Geneseo Mathematics Colloquium Schedule
Thursday, March 22, 2:30-3:30pm
Sedar Ngoma, SUNY Geneseo
An overview of inverse problems
When solving (or approximating solutions of) problems in science, engineering, mathematics, and many other fields, a process called a model is described in detail and an appropriate input called a cause is supplied. One is then required to find the unique output (or solution) called effect. This is known as direct or forward problems. In direct problems the media properties of a given model described by equations (for example, equation coefficients) are assumed to be known. However, media properties are often not readily observable. This lack of specification in the model leads to inverse problems, in which one is required to find the cause of the effect given the effect. For instance, one can try to determine the equation coefficients (which usually represent important media properties) from the information about solutions of the direct problem. One of the drawbacks of inverse problems is that their (approximate) solutions are almost always ill-posed in the sense that they may not be unique or stable. In this talk we introduce inverse problems, investigate some examples, and describe analytically and numerically a regularization technique used to combat instability in the solutions.
Friday, February 16, 2:30-3:30pm
Billy Jackson, SUNY Geneseo
From Differential Equations to Difference Equations
and Everything Else in Between
Of all areas of modern mathematics, differential equations is perhaps one of the oldest, of course going back to the days of Leibniz and Newton. Mathematicians in the latter half of the 20th century also began examining difference equations in detail as these structures became important in numerically solving differential equations with the advent in computing that took place. There are many similarities between discrete and continuous dynamical systems, but there are often many differences as well. For example, solutions to the usual logistic equation can exhibit chaotic behavior in discrete systems while simultaneously not being chaotic in their continuous counterparts.
In 1988, a German mathematician by the name of Stefan Hilger constructed a framework that we now call analysis on time scales which explains the similarities and differences between the two extremes: in addition to this unification aspect, we can now extend the analysis to other dynamical systems as well. For example, systems containing hybrid continuous and discrete components, fractal structures, or non-uniformly spaced domains are all ripe for exploration within this framework that will allow for a single analysis to treat them all.
In this talk, I will present Hilger's framework and then discuss applications of the work in one of my recent projects focusing on stability analysis and control theory from an engineering perspective. Most details will hopefully be accessible to those with at least a slight background in differential equations and an adequate preparation in analysis.
Thursday, February 1, 2:30-3:30pm
Yesim Demiroglu, University of Rochester
Waring's Problem over Finite Fields
Since Edward Waring stated his famous conjecture in his book "Meditationes Algebraicae" in 1770, Waring's problem has been of particular interest to mathematicians. As Charles Small put it in his survey "Indeed, it is one of those nasty gems, like Fermat's Last Theorem, which begins with a simply-stated assertion about natural numbers, and leads quickly into deep water.". The first half of our journey will be a quick introduction to Waring's problem in its original form together with its variants and a summary of some of the results in the literature. The second half of the talk is devoted to our contribution to this problem, namely how one can use Cayley digraphs and graph theoretical methods to solve Waring's problem over finite fields.
Friday, January 26, 2:30-3:30pm
Pamela E. Harris, Williams College
Invisible Lattice Points
This talk is about the invisibility of points on the integer lattice Z x Z, where we think of these points as (infinitely thin) trees. Standing at the origin one may notice that the tree at the integer lattice point (1, 1) blocks from view the trees at (2, 2), (3, 3), and, more generally, at (n,n) for any positive integer n. In fact any tree at (l,m) will be invisible from the origin whenever l and m share any divisor d, since the tree at (l/D,m/D), where D = gcd(l, m) blocks (l, m) from view. With this fact at hand, we will investigate the following questions. If the lines of sight are straight lines through the origin, then what is the probability that the tree at (l, m) is visible? Meaning, that the tree (l, m) is not blocked from view by a tree in front of it. Is possible for us to find forests of trees (rectangular regions of adjacent lattice points) in which all trees are invisible? If it is possible to find such forests, how large can those forests be? What happens if the lines of sight are no longer straight lines through the origin, i.e. functions of the form f(x) = ax with a ∈ Q, but instead are functions of the form f(x) = ax^b with b a positive integer and a ∈ Q? Along this mathematical journey, I will also discuss invisibility as it deals with the underrepresentation of women and minorities in the mathematical sciences and I will share the work I have done to help bring more visibility to the mathematical contributions of Latinx and Hispanic Mathematicians. Math work is joint with Bethany Kubik, Edray Goins, and Aba Mbirika. Diversity work with Alexander Diaz-Lopez, Alicia Prieto Langarica, and Gabriel Sosa.
