| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 19, 2018 |
| Latest Amendment Date: | July 19, 2018 |
| Award Number: | 1802371 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Adriana Salerno
asalerno@nsf.gov (703)292-2271 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | October 1, 2018 |
| End Date: | September 30, 2021 (Estimated) |
| Total Intended Award Amount: | $150,100.00 |
| Total Awarded Amount to Date: | $150,100.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
1850 RESEARCH PARK DR STE 300 DAVIS CA US 95618-6153 (530)754-7700 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
1 Shields ave Davis CA US 95616-5294 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): |
ALGEBRA,NUMBER THEORY,AND COM, Combinatorics |
| Primary Program Source: |
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| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This research project concerns algebraic aspects of a range of surprising recent conjectures due to several authors, relating topics in algebraic geometry, the algebraic theory of knots, topology, difficult combinatorial (counting) problems, number theory, and mathematical physics. Conjectures that relate such a broad range of topics are often especially compelling to mathematicians, and can lead to particularly powerful results. To give one example, mathematicians often find it useful to "count points" on spaces that parametrize mathematical objects. So-called HLV conjectures mentioned below predict not only a formula for doing this for certain spaces commonly called "character varieties," but also generalize them in a way that is connected to their topology. Other conjectures in this family connect closely related spaces with some extremely compelling formulas involving diagrams, called parking functions, that are elementary to test by hand. While this project is based on algebraic methods, a full mathematical understanding of this topic is expected to reveal the geometry behind many deep open problems, some of which have roots in physics. The investigator also plans to involve undergraduate and graduate researchers in the project. This activity will focus on combinatorial methods that require minimal student prerequisites, and on the creation of algebra software for conducting computational experiments, an especially effective approach for student researchers unfamiliar with these topics. The development of general computer software is another impact of this project, which is expected to be useful to researchers in computational fields.
Some of the topics this project examines are the cohomology of the affine Springer fiber, Khovanov-Rozanksy knot invariants, some famous conjectures of Hausel, Letellier, and Rodriguez-Villegas (HLV), conjectures relating four-dimensional gauge theory to conformal theory due to Alday, Gaiotto, and Tachikawa (AGT), and related combinatorial extensions of the proof of the shuffle conjecture, such as the nabla-positivity conjecture. On one side of these conjectures, nearly all these topics have in common (conjectured) relationships with sheaves on the Hilbert scheme of points in the complex plane. On the other side, they are connected by the presence of a Riemann surface whose significance is hidden on the Hilbert scheme side, except through formulas. For instance, this Riemann surface would be the punctured disc C^* in the example of the Springer fiber, the punctured surface of genus g defining the character variety in the case of the HLV conjectures, or the two-dimensional surface on which the conformal field theory takes place in the case of AGT. The goal of this project is to make progress towards mathematical proofs of these conjectures, discover new ones, and ultimately understand the general mathematical picture. A major aspect of the approach is to extrapolate from explicit combinatorial formulas when they are available, such as the sort that appear in the shuffle conjecture, often called "nabla formulas" in Macdonald theory. Understanding this connection is of considerable interest to number theory, algebraic geometry, and combinatorics. A second aspect is the creation of sophisticated computer software for testing new conjectures, as well as for generating data to make predictions about the general relationship with geometry.
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.
PROJECT OUTCOMES REPORT
Disclaimer
This 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.
This project has produced results towards open problems in the area of Algebraic Combinatorics, especially the area of Macdonald theory and q,t-Catalan type formulas, and Algebraic Geometry. In these problems, one has numerical data that may be generated experimentally on a computer using elementary linear algebra. A priori, the resulting numbers may be fractional, but due to a hidden relationship with geometry, topology, and representation theory, the results turn out in many cases to be nonnegative integers. One example involves the``nabla operator,'' which is a family of matrices that may be generated by the Gram-Schmidt algorithm, whose entries produce the the combinatorics of the q,t-Catalan numbers, and more generally that of the Shuffle Theorem. Part of this project has investigated a generalization of these combinatorics related to a problem known as the ``nabla positivity conjecture,'' in which both postive and negative integers appear in a subtle but regular pattern. The PI and collaborators have discovered a ``categorification'' of this problem in which the signs have a topological interpretation in terms of odd homology groups of a well known algebraic variety which appears in representation theory. Odd homology groups may be interpretted as loops, or other odd-dimensional subspaces which cannot be continuously deformed to a point within a given geometric space. Another is related to the ``diagonal harmonics,'' which is a vector space of polynomials satisfying certain differential equations, one for each number n. The dimension of this space may be easily observed experimentally to have the formula of (n+1)^(n-1) for small values of n, but the first proof of this fact required sophisticated Algebraic Geometry related to the Hilbert scheme of points in the complex plane. Another achievement of the PI and collaborators was the first discovery of a basis of monomials of an equivalent vector space known as the ``diagonal coinvariant algebra,'' again using a geometric argument. Identitifying the underlying geometry behind the type of numerical data that appears in thie subject is of great interest, first because it may lead to the solution previously unsolved problems in combinatorics, and second because the combinatorics often lead to a new understanding of the geometric objects in question, which in many cases are well-studied objects of independent interest in a range of other fields.
Last Modified: 02/25/2023
Modified by: Erik Carlsson
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