Date: | Jun 13 (Monday), 2011 |
Time: | 11am - 5:30pm ( Dinner from 6pm ) |
Location: | College of Natural Sciences, Building 1, Room 313 |
Schedule | ||
11:00 - 11:50 |
Yoshio Sano National Institute of Informatics, Japan | On the configuration algebra associated with colored graphs (Part 2 of 2) |
12:00 | Lunch | |
1:30 -2:20 |
Lou Shapiro SungKyunKwan University / Howard University | The Riordan group and generating functions |
2:30 -3:20 |
Jong Yoon Hyun Ehwa University | MacWilliams duality and Gleason-type theorem on self-dual bent functions |
3:40 - 4:30 |
Sangjune Lee Emory University, USA | The maximum size of a Sidon set contained in a sparse random set of integers |
4:40 - 5:30 |
Taoyang Wu National University of Singapore | On Dress's optimal realization conjectures |
6:00 | Dinner |
Abstracts
On the configuration algebra associated with colored graphs, Part II
Let be a finite color set and let be a nonnegative integer. A -configuration is an isomorphism class of a finite -edge-colored graph such that the valency of each vertex is at most . Let ConfConf denote the set of all -configurations. Let be a commutative associative ring with unit. The free -module Conf generated by Conf naturally become an algebra by using a semigroup structure on Conf defined by , where denotes the disjoint union. For , let be the ideal of Conf generated by Conf. Then the completion Conf Conf is called the (completed) configuration algebra (over ). For configurations , and , the covering coefficient is defined to be the number of tuples such that , , and , where and means that is an induced subgraph of . By letting an extension of a map defined by for Conf be a coproduct and letting an extension of a map defined by if and if Conf be a counit, the configuration algebra Conf (and also Conf) becomes a coalgebra and furthermore a bialgebra. Moreover there exists an antipodal map ConfConf and so Conf is a Hopf algebra.
In this talk, we consider about group-like elements and Lie-like elements in the configuration Hopf algebra. An element in Conf is called a group-like element if and , and Conf is called a Lie-like element if and . For a configuration , , where , is called the growth function of . When , we define for Conf with , where denotes the coefficient of in . For a configuration , is called the logarithmic growth function of . It will be shown that is a group-like element and is a Lie-like element in the configuration Hopf algebra for any Conf.
The Riordan group and generating functions
MacWilliams duality and Gleason-type theorem on self-dual bent functions
The maximum size of a Sidon set contained in a sparse random set of integers
A set of integers is a Sidon set if all the sums , with and , , are distinct. In the 1940s, Chowla, Erdos and Turán showed that the maximum possible size of a Sidon set contained in is . We study Sidon sets contained in sparse random sets of integers, replacing the `dense environment' by a sparse, random subset of .
Let be a uniformly chosen, random -element subset of . Let . An abridged version of our results states as follows. Fix a constant and suppose . Then there is a constant for which almost surely. The function is a continuous, piecewise linear function of , not differentiable at two points: and ; between those two points, the function is constant. This is joint work with Yoshiharu Kohayakawa and Vojtech Rödl.
On Dress's optimal realization conjectures
Although Conjecture I and a stronger version of Conjecture II have been disproven, in this talk I will report some recent work on the positive results related to these two conjectures.
This is joint work with Sven Herrmann, Jack Koolen, Alice Lesser, and Vincent Moulton.