Combinatorics, Algebra,
Number Theory Seminar
Spring 2012 Archive:

April 23

Tathagata Basak: Lattices from finite projective geometry.

Abstract: I shall describe a construction of a family of complex Hermitian Lorentzian lattices using the geometry of finite projective planes.

The reflection group of the "smallest" such example seems to be related to the monster simple group.

Using combinatorics of the finite projective plane, we can prove that infinitely many of these lattices are "modular."

In some cases, our construction leads to positive definite self-dual lattices having the symmetries of the finite projective plane.

In the "smallest" example, the positive definite lattice we get is known to provide the densest way to pack spheres in 24-dimensional space.

April 16

Jie Ling (University of Wisconsin - Madison): Arithmetic intersection on toric schemes and resultants.

Abstract: Let  K  be a number field and  O_K  its ring of integers. In this talk, I will present an arithmetic analog of Bernstein's theorem. Fix a finite set  A  in  Zⁿ. Let  (f₀,..., f_n)  be  n + 1  Laurent polynomials in  n  variables, and with integral coefficients. We assume that  f_i  is supported in  A. Let  F_i  be the homogenization of  f_i. When the associated toric scheme  X_A  (over  O_K ) is smooth and projective at a generic fiber, the arithmetic intersection number of  (F₀,..., F_n)  in the toric scheme is given by the norm of the resultants, i.e.

(F₀, . . . , F_n)_XA = log |N_K/Q(Res_{A(F0, . . . , Fn)} )|.

April 9

Holly Swisher (Oregon State University): Mock theta functions and shadows.

Abstract: Mock theta functions were first introduced in Ramanujan's last letter to Hardy. They are now known to be the holomorphic parts of harmonic weak Maass forms, thanks to recent work by Zwegers. A mock theta function has an associated shadow function, which is a cusp form, and there are some interesting examples in partition theory showing an arithmetic relationship between a mock theta function and its shadow. We discuss some of these examples, and further constructions of shadows.

March 26, April 2

Jonas Kibelbek: Congruences arising from weight 2 cusp forms.

Abstract: To every finite-index subgroup of  SL₂(Z), we associate a space of cusp forms and a modular curve. The Fourier coefficients of the cusp forms encode arithmetic information about the curve: the number of points modulo  p  and the eigenvalues of Frobenius. In the congruence subgroup case, this is the Atkin-Lehner-Li theory of newforms; in the noncongruence case, it is the theory of Atkin-Swinnerton-Dyer congruences. We discuss this structure in terms of formal group laws, and we use formal group isomorphisms to prove some new results and study some interesting examples.

The first week, I will give background on cusp forms and modular curves, and introduce the connection to formal groups. The second week, I will give additional background on formal groups and what they reveal about cusp forms.

March 19

Tathagata Basak: Lattices from finite geometries (preview).

March 12

Spring break: no seminar

March 5

Eli Stines: Strange Division or: How I Learned to Stop Worrying and Love the Zero.

Abstract: This discussion is a description of a recent attempt to define division by zero axiomatically, in terms of an extension of implication from the Boolean algebra  Z/₂ . We investigate the associated ring structure derived from subtraction and implication as binary operations. We will discuss the axiomatic construction of the variety in question, and some results on what kinds of rings are included within it.

February 20, 27

Candidate talks: no seminar

February 13

Deepak Naidu (Northern Illinois University): Deformation theory and quantum Drinfeld Hecke algebras.

Abstract: A deformation of an algebra is a 1-parameter family of algebras possessing some basic properties of the original algebra. In my talk, I will begin by giving a brief introduction to the deformation theory of associative algebras. In particular, I will explain how deformations of an associative algebra are governed by its Hochschild cohomology.

I will then discuss quantum Drinfeld Hecke algebras, which are generalizations of Drinfeld Hecke algebras in which polynomial rings are replaced by quantum polynomial rings. I will explain how these algebras can be identified as deformations of certain skew group algebras. This is joint work with Sarah Witherspoon (Texas A&M).

February 6

Anna Romanowska (ISU/Warsaw University of Technology): Isomorphism problems for generalized convex sets.

Abstract: I will first recall some necessary definitions and facts concerning affine spaces, generalized convex sets, and barycentric algebras. Then I will concentrate on the following problem:

When are two different generalized convex sets of similar type isomorphic (as barycentric algebras)?

I will discuss several instances of this problem, and formulate some results providing answers to the question in the case of certain (generalized) convex sets. These results will then be illustrated by some examples and counterexamples.

January 23, 30

Jonathan Smith: Engel groups and Bruck loops.

Abstract: On the real line, the reflection in the point  a  of a point  b  is given as  2a - b. More generally, in a group, Moufang loop, or diassociative loop  M, one may define the core operation as  ab^-1a. In a Moufang loop on which squaring is bijective, Bruck showed that the core is a quasigroup, isotopic to a Bol loop with the automorphic inverse property, or what is nowadays known as a Bruck loop. Glauberman used this Bruck loop structure to obtain many results about finite Moufang loops of odd order, including their solvability and the validity of Sylow's and Hall's Theorems. Since then, Bruck loops have gained additional importance through their natural occurrence in many areas, such as geometry, matrix decompositions, and special relativity theory.

The talk investigates the extension of Bruck's construction to loops, not necessarily Moufang, in which the squaring map is not necessarily bijective. Surjectivity is required, so a loop is said to be quadratically closed if the squaring map is surjective. For a quadratically closed loop, we consider the history space, a space of doubly-infinite sequences of loop elements such that each element in the sequence is succeeded by its square. When the loop is diassociative, and satisfies the 2-Engel condition under which elements commute with their conjugates, it is shown that the history space carries a loop structure with the right inverse and automorphic inverse properties, and forms a Bruck loop if the original loop is Moufang. With a quadratically closed field  K  as ground field, matrix groups  A(K), B(K), C(K)  of respective degrees 1, 2, and 3 are shown to satisfy the properties needed for the Bruck loop construction to work. Over the ground field  C  of complex numbers, the construction with  A(C)  offers a new setting for complex reflections and midpoints on the unit circle, and the Riemann surface of the square root function.

January 16

King Holiday: no seminar.

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