Fall 2008
Fridays, Milner 317, 3:00–3:50 p.m.
[Day] August 29 [Time] 3:00–3:50 [Name] Organizational Meeting [University] [Title] [Abstract] [Comment]
[Day] September 5 [Time] 3:00–3:50 [Name] Aaron Lauve [University] (TAMU) [Title] The Markoff condition and central words [Abstract] During his study of binary quadratic forms aX2 + bXY + cY2 in the late 1800s, Markoff introduced necessary and sufficient conditions on a reduced form f so that its minimum is less than 3. He phrased his condition in terms certain forbidden patterns of a bi-infinite word associated to f. From the Markoff condition one gets a large class of binary words that we call Markoff words. In this talk, we will see that the Markoff words are palindromes, and indeed are the important central words from the theory of Sturmian sequences. (This is joint work with Amy Glen and Franco Saliola.) [Comment]
[Day] September 12 [Time] 3:00–3:50 [Name] Chris Hillar [University] (TAMU) [Title] Polynomial recurrences and cyclic resultants [Abstract] Given a monic univariate polynomial f(x) of degree d, the m-th cyclic resultant of f is r_m = Resultant(f,x^m-1). These sequences of numbers give coarse information about certain dynamcal systems (toral endomorphims), but they also arise in many other contexts such as number theory, Lagrangian mechanics, varieties of amoebas, and quantum computation. It is an important open question how many such resultants (the coarse data) determine the given polynomial f (the dynamical system). Sturmfels and Zworski have conjectured that d+1 resultants suffice, however, the best current bound is 2^{d+1}. We discuss the most recent attacks on this problem. In the process we will explain how binomial factorizations in group algebras, polynomial recurrences, and Toepletz determinant factorizations play surprising special roles. (partly joint with Lionel Levine, MIT). [Comment] Hurricane Ike'd
[Day] September 19 [Time] 3:00–3:50 [Name] Stefan Forcey [University] (Tennessee State University) [Title] Positrons, polytopes, and antipodes. [Abstract] The process of renormalization lets us use Feynman diagrams to calculate the precise strengths of many forces of nature, despite the suspicious subtraction of infinities. Kreimer and Connes found a way to mathematically model the process using the antipode of a graded Hopf algebra. Their algebra turns out to be fundamental in mathematics as well as physics, as part of a larger family of algebras based on combinatorial structure. I'll show a new pictorial way of looking at that family and its inner relations. Then come introductions of new family members and lots of questions about how they fit in. One question is about the significance of the fact that these algebras, new and old, all come from the vertices of convex polytopes! If time permits we can go on to discuss multicolored versions and potential modules over the algebras already introduced. [Comment]
[Day] September 26 [Time] 3:00–3:50 [Name] Khaled Al-Takhman [University] (Birzeit University & TAMU) [Title] Corings and Comodules [Abstract] Basic properties of corings and comodules over them will be given. The comatrix coring induced from a finitely generated projective module will be invistigated. Every comatrix coring induces a pair of functors between certain comodule categories, properties of these functors, like being Frobenius, quasi-Frobenius will be established. At the end of the talk, some work in progress will be presented. [Comment]
[Day] October 3 [Time] 3:00–3:50 [Name] Dmitri Nikshych [University] (University of New Hampshire) [Title] Weakly group-theoretical and solvable fusion categories [Abstract] A fusion category (i.e., a finite semisimple tensor category) is called weakly group-theoretical (respectively, solvable) if it can be obtained by a certain iterative procedure using finite groups (respectively, cyclic groups). All known examples of semisimple (quasi-) Hopf algebras have weakly group-theoretical representation categories. We prove a categorical analogue of Burnside's theorem for finite groups, saying that a fusion category of dimension pnqm, where p and q are primes, is solvable. We also establish a Frobenius property of a weakly group-theoretical fusion category, i.e., that dimensions of simple objects of such a category C divide the dimension of C. We apply these results to classification of semisimple Hopf algebras of small dimension. (This is a joint work with P. Etingof and V. Ostrik.) [Comment]
[Day] October 10 [Time] 3:00–3:50 [Name] Alison Marr [University] (Southwestern University, Texas) [Title] A Magical Tour of Various Magic-Type Labelings [Abstract] This talk will discuss three new types of magic labelings: bimagic labelings of graphs, magic labelings of digraphs, and magic vertex labelings of digraphs. We will examine the definitions, some examples, and some properties of each type of labeling. In addition, open questions will be posed. [Comment]
[Day] October 17 [Time] 3:00–3:50 [Name] Deepak Naidu [University] (TAMU) [Title] Fusion subcategories of Rep(Dω(G)) [Abstract] We will begin by briefly discussing the notions of fusion category, braided category, and modular category. Then we will consider the modular category arising from the twisted Drinfeld double of a finite group. We will describe all fusion subcategories of this category. As a consequence, we obtain a description of all group-theoretical braided fusion categories. This talk is based on a joint work with D. Nikshych and S. Witherspoon. [Comment]
