Spring 2008, Fridays, Milner 317, 3:00-3:50 PM

[Day] January 18 [Time] 3:00-4:00 [Name] <a href="http://www.math.tamu.edu/~maguiar/" class="not-link"  >Marcelo Aguiar</a> [University] TAMU [Title] Bimonoids in species [Abstract] This talk is addressed to graduate students interested in algebra and combinatorics.<br><br> The category of species furnishes the right framework for the study of algebraic structures associated to combinatorial objects. One of the advantages of the notion of species is its simplicity: roughly, a species is a family of vector spaces, one space for each finite set. We will define species and concentrate on the notion of "bimonoid" in the category of species, illustrating it with several examples of a combinatorial nature. <br><br> Little background will be needed to follow the talk. [Comment]

[Day] January 25 [Time] 3:00-4:00 [Name] <a href="http://www.math.tamu.edu/~chillar/" class="not-link"  >Christopher Hillar</a> [University] TAMU [Title] A finiteness question of Sturmfels [Abstract] In chemistry and algebraic statistics, a motivating problem is to determine the algebraic relations between (a possibly infinite number of) experimental measurements. In this regard, Sturmfels has asked whether, up to symmetry, there are finitely many of them that generate the others.  We discuss the mathematics of this problem and an approach by Aschenbrenner and myself for solving it.  It should be accessible to a wide audience and involves a nice interplay between combinatorics and algebra.  Several open questions will also be discussed. [Comment]

[Day] February 1 [Time] 3:00-4:00 [Name] <a href="http://www.math.tamu.edu/~rowell/" class="not-link"  >Eric Rowell</a> [University] TAMU [Title] Unitarizability of pre-modular categories [Abstract] I will discuss the problem of unitarizability of categories associated with quantum groups at roots of unity.  In particular, I will describe the recently completed solution to this problem in the non-simply-laced cases.   This work is motivated by the application of modular categories to quantum systems of interest in topological quantum computation. I will speak on this application in the Math. Phys. seminar next week. [Comment] 

[Day] February 4 [Time] 3:00-4:00 [Name] <a href="http://home.gwu.edu/~wschmitt/" class="not-link"  >Bill Schmitt</a> [University] George Washington [Title] The Whitney algebra of a matroid [Abstract] The Whitney algebra is the analog for a matroid of the exterior algebra of a vector space. In this talk we describe the construction of the Whitney algebra and investigate its basic properties. In particular, it is a universal object with respect to vector representations of a matroid, and it is equipped with an operation that corresponds, under any representation, to Grassman's regressive product, simultaneously giving the join and meet of any pair of flats.  This is joint work with Henry Crapo. [Comment] (Special Time on Monday!)

[Day] February 8 [Time] 3:00-4:00 [Name] Geir Helleloid [University] UT Austin [Title] Automorphism Groups of Finite p-Groups [Abstract] It is reasonably difficult to find examples of finite p-groups whose automorphism group is also a p-group.  Yet I'll present a theorem which, in some asymptotic sense, shows that the automorphism group of a finite p-group is almost always a p-group.  The proof relies on a variety of combinatorial topics: counting subgroups of a p-group; analyzing the lower p-series of a free group via its connection to the free Lie algebra; and counting submodules of a module using Hall polynomials.  This is joint work with Ursula Martin. [Comment]

[Day] February 15 [Time]  [Name]  [University]  [Title]  [Abstract]  [Comment] --No Talk (Hiring Colloquium)--

[Day] February 22 [Time] 3:00-4:00 [Name] Deepak Naidu [University] UNH [Title] Lagrangian subcategories of twisted quantum doubles of finite groups [Abstract] We will begin the talk by recalling the notion of a fusion category and related concepts. We will then present a classification of Lagrangian subcategories of the representation category of a twisted quantum double of a finite group. In view of results of arxiv:0704.0195v2 [math.QA] this gives a complete description of all braided tensor equivalent pairs of twisted quantum doubles of finite groups. We also establish a canonical bijection between Lagrangian subcategories of the representation category of a twisted quantum double of a finite group G and module categories over the category of twisted G-graded vector spaces such that the dual fusion category is pointed. As a consequence, we obtain that two group-theoretical fusion categories are weakly Morita equivalent if and only if their centers are equivalent as braided tensor categories. <br><br>This talk is based on a joint work with Dmitri Nikshych (Preprint: arXiv:0705.0665v1 [math.QA]). [Comment]

[Day] February 25 [Time] 3:00-4:00 [Name] Charles Conley [University] U. North Texas [Title] Contact vector fields on the superline [Abstract] I will discuss several problems arising from the action of the Lie superalgebra K of contact vector fields on modules of differential operators over the superline.  K contains the conformal subalgebra, a copy of osp(1|2).  The focus will be on the action of K with respect to the decompositions of the differential operator modules under the action of the conformal subalgebra. There are applications to 1-cohomology, equivalences between subquotients of differential operator modules, and bounded modules of finite length. [Comment] (Room Milner 216 on Monday!)