Friday, December 1, 2:30-3:30pm
Joshua Hallam, Wake Forest University
Graph Coloring and Counting Increasing Forests
One of the most famous problems in combinatorics is the four color problem. It asks the following. Given any map, for example a map of the states of a country, is it always possible to color the map with at most 4 colors so that no two states that share a border are colored with the same color? In 1912 George Birkhoff introduced the chromatic polynomial as a tool to try to solve this problem. Although it was not used in the proof of the four color theorem, the chromatic polynomial has many remarkable properties. In this talk, we will discuss the chromatic polynomial and see a surprising connection between its coefficients and objects called increasing forests. I will also discuss some of the research two undergraduate students conducted with me over the summer related to this work and discuss some open problems.
Monday, November 27, 4:00-5:00pm
Colby Long, Mathematical Biosciences Institute,
the Ohio State University
Reconstructing the Past with Math: Evolutionary trees from
Understanding the evolutionary relationships between species is vitally important for species conservation, epidemiology, and evolutionary biology. With modern gene-sequencing techniques, we now have access to copious amounts of DNA sequence data. The field of phylogenetics is focused on turning this data into phylogenetic trees, graphs that explain how species have evolved and how they are related. In this talk, we will explore some of the novel mathematical ideas used in phylogenetics to extract evidence of the past from DNA sequences. We will also discuss some of the challenges and open problems for future mathematicians in the field.
Monday, November 20, 4:00-5:00pm
Jamie Juul, Amherst College
The Birthday Problem and a Problem in Arithmetic Dynamics
In a group of n people, what is the probability that two people share a birthday? Suppose we have a function mapping from a set S with d elements back to itself. If we pick a point from the set and apply the map over and over again what is the probability that we eventually return to the point we started with? The first question is the classical birthday problem, the second is a question from arithmetic dynamics. We will discuss these questions and how they relate to each other.
Thursday, November 16, 4:00-5:00pm
Katelynn Kochalski, SUNY Cortland
An Introduction to Queueing Theory
Have you ever wondered how grocery stores decide the number of cashiers to staff on a Sunday afternoon? Or how the TSA determines the number of agents needed during a given shift? These questions both relate back to queueing theory. This talk will explore some basic concepts in queueing theory. In particular, we will investigate properties of a queue where key characteristics of each customer are given in a random but suitably nice fashion. This will allow us to make predictions about the number of customers in the queue and the length of time each customer will be in the system using our knowledge of Calculus II in a surprising way.
Wednesday, November 15, 2:30-3:45pm
Jeff Johannes and Gary Towsley, SUNY Geneseo
A CONCISEHISTORY OF CALCULUS
A lively overview of over two thousand years of calculus history. Not only who-did-what along the way, but the cultural and sociological causes and effects of the calculus. Strongly recommended for anyone who has taken or is taking calculus.
Thursday, November 2, 4:00-4:50pm
Raymond Cook, AIR Worldwide
Introduction to Catastrophe Modeling
In the case of rare but severe events, historical loss information and traditional actuarial methods are inadequate for assessing future loss potential. AIR Worldwide developed probabilistic models that help organizations prepare for the financial impacts of catastrophes—before they occur. These probabilistic models are effectively robust Monte Carlo simulations of how catastrophe losses might develop over the next 12 months. In this talk, I will discuss why catastrophe modeling is an integral part of the insurance industry and introduce a few key concepts in catastrophe modeling.
Thursday, October 19, 4:00-4:50pm
Yusuf Bilgic, SUNY Geneseo
The mathematics behind emerging data science and machine learning
Statistics, mathematics, data management and computer programming have been merged in the practices of data science and machine learning, two growing interdisciplinary sciences. The talk covers technical and nontechnical discussions about some of the new developments in data science and machine learning in the data era through the window of a statistician who works at a mathematics department and has designed a machine learning course. Putting aside brilliant career opportunities for applied mathematicians and statisticians who are excelled with the toolboxes in data and algorithmic modeling, I emphasize that the strong foundation of mathematics plays the pivotal role in the data era. In this talk, I will share some mathematical details about three “winner” algorithms for classification, Linear Discriminant Analysis (LDA), Neural Networks (NN) and Support Vector Machines (SVM). I will also attempt to predict the next 50 years of the directions of data and algorithmic modeling without using any winner algorithms.
