Skip to Content
Search
A A A






Discrete Math Day

Discrete Mathematics Day

at Misericordia University

September 26, 2009

(Part of the Discrete Mathematics Day in the North-East conference series)

 

 Talks: 

Speaker: Laura Anderson (Binghamton University)
Title: Topological representations of matroids.
Abstract: A matroid is a combinatorial analog to an arrangement of hyperplanes over an arbitrary field.  An oriented matroid is a matroid with extra structure that models an arrangement of hyperplanes specifically over the reals.  A central result in oriented matroid theory is the Topological Representation Theorem, which says that oriented matroids are, in a topological sense, a very good model for real hyperplane arrangements.  Specifically, any oriented matroid can be represented by an arrangement of topological equators on a sphere.
 
There is no reason to expect a similar result for general matroids. For instance, if a matroid M arises from an arrangement of hyperplanes over some finite field  F, why should there be anything like an arrangement of equators on the sphere representing M? However, several years ago Ed Swartz proved a truly weird "Topological Representation Theorem for Matroids", representing any matroid by an arrangement of homotopy spheres. His result was both surprising and mysterious, in that his construction is not explicit and has topological steps that are hard to reconcile with matroid intuition. I will describe a new construction of representation of matroids by homotopy spheres which is (almost) canonical, completely explicit, and indeed rather simple.

Speaker: Steven Dougherty (University of Scranton)
Title: The wonderful world of self-dual codes
Abstract: We will show the underlying theory of self-dual codes over rings and explore their connection to number theory and design theory.
 
 
Speaker: Daniel Slilaty (Wright University)
Title: Signed-graphic matroids
Abstract: A signed graph is a pair (G,σ) in which σ is a labeling of the edges of G with elements of the multiplicative group {+1,-1}.  A cycle (i.e., a simple closed path) in G is called positive if the product of labels on its eges is positive.  Otherwise the circle is called negagtive.  A subgraph of (G,σ) is called balanced when all of its circles are positive.  There is a matroid M(G,σ) whose elements are the edges of G.  Given a subset of edges of G, the matroid M(G,σ) is defined by the rank function r(X) = vX-bX in which vX is the number of vertices incident to edges in X and bX is the number of balanced components of the subgraph defined by X.
     Signed graphs and their matroids are appearing in many important areas of general matroid theory and linear programming.  In this talk we will begin with an overview of how graphs and signed graphs fit into linear programming and general matroid theory and then talk about some recent structural results about signed graphs and directions of current research.
 
Speaker: Catherine Stenson (Juniata College)
Title: Weighted Voting Systems and Slices of Cubes
Abstract: The states in the U.S. Electoral College, the parties in the Israeli Knesset, and the shareholders in a company are all players in weighted voting systems.  Each player has a certain number of votes and casts them all for or against a proposal.  The proposal passes if the total number of votes exceeds some quota.  A player’s influence is measured by the Banzhaf Power Index, which counts the ways in which a player’s vote can be critical to passing a proposal.  We interpret weighted voting systems as slices of cubes, and we explore the effect of changes in the quota on the Banzhaf Power Index.
 
 
Speaker: Lorenzo Traldi (Lafayette College)
Title: Generalized Dice Games and Voting
Abstract:  A simple dice game involves two players who roll identical dice; if the results are different then the winner is the player whose die has rolled the higher value. In the long run, the two players will tend to have equal numbers of wins and losses. In a generalized dice game, instead, the two players roll non-identical dice; these games were discovered by Efron and popularized by Gardner in Scientific American. In the long run the players need not tend to have equal numbers of wins and losses, even if their rolls have the same expected value. Generalized dice provide a mechanism for assessing the relative strengths of two candidates in a multi-candidate election: each candidate is associated to a die that lists the voters' rankings of that candidate.
    A dice family D(n,a,b,s) contains all the n-sided dice involving integers between a and b whose sum is s. The elements of D(n,a,b,s) represent all the ways n voters could give a candidate the average rating s/n. In accordance with Arrow's Theorem, these families are surprisingly complicated. Representing them in the natural way with (directed) graphs allows us to describe these complications using ideas from graph theory, including connectivity, cycle structure, and diameter. Some results have been obtained regarding these graphs, but there's a lot we don't know.
 
Registration will start at 9:00 with the first talk at 10:00.  The day will be in Insalaco Hall, rooms 216/217.
 
General Information:

As with other Discrete Math Day conferences, there is no registration fee, but pre-registering is strongly encouraged (especially if you want to be guaranteed lunch!).

