## Geneseo Mathematics Colloquium Schedule |

Friday, January 29, 3:00-3:50pm

Newton 204

Knots are everywhere - in your shoelaces, neckties, and usually your earphones. The field of knot theory is devoted to the study of questions like: Are there infinitely many knots? When are two knots the same? How can we tell if one knot is more or less knotted than another? I will discuss the basics of knot theory along with a family of natural functions on the set of all knots. We will then investigate a recent conjecture that this family of functions gives the set of knots (modulo an equivalence relation) the structure of a fractal.

Monday, February 22, 4:00-4:50pm

Newton 201

Graph theory is an increasingly useful tool to model and analyze coupled dynamic systems in the sciences and engineering. Examples include communication satellites, social interaction sites like Facebook and Twitter, biological processes such as yeast protein interactions and gene regulatory networks, and many others. In this talk, I will present how algebraic graph theory is being used to analyze the ability to control these networked coupled systems.

Tuesday, March 1, 4:00-4:50pm

Newton 214

Suppose that you walk into the Fraser Study Area and find that a strange machine has been installed. It asks you to input a matrix. So, you give it one, it shakes and rattles a bit, and then it gives you a new matrix. Curious, you inspect the device and find the following message:

"This machine is linear and it does not change eigenvalues."

Is this enough information to figure out how the machine works? Come and find out the answer! Along the way, we will discuss the history of such machines, the current ones being studied, and their use in quantum computing.

Thursday, March 3, 4:00-4:50pm

Newton 214

The change in both the nature and the content of the natural sciences beginning with Copernicus' consideration of a different underlying dynamic for the motions of the heavenly bodies to the culmination of the process in Newton's Principia has been called the "Scientific Revolution". In recent years the concept of the "Scientific Revolution" has come under attack by historians and philosophers of science (and others). The chief criticism is that the accepted story of the revolution is far too simple. A major component of the scientific revolution was the alteration in the way mathematics was used in science, particularly in Physics. Newton and Galileo were two of the principle figures in this revolution but the way they each used mathematics was very different and often at odds with the accepted story of the revolution. This talk will deal with just how each of these figures used mathematics in advancing a new Physics.

Newton 201

Boundary value problem is an important topic in differential equation theory and has been a focus of research for decades. In this talk, some basic concepts, approaches, and challenges in this area will be presented. Potential research projects for students, especially undergraduate students, will be discussed as well. This talk will be appropriate for undergraduate students and graduate students who have an interest in differential equations.

Wednesday, March 30, 3:30-4:20pm **Walter Gerych, SUNY Geneseo PRISM**### REU Talk

Wednesday, April 6, 3:30-4:20pm **Stephanie Allen, SUNY Geneseo PRISM**### REU Talk

Thursday, April 14, 3:30-4:20pm **Megan Brunner, SUNY Geneseo PRISM**### REU Talk

**Brendan Murphy, University of Rochester (****SUNY Geneseo, Class of 2010**)### The sum-product problem

South 328

My team, under the mentorship of Kansas State's Dr. LeCrone, worked to develop a method of approximating Mean Curvature Flow with obstacles in the plane using cellular automata. Mean curvature flow is a well understood type of geometric flow of surfaces. However, when obstacles are placed in such a way as to obstruct the flow of the surfaces the system is no longer easily understood analytically. To study flow with obstacles, we developed a discrete model of such flow using cellular automata. Cellular automata are grids of discrete cells in which each cell can take on one of a finite number of states. The cells then transition to different states in "generations" over discrete time steps. Our goal was to develop updating rules in such a way that the cells would model mean curvature flow.

South 328

Identifying those groups in economic need is a process that can be approached in several different ways. For the county of Arlington, Virginia – one of the wealthiest counties in the US – I undertook two analyses to accomplish this goal. In the first, I looked at household-level Census data and ran a series of confidence intervals with the intent to find which household groups were overrepresented (and underrepresented) in the bottom portion of the income distribution. Group descriptors included racial composition of households, household language, veteran status, etc. In the second analysis, I examined Census tract-level data and used factor analysis (a statistical method that finds patterns in data) to rank the tracks of Arlington County by economic prosperity. I identified the tracts at the bottom of this ranking and sought to discover which groups of individuals (using the same characteristics as in the first analysis) were overrepresented in these tracts. Therefore, one analysis approaches the important question of ‘who is in economic need?’ from a micro point of view, while the other analysis approaches this question from a more macro point of view.

Thursday, April 7, 4:00-4:50pm **Amanda Tucker, University of Rochester**### Multiple Zeta Values

Newton 203

What's the difference between 3, 1/3, the sum from n=1 to infinity of 1/n^3, and pi? We will start with a discussion of what it means for a number to be integral, rational, algebraic, or transcendental. We will talk about the history of the multiple zeta values (certain real numbers) and some open problems about them. You will leave this talk knowing more about the real numbers than you thought was possible, but also with more questions about the real numbers than you thought was possible! Some familiarity with vector spaces and geometric series will help, but by no means is necessary for getting something out of this talk. A deck of cards will be involved.

South 328

This research is a reflection of the need for education to prepare students for either careers or college. As the world is becoming increasingly more globalized, it is more necessary for high school graduates to have basic understandings of other cultures and subcultures. We worked with three middle schools in Connecticut that are implementing interdisciplinary units that have components of multicultural knowledge, building "intercultural competence," or ICC, in their students. I worked to develop assessment tools for these teachers and school districts. The tools will measure specific qualities we determined to show "high ICC levels" in middle school students. There is a strong theoretical background influencing these decisions and the creation of the tools, which have multiple uses within classroom environments to gain a comprehensive understanding of the students' skills.

Megan will also introduce the research she did the previous summer at Kansas State University that focused on graph theory and combinatorial optimization.

Wednesday, April 27, 2:45-3:35pm

Newton 204

Erdos and Szemeredi conjectured that for a finite sequence of integers *a*_{1} < a_{2} < ... < a_{n}, the number of integers of the form *a*_{i}+a_{j}, *a*_{i}a_{j}, 1 ≤*n*^{2-ε }for any ε > 0. This is known as the "sum-product problem".** **** **

Despite three decades of effort, this conjecture is far from being resolved. In fact, work on the sum-product problem and related problems has grown into a field known as "arithmetic combinatorics", which combines elements of combinatorics, number theory, and algebra.

We will place the sum-product conjecture in context and examine the evidence that led Erdos and Szemeredi to pose their conjecture, as well as survey partial progress, variations, and generalizations. Along the way we will see a few gems, including Erdos' "multiplication table theorem", which answers the question "How many distinct entries are in an*n* by *n* multiplication table?". If time permits, we will discuss applications of sum-product type results to theoretical computer science.

This talk will be accessible to anyone who knows algebra and coordinate geometry. However, knowledge of sets, limits, and experience with proofs may be helpful.

` `

i ≤` `

j ≤` `

n is more than Despite three decades of effort, this conjecture is far from being resolved. In fact, work on the sum-product problem and related problems has grown into a field known as "arithmetic combinatorics", which combines elements of combinatorics, number theory, and algebra.

We will place the sum-product conjecture in context and examine the evidence that led Erdos and Szemeredi to pose their conjecture, as well as survey partial progress, variations, and generalizations. Along the way we will see a few gems, including Erdos' "multiplication table theorem", which answers the question "How many distinct entries are in an

This talk will be accessible to anyone who knows algebra and coordinate geometry. However, knowledge of sets, limits, and experience with proofs may be helpful.