigator is working on two projects.
On the one hand, he tries to extend his results on
Weil-etale cohomology and special values of zeta functions
from smooth projective varieties over finite fields
to general varieties over finite fields, and to varieties
over local fields. The other project is the examination
of properties of the de Rham-Witt complex and topological
cyclic homology of smooth varieties over complete
discrete valuation rings. Topological trace homology TR
has a Frobenius operator, a Galois action and a filtration
analog to Fontaine's functor. The investigator wants
to exploit this structure to construct etale and crystalline
cohomology, and to apply this to arithmetic problems.


In arithmetic algebraic geometry,
solutions of polynomial equations are studied.
Even though this field is more than two thousand years old,
it turned out recently that there is a variety of applications
to cryptography. One method to study a solution set of polynomials
is to associate invariants called cohomology and zeta functions
to it, and then study those invariants instead.
Since the invariants are defined in very different ways, finding
relationships between them allows to translate knowledge on
one into knowledge on the other. The investigator studies
the relationship between zeta functions and a new cohomology,
called Weil-etale cohomology, on the one hand, and a new
invariant called topological cyclic homology on the other hand.

This grant will support US-based speakers and other US-participants to attend the conference, Regulators III, to take place in July 2010 at the Centre de Recerca Matematica (CRM) in Barcelona, Spain. This is the culminating conference of a year-long program in arithmetic geometry at the CRM, and will attract many of the best researchers in the world. The funding will allow US-based researchers to learn of the latest developments in a field in which mathematicians from Europe and Asia have been playing an increasing role over the last 10 years. Recent developments in the subject include work on Beilinson?s conjecture on special values of L-functions of certain automorphic representations of a symplectic group in four variables, and the lack of surjectivity of certain regulators for generic surfaces of degree at least 5 in projective 3-space over p-adic fields.



Regulators help to measure the size of objects in arithmetic and geometry, and they are a unifying theme in the subject. The definition and computation of regulators is a fundamental part of several parts of mathematics. One recent theme in the subject has been to transport techniques from arithmetic to geometric situations, and vice-versa. These methods have allowed us to demonstrate the existence of interesting elements of important arithmetic and geometric objects.

The fifth Mathematical Field of Dreams Conference will be held at Arizona State University in Tempe, Arizona on October 14-16, 2011 with the support of the National Science Foundation. The goal of this project/conference is to inspire and facilitate the access of students underrepresented in quantitative disciplines, to doctoral programs in the mathematical sciences while providing a forum where the knowledge and information that facilitates the selection of and application to relevant graduate programs is provided. The Field of Dreams Conference is one of the principal activities of the National Alliance for Doctoral Studies in the Mathematical Sciences (http://www.mathalliance.org/), a conglomerate of colleges and universities dedicated to encouraging and supporting students seeking doctoral degrees in the mathematical sciences. As with previous Field of Dreams Conferences, the 2011 meeting will allow students to identify opportunities for graduate study in mathematics, applied mathematics, computational biology, statistics, biostatistics and bioinformatics, mathematics education, and others. Participants will include approximately 100 undergraduate students recruited nationwide from groups underrepresented in the mathematical sciences, 30 undergraduate mentors, and at least 22 faculty members from the National Alliance graduate programs. Priority for support will be given to students from minority-serving institutions that lack extensive resources and programs in the mathematical sciences, or related research programs.

