College of Charleston

Charleston's Regional Undergraduate Math Conference

October 18, 2008

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Abstracts

Jeffrey Arredondo (University of South Carolina): Mathematics of the Human Brain

Our project explores the application of game theory and graph theory to neuron communication inside the human brain. I will discuss the modeling and proposed payoff functions in the directed graphs of neural networks. Since our project is still in progress, most of the discussion will revolve around current methods and desired outcomes.

Hau Chan (College of Charleston): Beautifully Nested Balanced Incomplete Block Designs

Beautifully Nested Balanced Incomplete Block Designs, BNBIBD(v,k,λ,k1,λ1), are defined and it is proven that the necessary conditions are sufficient for the existence of BNBIBDs with block size k=3 and k1=2. We also prove that the necessary conditions are sufficient for the existence of BNBIBDs with k=4 and k1=2 except possibly for eleven exceptions. Several infinite families of BNBIBDs with block size 4 are constructed and non-existence of BNBIBD(7,4,2,3,1) is given.

Sjuvon Chung (Rutgers University): Symmetry of the category of modules for a vertex operator algebra, and Fusion Rules

Recently, in a joint project with Christopher Sadowski, William Cook, and Yi-Zhi Huang, we constructed and studied certain symmetries on the category of vertex operator algebra modules. Using these symmetries, we propose a method to estimate fusion rules.

Taylor Hamrick (College of Charleston): Zipf's Law and Avoidance of Excessive Synonymy

Zipf’s law in linguistics states that if words in a language are ranked by their frequency of use, the rank will be inversely proportional to the frequency, usually modeled by fk = k-B. In a recent paper by D. Manin, it is suggested that the ‘extent’ of a words meaning is related to its frequency of use. In a natural language, synonymy will be avoided by ‘expanding’ and ‘contracting’ meanings, which should lead to a Zipfian distribution. We show that an idealized model creates such a Zipfian distribution. We also test multiple methods for modeling this phenomenon in language and test the results in other languages.

Hudson Harper (University of South Carolina): Modeling Plant Phyllotaxis via Primordia Growth

This talk will focus on the appearance of spirals with frequencies related to the Fibonacci sequence in plants. A means for categorizing plants, phyllotaxis, is directly related to counting these spirals. One way to study and count these spirals is to look at plant primordia. In this talk, I will define plant phyllotaxis, the relation between phyllotaxis and the Fibonacci sequence, and the importance of plant primordia growth in understanding phyllotaxis. Classical models for regular primordia growth using dynamical systems and bifurcation theory will be discussed. Primarily, more recent work on modeling irregular primordia growth including computer simulation of growth will be discussed. Last of all, since this is ongoing research, I will mention current efforts and expected results.

Michael Lane (Charleston Southern University): Two Proofs to the Four Color Theorem

A brief history and overview of the Four Color Theorem along with outlines of the Haken-Appel Proof (1977) and the more recent Robertson/Sanders/Seymour/Thomas Proof (1996)

Patrick Moran (College of Charleston): Finding Maximum Box-Free Subsets of Grids

A box-free subset of a 2-dimensional grid is a subset of that grid in which no four points are the four corners of a rectangle. For an mxn grid, how big can such a subset be? Our approach to the problem is discussed, including our contribution to the problem - a new upper bound that is asymptotically optimal. This upper bound reveals a connection to finite projective planes, which will be explained and explored in future research.

Christopher Sadowski (Rutgers University): Standard Affine Lie Algebra Modules, Vertex Operator Algebras, and the Function Δ(H,x)

There are certain modules for affine Lie algebras which happen to be modules for a vertex operator algebra also. We examined how the module structure changes when our vertex operator is changed by a certain function called delta, and when our modules are what are called "standard" or "highest weight integrable" irreducible modules for an affine Lie algebra. It is well know that elements of the coroot lattice, when applied with this delta, cause delta to give us a module isomorphic to the module with which we started. We examined the more general case, in which we used elements of the coweight lattice (an important set containing the coroot lattice as a subset) with delta and determined what module structure delta had given us.

Tony Zamberlan (Augusta State University): The Theta Complex of Certain Families of Hypergraphs

This talk will focus on the results stemming from the REU I participated in this summer at the University of Tennessee in Knoxville. When necessary, requisite definitions and examples are provided for those whom this subject is completely foreign. Ideas from graph theory, combinatorics, algebra, and topology are discussed, as well as an interesting connection to a famous problem in computer science.