Thursday, September 28, 4:00-4:50pm
Dane Taylor, University of Buffalo
Centrality Analysis for Time-Varying Social Networks (and Beyond)
Many datasets are naturally represented by networks in which nodes represent objects and edges represent connections between the objects. Some examples include persons connected by friendships, neurons connected by axons, and webpages connected by hyperlinks. Given a network, an important problem is “centrality analysis” in which the goal is to identify how central (i.e., important) each node is. Centrality in social networks, for example, can reveal which persons are most important and/or influential. One of the leading approaches to centrality is Google’s PageRank algorithm, which identifies the most important webpages on the internet and has also been applied to diverse datasets ranging from gene-correlation networks to Facebook friendships. While there are many methods for centrality analysis, how to study networks that change over time has remained problematic. I will describe some of the relevant issues and present an approach we recently developed for a broad class of centralities known as eigenvector centrality. Using this approach, I will explore a few applications including data analytics for the Mathematics Genealogy Project, actor co-starring during the Golden Age of Hollywood, and citations between Supreme Court Decisions.
Monday, April 17, 2:30 pm
Katie Gayvert, Weill Medical College, Cornell University
Precision Medicine in the Age of "Big Data": Leveraging machine learning and Genomics for drug discoveries
Over the past few decades, great strides have been made in the treatment of cancer through the adoption of precision medicine approaches. One major effort of precision medicine is the greater application of targeted therapies, which seek to selectively kill tumor cells. However there are many challenges associated with the development and application of these therapies, including identification of tractable targets and the inevitable development of drug resistance. Furthermore the clinical trial failure rate continues to rise, which remains a barrier in the development of novel targeted therapies. Integration of extensive genomics datasets with large drug databases allows us to begin to address these problems. To this end, we have leveraged machine learning and “Big Data” to develop in silico drug discovery and toxicity prediction methods. I will give an overview of the predictive models developed in our lab and the underlying mathematical models that they employ. Approaches such as these have the potential to make a direct impact on how patients are treated, as well as to help prioritize and guide additional focused studies.
Wednesday, April 13, 2:45 pm
Robert Stephens, SUNY Geneseo
QUATERNIONS AND NUMBER SYSTEMS
The scene will be set by looking at the properties of familiar number systems like, for example, integers, rational numbers, and complex numbers. We then introduce the quaternions, which are an extension of the complex numbers. Some of the algebraic properties and uses of the quaternions will be discussed. One such use is calculating the rotation of objects, such as computer generated objects. Euler angles are another way to rotate objects, and we will see some of the difficulties that can arise. If time permits, we will extend the quaternions to new number systems, each of which have their own interesting algebraic properties.
Friday, April 7, 2:30 pm
Chad Magnum, Niagara University, Lewiston, NY
STRUCTURE AND SYMMETRY: AN INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY
Mathematical structure and symmetry have long fascinated the human mind. An in-depth study of these topics has helped to lead to quite a bit of the modern mathematical and scientific sophistication we enjoy today. One manifestation of this study is Lie algebra representation theory, which has been significant in various areas of mathematics and physics for several decades. In this talk, I will discuss how one can view Lie algebra representation theory from a perspective which is grounded in natural intuition of structure and symmetry. It is my hope that this discussion gives the audience a taste of the topic, provides context for how to demystify a potentially daunting subject, and stimulates enthusiasm for this fascinating field of mathematics.