The US Department of Labor reports that jobs for mathematicians are expected to increase more than 20% from 2008 to 2018. http://www.bls.gov/oco/ocos043.htm Further, "Ph.D. holders with a strong background in mathematics and another discipline ? and who apply mathematical theory to real world problems will have the best job prospects" (Ibid.) There is a shortage of U.S. citizen and permanent residents applying to doctoral programs in the mathematical sciences; the situation is worse for students from U.S. underrepresented minority groups. The Field of Dreams Conference introduces underrepresented undergraduates and their faculty mentors to relevant, exciting, and important research in mathematical sciences and/or to scientific research programs that required the participation of students with strong quantitative/computational training. The Field of Dreams Conference provides students with the knowledge and tools required to apply for doctoral studies in the mathematical sciences. At the Field of Dreams Conference students will have the opportunity to meet and interact with faculty, graduate students, and other undergraduates doing quantitative scientific research. Students will be encouraged to apply to graduate school and their faculty mentors will be provided with information that they may disseminate at their home institution. Therefore, doctoral studies in the mathematical sciences information will spread beyond the conference throughout National Alliance for Doctoral Studies in the Mathematical Sciences (http://www.mathalliance.org/). The participation of both undergraduate mentors and graduate faculty from the Alliance facilitates (i) the transition from undergraduate to graduate programs and (ii) the establishment of partnerships (research, curricular development and coordination). The Field of Dreams Conference will continue to be an important driving force in increasing the number of underrepresented U.S. students obtaining doctoral degrees in mathematical sciences.

The Pathways to Calculus: Disseminating and Scaling a Professional Development Model for Precalculus Level Instruction Phase II MSP involves four Core Partners; Arizona State University, as the lead, and the Mesa, Chandler, and Scottsdale school districts. Supporting partners are: Brigham Young University, the University of Northern Colorado, the University of Georgia, Northern Arizona University, and Scottsdale Community College.

This Phase II project builds on the work of Project Pathways, a targeted MSP that identified attributes of professional development for secondary mathematics and science teachers that resulted in substantive and sustained improvements in student learning, as documented by student performance on district exams, state exams and research-based tools. The Phase II project leverages the research-based processes and tools that emerged in Phase I research to be highly effective for shifting teachers' instruction to be more inquiry-based and conceptually oriented. Phase II builds on Phase I findings in five broad categories that are critical for supporting mathematics teachers to realize significant shifts in their students' learning of key ideas of mathematics. These are teachers': 1) knowledge of the mathematics they teach; 2) beliefs about what constitutes effective mathematics learning and teaching; 3) ability to engage in reflection on student thinking and learning in relation to their teaching; 4) use of curricular support materials that promote inquiry-based and conceptually oriented instruction; and 5) participation in Pathways Professional Learning Communities (PLCs).

Phase I resulted in the development of the Pathways Precalculus Professional Development Model (P3DM), which includes in-class student activities with detailed teacher notes, computer animations and assessments. This supported Precalculus teachers in making instructional transitions that realized significant gains in student learning. Phase II extends this work by scaling the P3DM in three ways: 1) implementation of the P3DM at the community college and university levels, 2) implementation of the model in larger classes; 3) engagement of school administrators, including department chairs, to support all teachers in Core Partner school districts (11 schools) in adopting P3DM in Precalculus. Phase II studies the process of scaling P3DM in each of the three ways, and continues to examine how the P3DM experience affects teachers' instruction in other Precalculus level courses such as algebra II, college algebra and trigonometry.

The Phase II Research Agenda addresses the following:
1. What institutional factors of a school inhibit or support quality implementation of Pathways Precalculus materials? What external resources are needed to mitigate inhibiting factors and capitalize on supportive factors?
2. What is the typical developmental trajectory of teachers understanding and taking ownership of Pathways Precalculus materials? What support do teachers need to enable this development?
3. What is the typical developmental trajectory of teachers' understanding of the core mathematical ideas in the Pathways Precalculus materials and ability to accurately assess students' reasoning about these ideas in the classroom setting? What support do teachers need to understand these concepts and effectively assess student reasoning?
4. What administrative support enables teachers to effectively implement Pathways Precalculus materials?

A minimum of 150 high school mathematics teachers will be involved in addition to university and community college instructors and faculty. The products of this research will contribute knowledge and tools for scaling up the P3DM in the Precalculus strand (precalculus, college algebra, trigonometry and high school algebra II) of mathematics. The investigations will also produce insights about the factors that contribute to a mathematics department's transformation to support students in developing the capacity and confidence to solve novel problems and construct deeper and more connected understanding of the central ideas of a course.