
Combinatorics/Algebra Seminar: Spring 2005 Archive

April 25

Doug Doan: Computational complexity of the modular Pascal triangle.

April 18

Stephanie Treneer (University of Illinois UrbanaChampaign): Congruences for the coefficients of weakly holomorphic modular forms
Abstract: Recent works have established linear congruences for certain arithmetic functions of interest to number theorists, such as the partition function and the traces of singular moduli. The method in each case is to express the arithmetic function as the Fourier coefficients of a particular modular form, and then use the theory of modular forms to derive the congruences. We show that this phenomenon is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form. In particular, we give linear congruences for a wide class of partition functions.

April 11

Charles Doran (University of Washington): String theory and mathematics.
Abstract: Why are mathematicians so excited about string theory? It is
clear why a physicist might be interested in a theory that seeks to unify
all the forces of nature, a theory whose stated goal is to obtain a
consistent set of equations explaining galaxies and subatomic particles
and everything in between. It's not so clear why mathematicians who study
geometry, topology, algebra, combinatorics, number theory, etc. should
care. The answer lies in the powerful application of ideas from string
theory within mathematics. Indeed string theory has pointed out deep
connections between whole branches of mathematics that were previously
thought unrelated. It is the goal of the speaker to convey the spirit of
these developments without the technical details.

April 4

Karl Mahlburg (University of WisconsinMadison): The AndrewsGarvanDyson crank and proofs of partition congruences.
Abstract: In 1944, Freeman Dyson conjectured the existence of an integervalued crank function for partitions that would provide a combinatorial proof of Ramanujan's congruence
p(11n + 6) = 0 mod 11
by dividing the partitions of 11n + 6 into 11 classes of equal size. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that
M(m,11,11n + 6) = p(11n + 6)/11 ,
where M(m,N,n) is the number of partitions of n for which the crank is m modulo N .
The main result of this talk proves a conjecture of Ono, which essentially asserts that Dyson's elusive crank is a universal combinatorial statistic that "explains" partition congruences of every known type.

March 28

Jonathan Smith: How to gauge the nonassociativity of a quasigroup.

March 21

James Fiedler: "Quasigroups and Topology" by I.M. James (contd.)

March 14

Spring Break: no seminar.

March 7

James Fiedler: "Quasigroups and Topology" by I.M. James.

February 28

Christian Roettger: Periodic points in discrete dynamical systems.

February 21

Jonathan Smith: Some combinatorial recurrences.
Abstract: By analogy with Stirling numbers, trirestricted numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most three elements. Trirestricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of trirestricted numbers of the second kind. A new threeterm recurrence relation for the trirestricted numbers of the second kind is presented, with a combinatorial proof. In answer to a problem that has remained open for several years, it is then shown by determinantal techniques that the trirestricted numbers of the first kind also satisfy a threeterm recurrence relation. This relation is used to obtain a reciprocity theorem connecting the two kinds of trirestricted number.

February 14

Anna Romanowska (Warsaw University of Technology): Some algebraic structures of logic programming.
Abstract: Bilattices were introduced by M. Ginsberg and M.
Fitting (19889) as a general framework for a variety of
applications such as truth maintenance systems, default inference
and logic programming, and investigated further by these and other
authors. The main feature of these algebras is that they have two
separate lattice structures defined on the same set. However,
different authors proposed different connections between the two
lattice structures. We will describe several notions of bilattices
used by different authors, provide their characterizations, and
investigate their structure. In particular, we show how the notions
of hyperidentities and of superproduct can be used to characterize
certain classes of bilattices.

February 7

Fernando Souza (University of Iowa): On integrals and trace functions for Hopfalgebra objects.
Abstract:
Hopf algebras (which include the quantum groups) appear in several areas
of mathematics, physics, and computer science. Hopfalgebra objects
generalize Hopf algebras from the category of vector spaces to monoidal
categories, i.e., categories with a (possibly quite abstract) notion of
tensor product, usually with some additional structure (e.g. a braiding,
which switches the entries of that tensor product.) These categories and
Hopfalgebra objects appear in a variety of contexts, including
lowdimensional topology, quantum theory, semantics (in logic and computer
science) and, of course, algebra and category theory. Case in point, the
interdisciplinary area called quantum topology, where Hopfalgebra objects
(Hopf algebras included) provide a wealth of invariants, and even
algebraizations and structural results for 3manifolds.
In this talk, we will first review Hopf algebras, and the importance of
the tensors known as integrals. Then we will summarize the basic
definition and facts about Hopfalgebra objects and the categories that
admit them Finally, we will discuss some approaches to integrals for
Hopfalgebra objects, focusing on the role of these tensors in
restricting the algebraic structure of those objects. Diagrammatic
methods play a key role in this topic due to universality and coherence
theorems that characterize diagrammatic Hopfalgebra objects as the
universal ones. This seminar is intended to be informative and accessible.

January 31

Bokhee Im (Chonnam National University): Thomas sums and bilinear forms.
Abstract: Comtrans algebras are ternary analogs of Lie algebras, and show up in many physical, geometric, and algebraic contexts. A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. We investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finitedimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the current methods involve only the theory of comtrans algebras.

January 24

Luiz Peresi (University of São Paulo): Elements of minimal degree in the center of the free alternative algebra.
Abstract:
It is unknown whether the center of the
free alternative algebra has a finite
number of generators, as a Tsubalgebra.
Therefore, it is of interest to find nonzero
elements of minimal degree in the center of
the free alternative algebra. This problem is
related to the problem of finding nonzero
elements in the annihilator of the free
Mal'tsev algebra.
Filippov (1999) conjectured that the minimal
degree of nonzero elements in the center of the
free alternative algebra is 7. Representing
identities by matrices we prove that this conjecture
is true. We find a basis for elements of degree 7.

January 17

King Holiday: no seminar.
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