TRUSTEES OF THE UNIVERSITY OF PENNSYLVANIA
In this project the PI will investigate the combinatorics associated with some useful polynomials in several variables which were introduced by I. G. Macdonald in 1988, and play a central role in algebraic combinatorics, with applications to a growing number of other fields including special functions, algebraic geometry, and mathematical physics. They are symmetric functions in a set of variables X={x_1,x_2,... ,x_n}. In 1995 Macdonald introduced ``nonsymmetric Macdonald polynomials", which have a number of intriguing analytic and algebraic properties.Their original definition was rather difficult and indirect, but in 2004 the PI found a nicecombinatorial formula for them, which was proved in subsequent joint work between Haiman, Loehr and the PI. The PI and others have been finding that the new combinatorics of Macdonald polynomials often leads to new formulas for related objects. For example, part of the award is being used by the PI and his collaborators, to investigate certain limiting cases of nonsymmetric Macdonald polynomials called Demazure characters and Demazure atoms, which are connected to a branch of algebra known as representation theory.
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| AWARD OVERVIEW |
| Award Number |
0901467 |
Funding Agency |
National Science Foundation |
| Total Award Amount |
$150,000 |
Project Location - City |
Philadelphia |
| Award Date |
07/30/2009 |
Project Location - State |
PA |
| Project Status |
Completed |
Project Location - Zip |
19104-6205
|
| Jobs Reported |
0.00 |
Congressional District |
02 |
| Project Location - Country |
US |
|
|
Recipient Information
(Grants)
| Recipient Information (Grants) |
|
Recipient Name
|
TRUSTEES OF THE UNIVERSITY OF PENNSYLVANIA |
| Recipient DUNS Number |
042250712
|
| Recipient Address |
3451 WALNUT ST |
| Recipient City |
PHILADELPHIA |
| Recipient State |
Pennsylvania |
| Recipient Zip |
19104-6205 |
| Recipient Congressional District |
02 |
| Recipient Country |
USA |
Required to Report Top 5
Highly Compensated Officials |
No |
Projects and Jobs Information
| Projects and Jobs Information |
| Project Title |
THE COMBINATORICS OF MACDONALD POLYNOMIALS AND RELATED OBJECTS |
| Project Status |
Completed |
| Final Project Report Submitted |
Yes |
| Project Activities Description |
Science & Technology, General/Other |
| Quarterly Activities/Project Description |
In joint work, D. Wagner, P. Branden, M. Visontai (one of the PI s Ph. D. students) and the PI proved the %22mutivariate MCP Conejcture%22 around Thanksgiving 2009. This was one of the projects listed in my original proposal. We have submitted a paper on it for publication. Visontai and the PI have a new preprint on stability of multivariate Eulerian polynomials, which has some connections to the MCP Theorem.While visiting UCSD on sabbatical during spring 2010, the PI found a new way of expressing the Hilbert series of the space of diagonal harmnonics DHn as a polynomial with integer coefficients. A unproved conjecture of the PI and N. Loehr gong back to 2001 gives this Hilbert series as s positive sum over lattice paths, and the PI s other Ph. D. student P. Levande is now working to try and show the new polynomial identity can be simplified to give the conjectured lattice path formula. A paper by the PI on this new formula has been accepted for publication by the journal Advances in Mathematics, and an extended abstract has been accepted as a talk at the international FPSAC 2011 conference to be held this summer in Iceland. Garsia, G. Xin and the PI have a new preprint building on this theme.The PI has several joint projects with J. Remmel of UCSD in various stages of completion. One project, also joint with the PI s former student M. Yoo, is on rook polynomials and has been submitted for publicaiton. Another, joint also with S. Mason (also a former Ph. D. student of the PI s) involves Mason s generalized RSK algorithm and is close to becoming a preprint. Finally, Remmel, the PI, and M. Can and F. Butler (two other former Ph. D. students of the PI) are wiriting a book on rook polynomials. We now have a draft of chapter 1, and outlines of 13 other chapters.In Fall 2009 J. Morse, M. Zabrocki, and the PI discovered an extension of the %22shufle conjecture%22, which gives a formula for the character of DHn in terms of lattice paths. |
| Jobs Created |
0.00 |
| Description of Jobs Created |
This project currently has no jobs created or retained. |
Purchaser Information
(Grants)
| Purchaser Information |
| Contracting Office ID |
Not Reported |
| Contracting Office Name |
Not Available |
| Contracting Office Region |
Not Available |
| TAS Major Program |
49-0101 |
| Award Information |
| Award Date |
07/30/2009 |
| Award Number |
0901467 |
| Order Number |
|
| Award Type |
Grants |
| Funding Agency ID |
49 |
| Funding Agency Name |
National Science Foundation |
| Funding Office Name |
Not Available |
| Awarding Agency ID |
49 |
| Awarding Agency Name |
National Science Foundation |
| Amount of Award |
$150,000 |
| Funds Invoiced/Received |
$150,000 |
| Expenditure Amount |
$150,000 |
| Infrastructure Expenditure Amount |
$0 |
| Infrastructure Purpose and Rationale |
Not Reported |
| Infrastructure Point of Contact Name |
Not Reported |
| Infrastructure Point of Contact Email |
Not Reported |
| Infrastructure Point of Contact Phone |
Not Reported |
| Infrastructure Point of Contact Address |
Not Reported |
| Infrastructure Point of Contact City |
Not Reported |
| Infrastructure Point of Contact State |
Not Reported |
| Infrastructure Point of Contact Zip |
Not Reported |
Product or Service Information
(Grants)
| Product or Service Information |
| Primary Activity Code |
U01 |
| Activity Description |
Science & Technology, General/Other |
| Sub-Awards Information |
| Sub-awards to Organizations |
0 |
| Sub-award Amounts to Organizations |
$0 |
| Sub-Awards to Individuals |
0 |
| Sub-Award Amounts to Individuals |
$0 |
| Number of Sub-awards less than $25,000/award |
0 |
| Amount of Sub-awards less than $25,000/award |
$0 |
| Number of payments to vendors greater than $25,000 |
0 |
| Total Amount of payments to vendors greater than $25,000/award |
$0 |
| Number of payments to vendors less than $25,000/award |
21 |
| Total Amount of payments to vendors less than $25,000/award |
$14,984 |
| Location Information |
| Latitude, Longitude |
39º 57' 10",
-75º 11' 34" |
| Congressional District |
02 |
| Address 1 |
|
| Address 2 |
|
| City |
Philadelphia |
| County |
Philadelphia |
| State |
PA |
| Zip |
19104-6205 |
|
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