[Day] February 29 [Time] 3:00-4:00 [Name] Muriel Livernet [University] MIT and Universite Paris 13 [Title] Posets, Groups and Hopf algebras associated to a set-operad [Abstract] In this talk we will review a  result of Bruno Vallette linking the notion of Koszul duality of operads and Cohen-MacCauley posets. We'll present in this context a joint work with F. Chapoton, where we compare two groups, one built directly from operads, and another one associated to the incidence Hopf algebra of a family of posets. This leads us to a new link between the Hopf algebra of Connes and Kreimer in renormalisation theory and operads built on rooted trees. [Comment] 

[Day] March 7 [Time] 3:00-4:00 [Name] Rob Ellis [University] IIT [Title] Recent results in liar games [Abstract] We consider a 2-person perfect information "liar" game, often called a Renyi-Ulam game.  The basic game is that of "twenty questions" played between questioner Paul and responder Carole.  Paul searches for a distinguished element x in a search space [n] by asking Yes-No questions of the form "is x in A", where A is a subset of n.  Carole responds "Yes" or "No", lying in up to some bounded number of responses.  If Paul's questions are all offline, this is equivalent to block coding, but we allow adaptivity when Paul may choose each question based on Carole's previous answers. We survey several results and techniques for the adaptive game, including the general case in which Carole's pattern of lying is constrained to come from a bounded set.  The talk will include joint work with Catherine Yan, Vadim Ponomarenko, and Kathryn Nyman. [Comment]

[Day] March 14 [Time]  [Name]  [University]  [Title]  [Abstract]  [Comment] --Spring Break--

[Day] March 17 [Time] 3:00-4:00 [Name] Bruce Reznick [University] UIUC [Title] On Hilbert's construction of positive polynomials which are not a sum of squares [Abstract] In 1888, Hilbert described how to find real polynomials which take only non-negative values, but are not a sum of squares of polynomials. His construction was so restrictive that no examples appeared until the 1960s, under a variation of his original plan. We revisit and generalize Hilbert's original construction and show how the underlying mechanism can be simplified and generalized. [Comment] (Joint with Algebraic Geometry on Monday!)

[Day] March 21 [Time] 3:00-4:00 [Name] Rosena Du [University] East China Normal University [Title] Counting Labelled Trees with Given Indegree Sequence [Abstract] For a labelled tree on the vertex set $[n]:=\{1,2,\ldots, n\}$, define the direction of each edge $ij$ to be $i\rightarrow j$ if $i &lt;   j$. The indegree sequence of $T$ can be considered as a partition $\lambda \vdash n-1$. The enumeration of trees with a given indegree sequence arises in counting secant planes of curves in projective spaces. Recently Ethan Cotterill conjectured a formula for the number of trees on $[n]$ with indegree sequence corresponding to a partition $\lambda$. In this talk I will give two proofs of Cotterill's conjecture: one is "semi-combinatorial" based on induction, the other is a bijective proof. (This is joint work with Jingbin Yin.) [Comment]

[Day] March 28 [Time] 3:00-4:00 [Name] Gregory Berkolaiko [University] TAMU [Title] The number of inequivalent minimal factorizations of an n-cycle [Abstract] We will discuss an elementary proof of the formula for the number of factorizations of the cycle (1,2,3,..n) into a product of transpositions. The proof uses a bijection between the factorizations and ternary trees.<br><br> The bijection is easy to visualize, which allows for an easy generalization: counting factorizations into a_k cycles of length k, for a suitable vector (a_2, a_3, ...).  Some harder generalizations will also be mentioned.<br><br>The problem arose in an applied context of semiclassical analysis of quantum transport through a chaotic cavity.  Time permitting, I will give a short elementary explanation of the physics involved. [Comment]

[Day] April 4 [Time] 3:00-4:00 [Name] Xingxing Yu [University] Georgia Institute of Technology [Title] On judicious partitions of graphs [Abstract] Judicious partition problems ask for partitions of the vertex set of a graphs so that several quantities are optimized simultaneously. I will discuss several judicious partition problems of Bollobas and  Scott, and present our recent results on these problems. This is joint work with Baogang Xu. [Comment]

[Day] April 9 [Time] 4:00-5:00 [Name] Mitja Mastnak [University] University of Waterloo [Title] On (combinatorial) Hopf algebras extensions [Abstract] I will describe an extension theory for combinatorial Hopf algebras. As an application I will explicitly describe a nice class of examples obtained as extensions of the Hopf algebra of symmetric functions  by itself. [Comment] (Special Time 4pm Milner 317 on Wednesday!)

[Day] April 11 [Time] 3:00-4:00 [Name] Vladimir Retakh [University] Rutgers [Title] Algebras Associated to Directed Acyclic Graphs [Abstract] We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices and edges. Each finite directed acyclic graph admits a structure of a  generalized layered graph. We construct linear bases in such algebras, compute their Hilbert series, and discuss their Koszulity and other properties. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings. [Comment]

[Day] April 18 [Time] 2:00-3:00 [Name] Hong-Jian Lai [University] West Virginia University [Title] Group Connectivity of Graphs [Abstract]  In his attempt to attack the (then) 4-color-conjecture, Tutte in 1950s discovered the nowhere zero flows, and founded the theory on nowhere zero flows of graphs. He proved that a plane map is 4-face-colorable if and only if the corresponding graph $G$ has a nowhere zero 4-flow. Tutte's fascinating conjectures on nowhere zero flows have drawn the attention of many researchers over the decades, and remain open as of today. Group connectivity of graphs is the nonhomogeneous version of the nowhere zero flows, and was first studied by Jaeger, Linial, Payan and Tarsi in 1992 ([J. Combinatorial Theory, Ser. B 56 (1992) 165-182]). In this talk, we shall survey the latest results and developments on the topic. [Comment] (Special Time 2pm)

*[Day] April 25 [Time] 3:00-4:00 [Name] Eric Rowell [University] TAMU [Title] Wild conjectures in topological quantum computation [Abstract] I will discuss some conjectures I have no business making that relate classical and quantum computational complexity to computational power of quantum computers and representations of the braid group. [Comment]