Awardee:  WASHINGTON UNIVERSITY, THE 
Doing Business As Name:  Washington University 
PD/PI: 

Award Date:  04/03/2014 
Estimated Total Award Amount:  $ 301,874 
Funds Obligated to Date: 
$
301,874

Start Date:  07/01/2014 
End Date:  06/30/2019 
Transaction Type:  Grant 
Agency:  NSF 
Awarding Agency Code:  4900 
Funding Agency Code:  4900 
CFDA Number:  47.049 
Primary Program Source:  040100 NSF RESEARCH & RELATED ACTIVIT 
Award Title or Description:  FRG: Collaborative Research: Hodge Theory, Moduli, and Representation Theory 
Federal Award ID Number:  1361147 
DUNS ID:  068552207 
Parent DUNS ID:  068552207 
Program:  ALGEBRA,NUMBER THEORY,AND COM 
Program Officer: 

Awardee Location  
Street:  CAMPUS BOX 1054 
City:  Saint Louis 
State:  MO 
ZIP:  631304862 
County:  Saint Louis 
Country:  US 
Awardee Cong. District:  01 
Primary Place of Performance  
Organization Name:  Washington University 
Street:  1 Brookings Drive 
City:  St. Louis 
State:  MO 
ZIP:  631304899 
County:  Saint Louis 
Country:  US 
Cong. District:  01 
Abstract at Time of Award  
The project will develop Hodge theory and apply it to problems in algebraic geometry, number theory and representation theory. The researchers intend to focus on four related topics: (1) MumfordTate (MT) domains, (2) moduli spaces, (3) algebraic cycles and the Hodge conjecture, and (4) mixed Hodge modules. (1) MT domains are classifying spaces of Hodge structures, and, roughly speaking, the boundary components of MumfordTate domains parametrize degenerations of Hodge structures. The PIs intend to advance number theory, representation theory and algebraic geometry by studying MumfordTate domains and their boundary components. For example, the PIs plan to extend work of Carayol, which seeks to associate Galois representations to automorphic representations whose archimedian component is a degenerate limit of discrete series. (2) The second topic concerns the realization of moduli spaces of geometric objects as quotients of discrete groups. An example of such a realization is the moduli space of nonhyperelliptic genus 3 curves, which can be realized as a ball quotient, where the 6 dimensional ball in question sits in the MT domain of K3 surfaces. However, there are not many examples of this type known. The PIs intend to look for more. (3) The third topic involves the approach to the Hodge conjecture via normal functions and their singularities due to Green and Griffiths. The PIs will develop this approach in several directions. For example, they will study the archimedean height function associated to a normal function, and they intend to study the nonreductive MT groups associated to normal functions. (4) Finally, the PIs will develop a flexible theory of complex variations of mixed Hodge modules and apply it to questions arising in representation theory. In particular, they would like to understand the structure of conformal blocks viewed as complex mixed Hodge modules on the moduli spaces of stable curves. Hodge theory is a central area of algebraic geometry with roots in the the classical (19th century) theory of special functions and period integrals. From a modern point of view, the goal of Hodge theory is to relate topological invariants of algebraic varieties to arithmetic and analytic invariants. The central notion is that of a Hodge structure on the cohomology groups of an algebraic variety. While the cohomology groups are purely topological, depending only on the shape of variety, the Hodge structure is a much more sensitive invariant. Consequently, the Hodge structure carries a great deal of important algebrogeometric and numbertheoretical information. The most famous unsolved problem in algebraic geometry is the Hodge conjecture, a question about the relationship between the Hodge structure of the cohomology groups of a variety and the existence of certain subvarieties. This focus on the relationship between topological objects and finer analytic invariants is typical of Hodge theory as a whole, and it is the main motivation for the research supported by this FRG. This research will consequently impact several areas of mathematics including number theory, algebraic geometry and representation theory. Owing to the number of techniques involved, the PIs have a diverse set of skills and points of view. An important component of the FRG will be devoted to conferences, which will exchange ideas between the PIs and train postdoctoral fellows and graduate students in a wide range of topics having to do with Hodge theory.  
Publications Produced as a Result of this ResearchNote: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval). Some links on this page may take you to nonfederal websites. Their policies may differ from this site.  
