Department Chair

  • Edwin Rogers, Ph.D.
    Professor of Mathematics
    Roger Bacon Hall 444
    (518) 783-2392
    rogers@siena.edu

Math Speaker Series

The Siena mathematics colloquium is held in Roger Bacon Hall at the time and day announced for each talk.
Refreshments are offered 30 minutes before each talk in the Mathematics Library on the fourth floor of RB hall.

FALL 2013

November 20, 2013
Speaker: Dr. Leila Khatami; Union College
Title: A Game of Commuting Matrices
Abstract:

    One of the first things that we learn about multiplication of matrices is that it is not "commutative", that is if A and B are two square matrices of the same size then in general A.B is not the same as B.A. However, we can easily think of pairs of matrices that do "commute", e.g. if one of the matrices is the identity matrix or the zero matrix, or if A and B are both the same, then we will have A.B=B.A. So it is natural to ask: which pairs of matrices do in fact commute? The goal of this talk is to present a refined version of this natural question, using one of the most important classical theorems in linear algebra, the Jordan Decomposition Theorem. We are going to introduce a game which will help us understand the main idea of Jordan decomposition theorem, as well as the question of commuting pairs of matrices.

November 12, 2013
Speaker: Francesca Romano ('2014); Siena College
Title: Using the Ammann-Beenker Tiling to Model Quasicrystals
Abstract:

    The Ammann-Beenker tiling is a mathematical model for quasicrystals with eightfold symmetry. Accurate approximations of the Laplacian of this octagonal tiling and the Hausdorff dimension provide physicists and chemists with information on the density of states and the movement of electrons in the quasicrystal. The Ammann-Beenker tiling can be constructed through a matching technique, substitution, or the cut and project method. Using MatLab, I will generate subsets of this tiling through the substitution method and then calculate the spectrum of the associated Laplacian. I will then use these calculations to approximate the Laplacian of the complete Ammann-Beenker tiling and approximate the Hausdorff dimension of the spectrum of this tiling.

November 5, 2013
Speaker: Dr. Graziano Vernizzi; Siena College
Title: Matrix Field Theory for RNA molecules
Abstract:

    In this seminar I will describe a recent application of random matrix field theory (RMT) to the problem of RNA folding. After reviewing some elementary properties of RNA molecules, I will show the intimate connection with RMT: any RNA secondary structure can be represented by planar diagrams which are naturally interpreted as the Feynman diagrams of a suitable field theory. Moreover, the classical topological expansion of RMT induces an elegant classification of all possible RNA pseudoknots according to their topological genus. I will present some models and algorithms to predict RNA structures with pseudoknots based on this approach.

October 29, 2013
Speaker: Dr. Keith Jones; SUNY Oneonta
Title: Routes to Infinity: Asymptotic Behavior in the Lamplighter Group
Abstract:

    Imagine a person, a "lamplighter", tasked with patrolling an infinitely long straight road with lamps placed at regular intervals. The lamplighter has two possible actions at a given point in time: take a step in one direction or the other, or toggle a lamp on or off. Hidden inside this scenario is an object known as the ``Lamplighter Group'' L2, which records all possible sequences of actions the lamplighter may take. When one sequence of actions is followed by another, they combine to form a third, and so there is algebra to be found here: this operation of combining sequences is akin to multiplication. Just as each real number has a reciprocal (i.e., a multiplicative inverse) each sequence has an inverse which undoes, that is cancels out, that sequence. However, unlike multiplication of real numbers, the operation of combining sequences is not commutative: clearly the act of first toggling a light then taking a step has a different effect than first taking a step then toggling a light.

    This operation imposes a complex and highly symmetric structure on the Lamplighter group. One way to understand this structure is by representing in the form of a graph, called a Cayley graph. In general, a group has many Cayley graphs, but a particularly nice Cayley graph for the Lamplighter group is known as the Diestel-Leader graph, DL(2,2), an important and beautiful mathematical object in its own right.

    In collaboration with Dr. Gregory Kesley of Trinity College, Hartford, CT, my research focuses on studying asymptotic properties of Diestel-Leader graphs. That is, we seek to understand all the different ways one can "go to infinity" within a Diestel-Leader graph. This information is encapsulated in a topological space known as the "boundary at infinity" of the graph, a generalization of the concept known to astronomers as the "Celestial Sphere". Our work shows that the boundary at infinity for the Diestel-Leader graph DL(2,2) is T1, and compact but not Hausdorff.

September 24, 2013
Speaker: Dr. Frank Morgan; Williams College
Title: Optimal Pentagonal Tilings
Abstract:

    Hales proved that the least-perimeter way to tile the plane with unit areas is by regular hexagons. What is the least-perimeter way to tile the plane with unit-area pentagons? We'll discuss some new results, examples, and open questions, including work by undergraduates.

