Algebra and Combinatorics Seminar 

Fridays, Milner 317 3:003:50 PM 
Cayley Graph of the Free Product Z_{3} * Z_{5 } 

The Algebra and Combinatorics seminar is devoted to studying algebra, combinatorics, their interconnection, and their relations to mathematics and applications. The seminar's organizers are Frank Sottile and Luis Garcia.  

January 18  Alvaro Rittatore, University of Uruguay  
The structure of algebraic monoids  
January 26  Fumei Lam, Brown University  
Polyhedral Study of Traveling Salesman Path Problems  
February 2  Volodymyr Kirichenko, Kyiv State University  
Quivers of rings  
February 9  Milena Hering, Institute of Mathematics and its Applications  
Syzygies of toric varieties  
February 16  TienYien Li, Michigan State University  
Numerical Calculation of the Jordan Normal Form  
February 23  Josephine Yu, University of California, Berkeley  
The Newton Polytope of the Implicit Equation  
March 2  Tatyana Chmutova, University of Michigan  
Twisted symplectic reflection algebras  
March 9  James Ruffo, Texas A&M University  
A straightening law for the Drinfel'd Lagrangian Grassmannian  
March 23  Tobias Hagge, Indiana University  
Graphical calculus methods for fusion categories  
March 30  Luis David Garcia, Texas A&M University  
Linear precision for multisided toric patches  
April 6  Chris Hillar, Texas A&M University  
Algebraic Characterizations of Uniquely Colorable Graphs  
April 13  Michael Anshelevich, Texas A&M University  
Lattice paths and orthogonal polynomials: a case study  
April 20  Eric Rowell, Texas A&M University  
Braid group representations from twisted quantum doubles of finite groups  
April 27  Laura Matusevich, Texas A&M University  
Combinatorics of binomial primary decomposition  
Abstract:
A classical theorem of Chevalley asserts that any connected
algebraic group is an extension of an abelian variety by a connected
affine group. In this talk we show an analogous result for normal
algebraic monoids (joint work with M. Brion), namely that a normal
irreducible algebraic monoid is obtained by the extension of an abelian
variety by a normal affine algebraic monoid. We begin by describing
some basic facts about the geometry of algebraic monoids (as embeddings
of their unit groups).
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Abstract:
In the Traveling Salesman Path Problem, we are given a set of cities,
traveling costs between city pairs and fixed source and destination
cities. The objective is to find a minimum cost path from the source to
destination visiting all cities exactly once. The problem is a
generalization of the Traveling Salesman Problem with many important
applications.
In this talk, we will discuss properties of the polytope corresponding to
a linear programming relaxation for the traveling salesman path problem.
We consider the set of traveling salesman path perfect graphs, graphs for
which the convex hull of incidence vectors of traveling salesman paths can
be described by linear inequalities. We show such graphs have a
description by way of forbidden minors and also characterize them
constructively. We extend this description to perfect graphs for the
2edge connected subgraph problem and discuss approximation algorithms for
these problems.
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Abstract:
The notion of a quiver for a finite dimensional algebra was
introduced by Gabriel in 1972 in connection with problems of the
representation theory of finite dimensional algebras. In 1975
the author carried over this notion to the case of semiperfect right
Noetherian rings and applied it to the structural theory of
rings, in particular, to the theory of serial rings. We consider
the different types of quivers of rings: prime quivers, Pierce
quivers, link graphs and others. Note the following results:
(1) a Noetherian semiperfect ring is indecomposable if and only if its
quiver is connected;
(2) the quiver of a Noetherian semiperfect and semiprime ring is strongly
connected;
(3) the quiver of a indecomposable right Artinian and hereditary ring is
acyclic, i.e., it does not have oriented cycles.
Quivers and prime quivers of semiperfect and semidistributive
(SPSD) rings are considered. We prove that for any semihereditary
SPSDring A there exists the classical ring of fractions A' and the
prime quiver of A coincides with the usual quiver of A'.
This is false for Noetherian semiperfect piecewise domains.
With any prime Noetherian SPSDring A with nonzero Jacobson
radical we associate its exponent matrix E(A). The quiver of A is
defined by E(A). Theory of nonnegative matrices is used for the study
of exponent matrices.
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Abstract:
Understanding the equations defining algebraic varieties and the
relations, or syzygies, between them is a classical problem
in algebraic geometry. Green showed that sufficient powers of
ample line bundles induce a projectively normal
embedding that is cut out
by quadratic equations and whose first q syzygies are linear.
In this talk I will present numerical criteria for line bundles
on toric varieties to satisfy this property. I will also discuss
criteria for the coordinate ring of such an embedding to be Koszul.
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Abstract:
Nontrivial Jordan Normal Forms are extremely difficult, if not
impossible, to compute numerically mainly due to the fact that multiple
eigenvalues are multiple roots of the characteristic polynomial of the
underlying matrix. We solve this problem via a recently established robust
algorithm that determines, with remarkable accuracy, the multiple roots and
multiplicity structure of a polynomial, even under perturbations.
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Abstract:
We apply tropical geometry to study the image of a map defined by
Laurent polynomials with generic coefficients. If this image is a
hypersurface then our approach gives a construction of its Newton
polytope. This talk should be accessible to a general mathematical
audience. No knowledge of tropical geometry will be assumed, and the
talk will include an introduction to this subject. This is joint work
