Apr 15, 2011 - Michelle Delcourt. College of Science ... About Me. What is Mathematical ... 2010 Georgia Tech REU (Graph Theory). 2009 LSU REU (Knot ...
Background Sum-Sets on Graphs Benefits to Research
Sum-Sets on Graphs Michelle Delcourt College of Science Student Advisory Board Georgia Institute of Technology
April 15, 2011
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
About Me What is Mathematical Research? Research in Mathematics How I Found My Advisor
Background
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
About Me What is Mathematical Research? Research in Mathematics How I Found My Advisor
About Me Fourth Year Discrete Math Major Will Attend the PhD Program at University of Illinois, Urbana-Champaign in Fall 2011 NSF Fellowship Winner Outstanding Undergraduate Researcher, College of Science President, Pi Mu Epsilon (Mathematical Honors Society) Participant in Three Research Experience for Undergraduates Programs: 2010 Georgia Tech REU (Graph Theory) 2009 LSU REU (Knot Theory) 2008 Clemson REU (Combinatorics) Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
About Me What is Mathematical Research? Research in Mathematics How I Found My Advisor
What is Mathematical Research? A Variety of Different Methods: Laboratory Research Computer Assisted Research (Algorithms, Parallel Computing, etc.) Theoretical Research
A Variety of Ways to Conduct Research: As an Individual As a Team As a Community (Polymath)
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
About Me What is Mathematical Research? Research in Mathematics How I Found My Advisor
Research in Mathematics Research in Theoretical Math Seems Different from Research in Other Disciplines: Read Papers and Understand Previous Results Create Conjectures, Lemmata, and Theorems from Scratch Find Applications for Existing Results
Mathematicians Often Assist Others with Problems in Related Fields: Physics Computer Science Biology Engineering Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
About Me What is Mathematical Research? Research in Mathematics How I Found My Advisor
How I Found My Advisor My advisor is Professor Xingxing Yu in the School of Math In Fall 2010 he taught MATH 4022 (Graph Theory) We solved a problem about a game played on 2-connected graphs posed in the American Mathematical Monthly He was my advisor for my senior projects and the 2011 Georgia Tech REU
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Formal Definitions Example of a Graph Sum-Set Sum-Set Example Results Applications
Sum-Sets on Graphs
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Formal Definitions Example of a Graph Sum-Set Sum-Set Example Results Applications
Formal Definitions Definition A graph is an ordered pair of disjoint sets G = (V , E) such that E ⊆ [V ]2 of unordered pairs of V . The set V is the set of vertices. The set E is the set of edges.
Definition An edge {x, y } ∈ E is said to join x and y ∈ V and is denoted xy ; x and y are said to be endpoints of edge xy . Note: xy and yx denote the same edge.
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Formal Definitions Example of a Graph Sum-Set Sum-Set Example Results Applications
Example of a Graph Below is a famous graph:
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Formal Definitions Example of a Graph Sum-Set Sum-Set Example Results Applications
Sum-set
Definition Let G = (V , E) be a graph. Given that S = (su )u∈V is an injective map of V into some ring R, we define the sum-set of G in the following additive combinatorial way: G
S + S= {su + sv : uv ∈ E}
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Formal Definitions Example of a Graph Sum-Set Sum-Set Example Results Applications
Sum-Set Example Here is an example:
y
y+z
x+y x
z x+y
2z
x+y-z 2z-x Michelle Delcourt
y+z Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Formal Definitions Example of a Graph Sum-Set Sum-Set Example Results Applications
Results
We were able to find the following sum-set lower bound. Let A be the adjacency matrix for G. For any odd integer k ≥ 3: 1/k G | S + S | ≥ Trace Ak .
This is an improvement over the best known lower bound.
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Formal Definitions Example of a Graph Sum-Set Sum-Set Example Results Applications
Applications Information is being transferred over a limited spectrum Optimization is necessary for global allocation Locally signals must be distinct to avoid signal conflict A maximum sum-set labeling yields an optimal solution Configurations for radio transmissions, Wi-Fi networks, etc.
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Conferences Publications
Benefits to Research
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Conferences Publications
Conferences and Seminars Georgia Tech REU Research Poster, Undergrad Research Spring Symposium (GT, April 2011) Poster, Joint Mathematics Meetings (New Orleans, January 2011)
LSU REU Research Poster, Undergrad Research Spring Symposium (GT, April 2010) Poster, Joint Mathematics Meetings (San Francisco, January 2010) Talk, Undergrad Research Seminar (GT, December 2009)
Clemson REU Research Talk, Mini-Conference on Additive Combinatorics (GT, June 2010) Poster, Parents Weekend (GT, September 2009) Poster, Joint Mathematics Meetings (Washington D.C., January 2009) Talk, Senior Seminar Series (GT, November 2008) Talk, Mini-Conference on Discrete Math and Algorithms (Clemson University, October 2008) Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Conferences Publications
Publications N. Calkin, J. Davis, M. Delcourt, Z. Engberg, J. Jacob, and K. James. Discrete Bernoulli Convolutions, Proceedings of the American Mathematical Society, 139 (2011), no. 5, 1579-1584. Available at: http://www.ams.org/journals/proc/2011-139-05/ S0002-9939-2010-10633-0/home.html
Michelle Delcourt
Sum-Sets on Graphs
Background Sum-Sets on Graphs Benefits to Research
Conferences Publications
Thank you for listening!
Michelle Delcourt
Sum-Sets on Graphs