     

SPRING 2013

April 5, 2013
Speaker: Dr. Len Putnick; Siena College
Title: Population Models with Mutualism
Abstract:

    Interactive dynamical systems appear in a large variety of applications. A common use of such systems is the study of how different species of animals or plants interact with each other. Two well-known examples are the predator-prey models and competing hunter models.

    There is a third type of relationship between two species, known as mutualism, which is less often studied. In this model, the two species rely on each other for their survival. We will explore a few simple mathematical models that might be used to study this type of dynamical system.

     

March 15, 2013
Speaker: Dr. Mohammad Javaheri; Siena College
Title: Paul Erdos, the traveling mathematician
Abstract:

    March 26 this year marks the 100th birthday of Hungarian mathematician Paul Erdos, the most prolific mathematician of all times. Erdos published more than 1,500 articles in collaboration with more than 500 mathematicians worldwide. In this talk, we give a short biography of his life, and along the way discuss his contributions to Number Theory, Combinatorics, and Graph Theory.

     

March 8, 2013
Speaker: Dr. Yu Sun; Siena College
Title: Consensus-Type Stochastic Approximation Algorithms
Abstract:

    The interest in this work is motivated by cooperative and coordinated control of unmanned aerial vehicles (UAVs). This work is concerned with asymptotic properties of consensus-type algorithms for networked systems whose topologies switch randomly. The regime- switching process is modeled as a discrete-time Markov chain with a finite state space. The consensus control is achieved by using stochastic approximation methods. Distinct convergence properties of three different scenarios with respect to the stationary measure of the Markov chain are studied. Simulation results are presented to demonstrate these findings.

     

February 15, 2013
Speaker: Francesca Romano (2014); Siena College
Title: More Explicit Formulas for Euler and Bernoulli Numbers
Abstract:

    In a previous Math Colloquium talk, Dr. David Vella (Skidmore College) explored the notion of a generating function as a way to bridge the gap between a continuous perspective and a discrete perspective on mathematics. This talk will extend Vella's work to the multivariable setting beginning with a multivariable chain rule formula, the generalized Faá di Bruno formula for higher partial derivatives. By directly considering Taylor coefficients and generating functions, we employ this generalized formula to obtain further identities for Euler and Bernoulli numbers.

     

February 1, 2013
Speaker: Dr. Drew Warner; RPI
Title: Event Chains and Inverse Problems with Applications to Neuroscience
Abstract:

    Over the last half century, mathematical modeling of the biological interaction of neurons has mainly comprised the field of Computational Neuroscience. Here we define a set of brain models that simulate biologically consistent neuronal interaction and use event-chain data to study behavioral characteristics. Randomly stimulating the brain model leads to data of successive neuronal firings. We are then able to study how well this data characterizes the brain models. In particular, we examine the effect of the size of the brain model, inhibitory and excitatory weight distributions, and damage effects. We find that modifying physical structure produces a measurable change in behavior.

    Of particular interest is whether brains subjected to learning can be distinguished from brains generated by statistical procedures. These learning algorithms adaptively modify the connection strengths, and allow for the creation of new connections. The event-chain data shows conclusively that brains that have undergone learning are significantly different. This unique structure and behavior of learned brains allows for identification based on purely behavioral characteristics; specifically, the event-chain data is sufficient to distinguish between learned and unlearned brain models

     

January 25, 2013
Speaker: Dr. David Clark; Randolph-Macon College
Title: Japanese Temple Geometry: A Tale of Math, Art, Religion, and History
Abstract:

    What is Japanese mathematics? During the Tokugawa Period (1603-1868), Japan was almost completely isolated from the West, including the products of the Western revolutions in math and science. At the same time, the Japanese witnessed a cultural renaissance in the visual and performing arts, music, fashion, and ceremony … and mathematics. New problems and solutions found surprising applications to the traditional Buddhist temples and Shinto shrines that pervade the Japanese landscape. By the end of the talk, I hope you’ll understand a bit of what makes a piece of mathematics Japanese, and how wasan (“wa” = Japanese, “san” = mathematics) became so delicately folded into 18th and 19th century Japanese culture.

     

FALL 2012

November 30, 2012
Speaker: Dr. Mary O'Keeffe; Founding advisor of Albany Area Math Circle, co-organizer of the Math Prize for Girls@MIT, and public policy economist at Union College
Title: Math is a super power! Share it! Marketing “Extreme Problem Solving” to the Broader Community
Abstract:

    On February 20, Siena College will host an exciting national math contest for high school students, the AMC10/12. The Mathematical Association of America has designed these problems to be a great learning experience for high school students to struggle with--but the problems are scary hard to students who have never encountered them before. (In fact, even accomplished adult professionals in STEM fields find them hard! High school teachers find them hard! College professors find them hard!) How do we encourage and support local high school students in embracing this contest as an opportunity to stretch themselves beyond their usual comfort zone and grow mathematically?

