Thursday September 30, 2010, 1:20 PM
Florian block (University of Michigan)
Computing Node Polynomials for Plane Curves
Enumeration of plane algebraic curves has a 150-year-old history. A
combinatorial approach to this problem, inspired by tropical geometry, was
recently suggested by Brugalle, Fomin, and Mikhalkin. I will explain this
approach and its applications to computing Gromov-Witten invariants (or
Severi degrees) of the complex projective plane, and their various
generalizations. According to Goettsche's conjecture (now a theorem),
these invariants are given by polynomials in the degree d of the curves
being counted, provided that d is sufficiently large. I will discuss how
to compute these "node polynomials," and how large d needs to be.
Thursday October 7, 2010, 1:20 PM
Sergi Elizalde (Dartmouth)
Permutations and beta-shifts
A permutation pi is realized by the shift on N letters if there is an
infinite word on an N-letter alphabet whose successive shifts by one
position are lexicographically in the same relative order as pi.
Understanding the set of permutations realized by shifts, as well as other
one-dimensional dynamical systems, is important because it provides tests
to distinguish deterministic sequences from random ones.
In this talk I will give a characterization of permutations realized by
shifts, and also by a natural generalization of them, where the instead of
N we have a real number beta.
Thursday October 14, 2010, 1:20 PM
Steven Sam (MIT)
Saturation theorems for the classical groups
Following Klyachko's solution of Horn's problem of
characterizing the eigenvalues of A+B in terms of the eigenvalues of
Hermitian matrices A and B, there has been interest in the so-called
saturation conjecture (now theorem). This says that c^\nu_{\lambda, \mu}
> 0 if and only if c^{N\nu}_{N\lambda, N\mu} > 0 for some N>0 where c is
the Littlewood-Richardson coefficient. Following work of Derksen-Weyman
and Schofield, I proved a generalization of this statement for
orthogonal and symplectic groups. The proof uses techniques from
invariant theory and quivers, so is not exactly combinatorial, so I will
try to emphasize the combinatorial parts and questions that arise from
the work and no prior knowledge of quivers or Lie theory will be
assumed.
Thursday October 21, 2010, 1:20 PM
Delong Meng (MIT)
Reduced decompositions and permutation patterns generalized to the higher Bruhat order
A reduced decomposition of a permutation is an expression of the permutation as a sequence
of adjacent transpositions. For example, (12, 13, 23, 14, 24) is a reduced decomposition
of 3421 because 3421 can be obtained from 1234 through this sequence of transpositions as
follow: {12}34 ---> 2{13}4 ---> {23}14 ---> 32{14} ---> 3{24}1 ---> 3421.
Reduced decompositions received considerable attention in a variety of contexts such as the
study of the Coxeter groups and the enumeration of pseudoline arrangements. In this talk,
we generalize the concept of reduced decompositions to the higher Bruhat order, a family of posets,
one of which is a partial order on the symmetric group. We also introduce generalized permutation
patterns---which turns out to be intrinsically related to subnetworks of a random sorting
network---into the picture. In particular, we geometrically interpret reduced decompositions
along with generalized permutation patterns through the cyclic hyperplane arrangements.
This talk is largely based on my research project from the Research Experience for
Undergraduates (REU) at the University of Minnesota Duluth this summer.
Thursday November 4, 2010, 1:20 PM
Amir Barghi (Dartmouth)
Firefighting on Random Geometric Graphs
In the Firefighter Problem which
was first introduced in 1995, a fire starts at a vertex of a
graph and in discrete time intervals spreads from
burned vertices to their neighbors, unless they are
protected by one of the
$f$ firefighters that are deployed every turn. Once
protected, a vertex remains protected.
We assume that the trees in a forest are randomly
distributed with a fixed
density and fire spreads from one tree to another if their
distance is less than one.
In this talk, we will discuss a technique from percolation
that helps us prove that
stopping the fire from spreading indefinitely, requires a
linear relation
between $f$ and the density of the forest.
This talk is the extended version of my earlier talk that I
gave at GSS this quarter.
Thursday November 11, 2010, 1:20 PM
Luis Serrano (Universite du Quebec a Montreal)
Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials
We exhibit a canonical connection between maximal (0,1)-fillings of a moon
polyomino avoiding north-east chains of a given length and reduced pipe dreams
of a certain permutation. Following this approach we show that the simplicial
complex of such maximal fillings is a vertex-decomposable and thus a shellable
sphere. In particular, this implies a positivity result for Schubert
polynomials. For Ferrers shapes, we moreover construct a bijection to maximal
fillings avoiding south-east chains of the same length which specializes to a
bijection between k-triangulations of the n-gon and k-fans of Dyck paths.
Using this, we translate a conjectured cyclic sieving phenomenon for
k-triangulations with rotation to the language of k-flagged tableaux with
promotion. This is based on http://arxiv.org/abs/1009.4690.
Thursday November 18, 2010, 1:20 PM
Joel Lewis (MIT)
Pattern Avoidance in Alternating Permutations
We give bijective proofs of pattern-avoidance results for a class of
permutations generalizing alternating (up-down) permutations. The bijections
employed include a modified form of the RSK insertion algorithm and recursive
bijections based on generating trees. As special cases, we show that the sets
A_{2n}(1234) and A_{2n}(2143) are in bijection with standard Young tableaux
(SYT) of shape (3^n).
Alternating permutations may be viewed as the reading words of SYT of a certain
skew shape. We extend our study to pattern avoidance in the reading words of
SYT of any skew shape. We show bijectively that the number of standard Young
tableaux of shape lambda/mu whose reading words avoid 213 is a natural
mu-analogue of the Catalan numbers. Similar results hold for the patterns 132,
231 and 312.
Thursday December 2, 2010, 1:20 PM
Nan Li (MIT)
Generalized Ehrhart polynomials
Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart
states that the number of lattice points in the dilations
$P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem
by allowing the vertices of P(n) to be arbitrary rational functions in
$n$. In this case we prove that the number of lattice points in P(n)
is a quasi-polynomial for $n$ sufficiently large. Our work was
motivated by a conjecture of Ehrhart on the number of solutions to
parametrized linear Diophantine equations whose coefficients are
polynomials in $n$, and we explain how these two problems are related.
Previous Terms:
Fall 2004.
Spring 2006.
Fall 2008
Winter 2009
Spring 2009
Fall 2009
Winter 2010
Spring 2010
If you are interested in speaking please email Sergi Elizalde, or Rosa Orellana.