Thursday, March 23, 2:30 pm
Ahmad Almomani, Clarkson University, Potsdam, NY
DERIVATIVE-FREE METHODS AND APPLICATIONS
Derivative-free methods are modern techniques used to solve optimization problems. In many practical applications the derivatives are not available or hard to compute due to a “black-box” or simulation-based formulation. Derivative-Free Optimization (DFO) methods are applicable for these kinds of problems as opposed to methods that employ derivatives. The need for DFO arises extensively across all engineering disciplines. In this talk, we compare and analyze the performance of different solvers for local and global optimization paired with different constraint handling techniques (penalty, barrier, and filter methods). Specifically we consider implicit filtering (IF), Nelder-Mead (NM), DIRECT (DIR), a genetic algorithm (GA), particle swarm optimization (PSO), and simulated annealing (SA). We use different values of the penalty or barrier parameters on both smooth and noisy problems. By using performance and data profiles together with a convergence test that measures the decrease in objective function value, we seek to identify which is the best solver and constraint method on which kind of optimization problem. We also hybridize the implicit filtering algorithm with the filter method for constraints to get a new algorithm. Additionally, we introduce a new algorithm that combines the filter method with PSO. We apply the new algorithms for local and global optimization on a Hydraulic Capture Problem from environmental engineering and a suite of test problems and show the new methods are competitive compared with classical approaches for handling constraints.
Thursday, March 2, 2:30 pm
Eleni Panagiotou, UC Santa Barbara, Santa Barbara, CA
QUANTIFYING ENTANGLEMENT IN PHYSICAL SYSTEMS
Many physical systems, such as biopolymers and polymer melts, are composed by macromolecules which cannot cross each other and attain entangled conformations. The entanglement complexity in these systems affects dramatically their mechanical properties. For their simulation, Periodic Boundary Conditions (PBC) are employed, which impose further complications. In this talk we will see methods by which one can measure entanglement in collections of open or closed curves in 3-space and in systems employing PBC. More precisely, using the Gauss linking number, we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing PBC and study its properties. We will see two applications of these measures. First, we will apply our entanglement measures to discuss the evolution of the dimensional character of the entanglement as a function of density in Olympic systems which model the behavior of DNA networks. Next, using Molecular Dynamics simulations, we will apply our measures to investigate how the entanglement of polymeric chains relates to bulk viscoelastic responses in polymeric materials. Our approaches provide new mathematical tools for characterizing the origins of the rheological responses of polymeric materials.
Monday, February 27, 2:30 pm
Yu-Min Chung, College of William and Mary, Williamsburg, VA
COMPUTATIONAL TOPOLOGY WITH APPLICATIONS IN THE NATURAL SCIENCES
Computational topology is a relatively young field in algebraic topology. Tools from computational topology have proven successful in many scientific disciplines, such as fluid dynamics, biology, material science, climatology, etc. In this talk, we will give a brief introduction to computational topology, focusing primarily on persistent homology. Applications to various datasets from cell biology and climatology will be presented to illustrate the methods.
Thursday, February 23, 2:30 pm
Sedar Ngoma, Auburn University, AL
RECOVERING A DIFFUSION COEFFICIENT IN A PARABOLIC EQUATION ARISING IN GEOCHRONOLOGY
We investigate a problem with applications in geochronology, a branch of geology in which the dating of rock formations and geological events are studied. In particular, we reconstruct the temperature history of rocks. Reconstructing the temperature history amounts to solving (and approximating solutions of) a time-dependent inverse diffusion coefficient problem for a parabolic partial differential equation with an integral constraint. We describe both the direct and the inverse problems, and under some condition, we show that the underlying problem applies to problems of continuous compounding of interest, population dynamics, and radioactive decay. We present some computations and several numerical experiments.
Friday, January 27, 3:00-4:00pm
Cesar Aguilar, SUNY Geneseo
THE PAGERANK ALGORITHM: THE MATHEMATICS BEHID GOOGLE'S SEARCH ENGINE AND ITS APPLICATIONS
In 1998, two graduate students from Stanford changed the way all of us search on the web by creating the search engine Google. Before 1998, search engines used mostly traditional text processing to find and display relevant pages after a search query. The founders of Google had a different idea: Why not model the web as a directed graph and compute the dominant eigenvector of the Google web-matrix to rank all web pages and incorporate this ranking when displaying search results? As you may verify for yourself by looking up the present market value of Google, this idea worked extremely well. In this talk, we will give an overview of Google's PageRank algorithm, some of its applications other than web page ranking, and some of the computational issues involved.
Wednesday, November 16, 2:30-3:45pm
Jeff Johannes and Gary Towsley, SUNY Geneseo
A CONCISEHISTORY OF CALCULUS
A lively overview of over two thousand years of calculus history. Not only who-did-what along the way, but the cultural and sociological causes and effects of the calculus. Strongly recommended for anyone who has taken or is taking calculus.