[Day] October 24 [Time] 3:00–3:50 [Name] Charles Doran [University] (University of Alberta, Canada) [Title] Algebraic Cycles, Regulator Periods, and Local Mirror Symmetry [Abstract] A number of constructions in the string theory literature in recent years are, in fact, closely related to quantities of interest to the algebraic-cycles community. A careful reinterpretation of these physical computations in the proper mathematical context clarifies the connection between physical predictions — involving for example the mirror map, prepotential, and asymptotic growth of Gromov-Witten invariants — and deep mathematical structures and conjectures. This is joint work with Matt Kerr. [Comment]
[Day] October 31 [Time] 3:00–3:50 [Name] Michael Anshelevich [University] (TAMU) [Title] Measures, orthogonal polynomials, and continued fractions [Abstract] This general-purpose talk will be more in the style of a working seminar, in the sense that it will contain a number of (elementary) concrete calculations and proofs. Most of the results I will explain were well-known 50 years ago, but may have fallen into utterly undeserved obscurity. Specifically, I will describe, and prove, the relationship between measures, orthogonal polynomials, and continued fractions. Some names associated with these topics are Stieltjes, Szego, Simon, and other people not beginning with an S. [Comment]
[Day] November 7 [Time] 3:00–3:50 [Name] Sarah Witherspoon [University] (TAMU) [Title] Quantum Symmetric Algebras [Abstract] A symmetric algebra on a vector space V is just a polynomial ring (in variables corresponding to a basis of V). This is a commutative algebra, and may be defined by generators in V, and relations yx = xy for all x and y in V. In this talk, we will put this example in a much bigger context, generalizing these relations to those coming from a braiding on tensor powers of V. The resulting algebras are called quantum symmetric algebras, or Nichols algebras, or Nichols-Woronowicz algebras. They have appeared in many places, including quantum groups, the cohomology of flag manifolds, and the recent classification, by Andruskiewitsch and Schneider, of finite dimensional pointed Hopf algebras. We will define quantum symmetric algebras and survey some of this history. Then we will present a series of Hopf algebras in positive characteristic, found in joint work with Aaron Lauve, via a new combinatorial approach to quantum symmetric algebras. [Comment]
[Day] November 14 [Time] 3:00–3:50 [Name] Dimitrije Kostic [University] [Title] The Combinatorics of Integer Points in a Certain Polytope [Abstract] Let (x1,...,xn) have nonnegative integer coordinates (and x1>0). Stanley and Pitman studied the set of integer points (y1,...,yn) satisfying y1+...+yk ≤ x1+...+xk and found a precise formula for this number in terms of the Ehrhart polynomial. We will explore further questions related to this set of points, particularly the q-distributions of partial sums of the yi and a surprising formula I stumbled across accidentally. [Comment]
[Day] November 19 [Time] 2:00–2:55 [Name] Mitja Mastnak [University] (Saint Mary's University) [Title] Hopf algebraic approach to the combinatorics of free probability [Abstract] In the talk I will try to explain how combinatorial Hopf algebras can be used to study joint distributions of k-tuples in a noncommutative probability space. In recent joint work with A. Nica we have constructed a Hopf algebra whose multiplication of characters corresponds to free multiplicative convolution of joint distributions. I will highlight the case k=1 when the combinatorial Hopf algebra in question is the well known Hopf algebra of symmetric functions. In this case several notions in free probability, such as the S-transform, its reciprocal 1/S, and its logarithm log S, relate in a natural sense to the sequences of complete, elementary and power sum symmetric functions. [Comment] Wed; 2:00; Milner 216
[Day] November 21 [Time] 3:00–3:50 [Name] Scott Chapman [University] (Sam Houston State University) [Title] An Introduction to the Theory of Non-unique Factorizations [Abstract] The study of unique factorization domains (or UFDs) played an important role in the development of Algebraic Number Theory and Commutative Algebra. In this talk, we consider integral domains and monoids which do not satisfy the unique factorization criteria. Over the past 30 years, the algebraic and combinatorial aspects of non-unique factorizations has been widely studied. After reviewing many of the basic results in this area, I will focus on several classes of monoids (numerical monoids, congruence monoids and block monoids) which have appeared frequently in the recent mathematical literature. [Comment]
*[Day] December 5 [Time] 3:00–3:50 [name] Federico Ardila [University] (San Francisco State University) [Title] Combinatorics and geometry of power ideals [Abstract] We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box splines. We pay special attention to a family of power ideals that arises naturally from a hyperplane arrangement A. We prove that their Hilbert series are determined by the combinatorics of A, and can be computed from its Tutte polynomial. We also obtain formulas for the Hilbert series of the resulting fat point ideals and zonotopal Cox rings. Our work unifies and generalizes results on power ideals obtained by Dahmen-Micchelli, de Boor-Ron, Holtz-Ron, Postnikov-Shapiro-Shapiro, and Sturmfels-Xu, among others. It also settles a conjecture of Holtz-Ron on the spline interpolation of functions on the lattice points of a zonotope. This is joint work with Alex Postnikov from MIT. [Comment]