with Bernd Sturmfels and Jenia Tevelev.
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Abstract:
Symplectic reflection algebras associated to a finite subgroup
G of Sp(U) were introduced by P. Etingof and V. Ginzburg in 2000. These
algebras and their representations turned out to be related to numerous other areas
of mathematics, and hence they were studied extensively during the last
several years.
We will consider two generalizations of symplectic reflection algebras.
The first one is a notion of symplectic reflection algebra associated to a
finite group G mapping, not necessarily injectively, to Sp(U). The second
one is a twisted symplectic reflection algebra associated to a finite group G
with its symplectic representation U and a twococycle $\psi$.
We will see that these two generalizations are connected: the case of
noninjective U can be reduced to the injective one, but in this
reduction the cocycle $\psi$ might become nontrivial even if at the
beginning it was trivial.
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Abstract:
We present a structured set of defining equations for the
Drinfel'd compactification of the space of rational maps of a
given degree in the Lagrangian Grassmannian. These equations
give a straightening law on a certain ordered set, which allows
the use of combinatorial arguments to establish useful geometric
properties of this compactification. For instance, its coordinate
ring is CohenMacaulay and Koszul.
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Abstract:
Fusion categories can be thought of as generalizations of the category
of finite dimensional vector spaces, or as generalizations of
representation categories of Hopf algebras. They have recently played
important roles in representation theory, conformal field theory and
low dimensional topology. No general classification is known, and it is
not clear to what extent results known for vector spaces and Hopf
algebra representations generalize to the fusion category context. One
useful fact is that only finitely many fusion categories exist with a
given tensor product structure. Finding these categories is possible in
principle, but difficult in practice.
In this talk I will introduce fusion categories and explain some of
their uses. I'll discuss some properties of the category of finite
dimensional vector spaces which are conjectured hold for fusion
categories, and the problem of classifying fusion categories with a
given tensor structure. Categorical language for fusion categories can
be replaced with elementary algebra; I will not assume substantial
knowledge of category theory.
This talk is based on joint work with SeungMoon Hong.
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Abstract:
In 2002, Krasauskas generalized the standard Bezier
and tensor product patches of geometric modeling to
multisided toric patches. While these offer the
promise of greater design flexibility, it is not clear
whether they possess the desirable properties of
the standard patches. One such property is linear
precision, which is the ability to replicate a linear
function.
In this talk, I will discuss work with Luis
Garcia on linear precision. Our geometric analysis
separates the shape of a patch (given by a polytope
and a system of weights) from its parametrization.
We first show that for any system of weights, there
is a unique parametrization by algebraic functions that
has linear precision. The existence of a rational
parametrization with linear precision becomes a geometric
statement concerning the intersection of a toric variety
with a particular linear subspace (which depends on the
system of weights).
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Abstract:
The study of graph colorability from an algebraic perspective
has introduced novel techniques and algorithms into the field. For
instance, kcolorability of a graph can be characterized in terms of
whether its graph polynomial is contained in a certain ideal. In this
talk, we interpret unique colorability in an analogous manner and use
Groebner basis techniques to prove an algebraic characterization for
uniquely kcolorable graphs. Our result also gives algorithms for testing
unique colorability. As an application, we verify a counterexample to a
conjecture of Xu concerning uniquely 3colorable graphs without triangles.
(Joint with Troels Windfeldt).
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Abstract:
One can calculate moments of measures using lattice paths and
orthogonal polynomials. I will explain this procedure for the specific
class of socalled free Meixner polynomials. In this particular case,
there are explicit combinatorial formulas not just for their moments but
for their free cumulants (to be defined) as well. At the end, I will
describe an alternative way to express these moments and cumulants, in
terms of operators.
The material of the talk is related to the Linear Analysis talk following
it, but the talks themselves will be independent.
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Abstract:
In joint work with Etingof and Witherspoon, we investigate the braid group
representations arising from categories of representations of twisted
quantum doubles of finite groups. For these categories, we show that the
resulting braid group representations always factor through finite groups,
in contrast to the categories associated with quantum groups at roots of
unity. We also show that in the case of pgroups, the corresponding pure
braid group representations factor through a finite pgroup, which answers
a question of Drinfeld.
If time permits, I will discuss some open questions and applications of
these results. A preprint can be found at:
http://arxiv.org/abs/math.QA/0703274
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Abstract:
Using examples, I will illustrate the main elements needed to
explicitly describe the primary components of a binomial ideal,
emphasizing the connections to combinatorics and (hypergeometric)
differential equations. This is joint work with Alicia Dickenstein and
Ezra Miller.
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