This conference is one of a series of one-day conferences, titled "Discrete Mathematics Days in the Northeast" and styled after the earlier "CoNE" meetings.
 
Discrete Math Days are funded in part by the National Security Agency.

 
Financial Support:

We will be able to offer some financial support to participants (who are U.S. citizens or permanent residents).  Priority will be given to students and post-docs.  Students and post-docs traveling more than 50 miles (each way) may apply for a $50 travel stipend.  Participants traveling more than 100 miles can apply for a lodging reimbursement.  Please indicate when you register if you are applying for travel support.


Registration:

To register for the conference, email Steven Tedford at stedford@misericordia.edu by September 18, 2009. 

Local Hotels/Motels: (Price Information is as of 08/05/09)

East Mountain Inn (1-800-780-7234):  Without Breakfast: $79.  With Breakfast: $84.  Mention Misericordia University for this special rate.

Red Roof Inn: ( 570-829-6422): $53.95-$59.95.

Days Inn (570-826-0111): $69.95.

Comfort Inn and Suites: $99.95-$139.95.

EconoLodge: $69.95-$89.95.

Quality Inn: $89.95.


Directions to Campus:

From Northwestern Pennsylvania
Follow Route 6 east to Tunkhannock.Take Route 29 South; stay on Route 309 South when Route 29 veers off; turn right at first traffic light in Dallas, Center Hill Road (about 17 miles from Tunkhannock). At the second stop sign, turn right onto Lake Street. The University entrance arch in on right.

From Binghamton/Montrose area
Follow Route 29 South through Tunkhannock. Take Route 29 South; stay on Route 309 South when Route 29 veers off; turn right at first traffic light in Dallas, Center Hill Road (about 17 miles from Tunkhannock). At the second stop sign, turn right onto Lake Street. The University entrance arch in on right.

From Philadelphia
Take Route 476 (The PA Turnpike Northeast Extension) To exit 105,Wilkes-Barre. Turn left onto Route 115 North; stay on 115 for about 4 miles to Route 309 North. (Route 115 will turn into Route 309) Stay on 309 for 9 miles to Dallas. In Dallas, continue on Route 309 North. Where Route 309 veers toward the right (just past the Dallas Shopping Center), continue straight on Route 415 for 1/4 mile to the second light (stay in right lane). At this light, bear to the right onto Lake Street (beyond the Sunoco gas station). Follow Lake Street for 1/2 mile, past Center Hill Road. The University entrance arch is on your right.
 
From Route 84 West
Route 84 West merges into Route 380 West. Follow 380 for about 3 miles; stay in left lane. Follow signs for 81 South; exit ramp is on left. Follow directions from Route 81 below.

From Route 76 East
Take Route 81 North. Follow directions from Route 81 below.
 
From Route 80 West
Take exit 284 (Blakeslee) onto Route 115 North about 20 miles to Route 309 North. Stay on 309 North for 9 miles to Dallas. In Dallas, continue on Route 309 North. Where Route 309 veers toward the right (just past the Dallas Shopping Center), continue straight on Route 415 for 1/4 mile to the second light (stay in right lane). At this light, bear to the right onto Lake Street (beyond the Sunoco gas station). Follow Lake Street for 1/2 mile, past Center Hill Road. The University entrance arch is on your right.

From Route 81 South and North
Take exit 170B Wilkes-Barre. Exit ramp merges with Route 309 North. Stay on 309 For 9 miles into Dallas. In Dallas, continue on Route 309 North. Where Route 309 veers toward the right (just past the Dallas Shopping Center), continue straight on Route 415 for 1/4 mile to the second light (stay in right lane). At this light, bear to the right onto Lake Street (beyond the Sunoco gas station). Follow Lake Street for 1/2 mile, past Center Hill Road. The University entrance arch is on your right.

To Insalaco Hall

Drive under the Arch and up the driveway.  Bear right at fork and drive past the Science building on the right hand side.  Park in the parking lot next to the Science building.  Insalaco Hall is across the street from the parking lot, the second building in.  The meeting is on the second floor of the building.

Local Organizer:

Steven Tedford (Misericordia University)

 Steering Committee:

Seth Chaiken, University at Albany (SUNY)  
 
Karen Collins, Wesleyan University

Cristian Lenart, University at Albany (SUNY)

 
Rosa Orellana, Dartmouth College

Lauren Rose, Bard College
 

 

 

 

 Campus Map

Map of Dallas Area