Wednesday, November 2, 2:30-3:30pm
Sean Nixon, SUNY Geneseo
OVERTHINKING IT: THE UNNECESSARY APPLICATION OF ADVANCED MATHEMATICS TO POPULAR CULTURE
Part 1: Modeling Politics in Marvel’s Civil War – In the 2016 film, Captain America: Civil War, the Marvel Cinematic Universe moved from the basic her vs. villain plot to a hero vs. hero story. The battle between good and evil to a philosophical debate over the proper level of government oversight. Understanding the politics involved in the Marvel Civil War takes us from one dimensional models and basic game theory to embedded manifolds and applications for computer visualization.
Part 2: Fractional Characterization in Inside Out – Writing a well rounded character is a lot like measuring the coast of Britain, or at least, that’s what Pixar’s Inside Out seems to be telling us. The Anthropomorphic personifications of Riley’s emotions give the audience a simplified representation for the struggles in a young girl’s coming of age. But, they also give a window into how the study of fractional geometry and fractals can help us understand emotional complexity.
Thursday, October 20, 4:00-4:50pm
Alex Rennet, University of Toronto, Mississauga
INCOMPLETENESS, HERCULES AND THE HYDRA
In this talk, I'll start with an introduction to Godel's Incompleteness Theorems. These theorems were a major turning point for our modern understanding of the logical underpinnings of mathematics. One of them can be phrased as follows: "Given any reasonable set of axioms for number theory, there are number-theoretic statements which are true but unprovable from those axioms." In the classic proofs of this theorem, a true but unprovable statement is constructed, called a 'Godel sentence'. However, such classically-constructed Godel sentences didn't seem to have any concrete mathematical meaning. This lead many to wonder whether there were any examples of natural, contentful statements about number theory which are both true and unprovable, like Godel sentences. A number of different positive answers were found. I will discuss my favourite, which involves Hercules and The Hydra fighting to the death, and some really, really large numbers. No particular background knowledge is required, but some basic knowledge of sets and axioms would be helpful.
Thursday, September 22, 4:00-4:50pm
Hossein Shahmohamad, RIT
GRAPH COLORINGS & CHROMATIC POLYNOMIAL: EQUIVALENCE, UNIQUENESS, ROOTS & MORE
This talk will introduce vertex colorings as well as the celebrated chromatic polynomial, which counts the number of proper vertex colorings of a graph given n colors. The chromatic polynomial was intended as a tool to solve the four color conjecture in the early 1900’s. Many interesting properties of this polynomial will be discussed. After introducing graph isomorphism, chromatic equivalence and chromatic uniqueness will be presented. Some open problems in this area will be discussed as well.
Friday, January 29, 3:00-3:50pm
Arunima Ray, Brandeis University (SUNY Geneseo, Class of 2009)
IS THE SET OF KNOTS FRACTAL?
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
Cesar Aguilar, California State University at Bakersfield
CONTROLLABILITY OF GRAPHS
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
Kristopher Lee, Iowa State University
THE EIGENVALUE MACHINE
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.
Physics Department Colloquium
Thursday, March 3, 4:00-4:50pm
Gary Towsley, SUNY Geneseo
MATHEMATICAL PHYSICS AND THE SCIENTIFIC REVOLUTION
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.
Friday, March 4, 2:30-3:20pm
Min Wang, Equifax Inc.
INTRODUCTION TO BOUNDARY VALUE PROBLEMS: FROM INTEGER TO FRACTION
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
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.
Wednesday, April 6, 3:30-4:20pm
Stephanie Allen, SUNY Geneseo PRISM
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, 4:00-4:50p
Amanda Tucker, University of Rochester
MULTIPLE ZETA VALUES
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.
Thursday, April 14, 3:30-4:20pm
Megan Brunner, SUNY Geneseo PRISM
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
Brendan Murphy, University of Rochester (SUNY Geneseo, Class of 2010)
THE SUM-PRODUCT PROBLEM
Erdos and Szemeredi conjectured that for a finite sequence of integers a1 < a2 < ... < an, the number of integers of the form ai+aj,aiaj, 1 =
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.