Sami H. Assaf

C. L. E. Moore Instructor
Massachusetts Institute of Technology


Courses  Research  Papers  Conferences  Photos



Office: Building 4, Room 182

Phone: (617) 258-6895

Email: sassaf [AT] math [DOT] mit [DOT] edu

Mailing Address:
    Department of Mathematics
    Massachusetts Institute of Technology
    77 Massachusetts Avenue
    Cambridge, MA 02139-4307

Curriculum Vita

Courses:
Fall 2008 18.02 Multivariable Calculus
Recitations MW 1pm & MW 2pm in room 2-131
Office Hours TW 10:30am - 11:30am in room 4-182

Research interests:

My research interests lie in the general areas of algebraic combinatorics and combinatorial representation theory. In particular, I am interested in symmetric functions, tableaux combinatorics and the representation theory of classical groups.

As a graduate student at the University of California Berkeley, I studied the Schur expansion for LLT polynomials, which are q-analogues of products of Schur functions introduced by Lascoux, Leclerc and Thibon in 1997. My research advisor is Mark Haiman. Here are my qualifying exam syllabus and my dissertation, along with a 3-page synopsis and a 9-page extended abstract.

My recent projects have included completing a combinatorial proof of Macdonald positivity based on ideas developed in my dissertation, exploring the connections between crystal graphs (which combinatorialize SL_n modules) and dual equivalence graphs (which combinatorialize S_n modules), investigating riffle shuffles of decks with repeated cards in collaboration with Persi Diaconis and K. Soundararajan at Stanford University, and searching for a 'kicking' basis for the Garsia-Haiman modules in collaboration with Adriano Garsia at UC San Diego.


Publications and preprints:

The preprints below are draft versions and may change over time; please consult the date (bottom of p. 1) for the latest version. Preprints are only posted after they have been submitted for publication. Once accepted, preprints will be posted to the arXiv.
(with Persi Diaconis and K. Soundararajan) A rule of thumb for riffle shuffling.
We study how many riffle shuffles are required to mix n cards if only certain features of the deck are of interest, e.g. suits disregarded or only the colors of interest. For a wide variety of features, the number of shuffles drops from 3/2 log_2 n to log_2 n. We derive closed formulae and an asymptotic `rule of thumb' formula which is remarkably accurate.
&bull  21 pages  &bull 
preprint  &bull
A generalized major index statistic. Seminaire Lotharingien de Combinatoire 60 (2008), Art. B60c, 13 pp. (electronic).   arVix:0807.0433
Inspired by the k-inversion statistic for LLT polynomials, we define the k-inversion number and k-descent set for words. Using these, we define a new statistic on words, called the k-major index, that interpolates between the major index and inversion number. We give a bijective proof that the k-major index is equidistributed with the major index, generalizing a classical result of Foata and rediscovering a result of Kadell. Inspired by recent work of Haglund and Stevens, we give a partial extension of these definitions and constructions to standard Young tableaux. Finally, we give an application to Macdonald polynomials made possible by connections with LLT polynomials.
A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs. Discrete Mathematics and Theoretical Computer Science, to appear as part of the conference proceedings for FPSAC 2008.   arVix:0804.1587
&bull  slides from MSRI Workshop (01/2008)  &bull
For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.
The Schur expansion of Macdonald polynomials.
Building on Haglund's combinatorial formula for the transformed Macdonald polynomials, we provide a purely combinatorial proof of Macdonald positivity using dual equivalence graphs and give a combinatorial formula for the coefficients in the Schur expansion.
&bull  9 pages  &bull  preprint  &bull  slides from BIRS Workshop (09/2007)  &bull
A combinatorial proof of LLT and Macdonald positivity.
We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomials.
&bull 24 pages  &bull  preprint  &bull

Upcoming conferences:

AMS Fall Western Section Meeting
     University of British Columbia, Vancouver, Canada
     October 4 - 5, 2008

MSRI Workshop: Classical Algebraic Geometry Today
     Mathematical Sciences Research Institute, Berkeley, CA
     January 26 - 30, 2009

MSRI Workshop: Combinatorial, Enumerative and Toric Geometry
     Mathematical Sciences Research Institute, Berkeley, CA
     March 23 - 27, 2009

AMS Spring Southeastern Section Meeting
     North Carolina State University, Raleigh, NC
     April 4 - 5, 2009

FPSAC 2009, 21st International conference on Formal Power Series and Algebraic Combinatorics
     Hagenberg, Austria
     July 20 - 24, 2009



Photos from recent adventures:

For videos, go to my YouTube channel SprocketEatsFraggles.


A tutorial on how to make a canvas shopping bag.





If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one.
--John Barrow