M. Kerr and C. Robles "Variations of Hodge structure and orbits in flag varieties" Advances in Math, v.315, 2017, p.27. doi:https://doi.org/10.1016/j.aim.2017.05.013 M. Kerr and C. Robles "Classification of smooth horizontal Schubert varieties" Eur. J. Math, v.3, 2017, p.289. doi:doi:10.1007/s408790170140x S. Bloch, M. Kerr and P. Vanhove "Local mirror symmetry and the sunset Feynman integral" Adv. Theor. Math. Phys., v.21, 2017, p.1373. doi:10.4310/ATMP.2017.v21.n6.a1 M. Kerr and Y. Yang "An explicit basis for the rational higher Chow groups of abelian number fields" Annals of Ktheory, v.3, 2018, p.173. doi:doi:10.2140/akt.2018.3.173 G. da Silva Jr., M. Kerr, and G. Pearlstein "Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution" Canadian Journal of Mathematics, v.68, 2016, p.280. doi:10.4153/CJM20150204 P. del Angel, C. Doran, J. Iyer, M. Kerr, J. Lewis, S. MuellerStach, and D. Patel "Specialization of cycles and the Ktheory elevator" Comm. in Number Theory and Physics, v.13, 2017, p.299. doi:10.4310/CNTP.2019.v13.n2.a2 M. Kerr and C. Robles "Classification of smooth horizontal Schubert varieties" European J. Math., v.3, 2017, p.27. doi:10.1007/s408790170140x G. da Silva, M. Kerr and G. Pearlstein "Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution" Canadian Journal of Mathematics, v.68, 2016, p.280. doi:http://dx.doi.org/10.4153/CJM20150204 Muxi Li "Integral regulators on higher Chow complexes" SIGMA, v.14, 2018, p.118. doi:10.3842/SIGMA.2018.118 R. Keast "CM Galois groups and MumfordTate domains" IMRN, v., 2016, p.. doi:10.1093/imrn/rnw018 M. Kerr and Y. Yang "An explicit basis for the rational higher Chow groups of abelian number fields" Annals of Ktheory, v.3, 2018, p.173. doi:10.2140/akt.2018.3.173 R. Keast and M. Kerr "Normal functions over locally symmetric varieties" SIGMA, v.14, 2018, p.116. doi:10.3842/SIGMA.2018.116 M. Kerr and G. Pearlstein "Boundary components of MumfordTate domains" Duke Math. J., v.165, 2016, p.661. G. da Silva Jr., M. Kerr, and G. Pearlstein "Arithmetic of degenerating principal variations of Hodge structure: examples arising from mirror symmetry and middle convolution" Canad. J. Math., v.68, 2016, p.280. doi:10.4153/CJM20150204 M. Kerr and G. Pearlstein "Boundary components of MumfordTate domains" Duke Mathematical Journal, v.165, 2016, p.661. M. Kerr and C. Robles "Variations of Hodge structure and orbits in flag varieties" Adv. in Math., v.315, 2017, p.27. doi:10.1016/j.aim.2017.05.013 S. Bloch, P. Vanhove, and M. Kerr "Local mirror symmetry and the sunset Feynman integral" Advances in Theoretical and Mathematical Physics, v.21, 2017, p.1373. doi:http://dx.doi.org/10.4310/ATMP.2017.v21.n6.a1 J. Burgos Gil, M. Kerr, J. Lewis, and P. Lopatto "Simplicial AbelJacobi maps and reciprocity laws" Journal of Algebraic Geometry, v.27, 2017, p.121. doi:https://doi.org/10.1090/jag/692 M. Kerr, J. Lewis and P. Lopatto "Simplicial AbelJacobi maps and reciprocity laws" J. Alg. Geom., v.27, 2018, p.121. doi:10.1090/jag/692  
Project Outcomes ReportDisclaimerThis Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.  
Summary: This report concerns the Washington University portion of an FRG project on Hodge theory and its applications to and interactions with representation theory, moduli theory, and algebraic cycles. As the part of algebraic geometry which converts families of polynomial equations (algebraic varieties) into linear algebra, differential equations, and period integrals, Hodge theory is uniquely equipped to explore the relationship between the topology, symmetries, and arithmetic of these families. The main results established concern: (i) degenerations of algebraic varieties and of Hodge structure, (ii) the differential constraints exerted by Lie theory on Hodge structures and algebraic cycles in families, (iii) the use of Hodge theory to evauate Feynman integrals in quantum field theory, and (iv) a complete algebrogeometric reinterpretation of Apery's irrationality proof for zeta(3). Intellectual Merit: The results (iii)(iv) hinge on the PI's formulas for Hodgetheoretic invariants of "higher" cycles related to algebraic Ktheory and motivic cohomology; (iv) finally completes a story begun decades ago by Beukers, Peters, and Stienstra, while the work on Feynman integrals (iii) is becoming wellcited and influential in physics. The PI's results on degenerations include a longsought classification of nilpotent cones (which appears to have implications in quantum gravity), and several generalizations of the ClemensSchmid sequence which improve our toolbox for relating the geometric and Hodgetheoretic boundaries in moduli theory. Broader Impact: During the award period, the PI was heavily involved in the training of graduate students in Hodge theory, graduating 5 Ph.D. students and coorganizing three major international conferences and other training events. In addition, the PI was a regular participant in the local math circle for junior high school students.
Last Modified: 07/10/2019 
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