Combinatorics/Algebra Seminar:
Spring 2005 Archive

April 25

Doug Doan: Computational complexity of the modular Pascal triangle.

April 18

Stephanie Treneer (University of Illinois Urbana-Champaign): 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 Wisconsin-Madison): The Andrews-Garvan-Dyson crank and proofs of partition congruences.

Abstract: In 1944, Freeman Dyson conjectured the existence of an integer-valued 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 non-associativity 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, tri-restricted 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. Tri-restricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of tri-restricted numbers of the second kind. A new three-term recurrence relation for the tri-restricted 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 tri-restricted numbers of the first kind also satisfy a three-term recurrence relation. This relation is used to obtain a reciprocity theorem connecting the two kinds of tri-restricted 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 (1988-9) 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 Hopf-algebra objects.

Abstract:

Hopf algebras (which include the quantum groups) appear in several areas of mathematics, physics, and computer science. Hopf-algebra 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 Hopf-algebra objects appear in a variety of contexts, including low-dimensional 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 Hopf-algebra objects (Hopf algebras included) provide a wealth of invariants, and even algebraizations and structural results for 3-manifolds.

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 Hopf-algebra objects and the categories that admit them Finally, we will discuss some approaches to integrals for Hopf-algebra 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 Hopf-algebra 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 finite-dimensional 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 T-subalgebra. 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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