    We are especially concerned about outreach to students in inner city schools, students who (like Mary herself, when she was in high school) may have no idea of the existence or value of a contest like this one, let alone have any idea how to prepare for it. Albany Area Math Circle has made some inroads on this problem with some fun ideas we would like to share. We are excited about the idea of recruiting interested Siena College students to join us in our "mathematical missionary" work.

    After the talk, interested Siena students, faculty, and community members are invited to join Albany Area Math Circle members in brainstorming how to implement our mathematical marketing ideas as we engage in an informal problem solving session. We will create (and eat!) mathematically tessellating and sphere packing polyhedral refreshments.

     

November 16, 2012
Speaker: Jon Bannon; Siena College
Title: Operator Algebras and Quantum Entanglement
Abstract:

    Quantum mechanics violently defies common sense, yet the mathematical description of the quantum world performs flawlessly. In this talk, we'll discuss the mathematical description of quantum mechanics, as well as some of its bizarre predictions and predicaments. Particularly we'll introduce the theory of operator algebras as a mathematical framework for quantum entanglement, which captures the imagination via phenomena such as quantum teleportation, quantum cryptography and superdense coding. After introducing the basics and recounting some recent progress, we'll move on to discuss some open problems.

     

November 2, 2012
Speaker: Zachary Kudlak; Mount Saint Mary College, NY
Title: Structure in Numbers
Abstract:

    Suppose that given any two distinct people, the two people are either a pair of mutual acquaintances, or a pair of complete strangers. Under these conditions, in any group of at least six people, there must be a subgroup of three or more people who are all mutual acquaintances, or there must be a subgroup of three or more people who are complete strangers. This is a typical result from the field of Ramsey Theory. During this talk, we will discover more about Ramsey Theory, and see how probability can be used to find a bound on the size of a group of people which contains either a subgroup of k mutual acquaintances or a subgroup of k complete strangers, for an integer k greater than 3. Thanks to social network cites, such as facebook, tracking who knows who, and who likes what, has become a booming area in social network analysis, a field built on the mathematics of graph theory.

     

October 26, 2012
Speaker: Nikolai Krylov; Siena College
Title: Bachet's equation: From Diophantus to modern times
Abstract:

    We will consider the Diophantine equation y^2 = x^3 + k for various values of k. Diophantus (c. 250 AD) used a clever trick to find an integer solution when k=-2, but it was proved only in 18th century that Diophantus' solution was the only integer solution for that k. Today we have well developed tools of algebraic number theory to attack such problems. I will discuss some of the tools and show how the arithmetic of quadratic fields can be used to determine all the solutions for certain classes of integers k.

     

October 10, 2012
Speaker: Andy Dorsett; Wolfram Research, Inc.
Title: Mathematica 8 in Education and Research
Abstract:

    This seminar will be given 100% in Mathematica and will show useful teaching and research examples for mathematics, the physical sciences, engineering, and business/economics. Ideas for creating universal examples in Mathematica that can be used by colleagues or students with no prior Mathematica experience will be a central theme.

    The content will help attendees with no prior experience get started with the Mathematica language and workflow. Since there is a large amount of new functionality in Version 8, most intermediate and advanced users who attend these talks report learning quite a bit as well. All attendees will receive an electronic copy of the examples, which can be adapted to individual projects.

     

October 5, 2012
Speaker: Vladimir Chernov; Dartmouth College, NH
Title: Link theory and general relativity
Abstract:

    We study surprising interactions between link theory and causal relations between events in general relativity. The talk is based on joint works with Stefan Nemirovski and Yuli Rudyak.

     

SPRING 2012

April 13, 2012
Speaker: Alan Taylor; Union College, NY
Title: On Making Honesty the Best Policy (Mathematically)
Abstract:

    We will consider a number of procedures that arise in contexts such as auctions, fair division, and voting. For each, we will ask two questions: (1) Is honesty the best policy? (2) If not, can we change the procedure, while preserving the spirit of the original, so that disingenuous behavior is no longer advantageous?

     

February 24, 2012
Speaker: David Vella; Skidmore College, NY
Title: Across the Great Divide: Generating Fun and Bridging the Gap between Two Competing Perspectives on Mathematics
Abstract:

    Do mathematicians do it continuously or do they do it discretely?  Some problems in mathematics are suited to be solved using the techniques of analysis (continuous functions, derivatives, integrals, composing functions.)   Other problems are more suited to be solved using the techniques of discrete mathematics or combinatorics (sets, sequences, recursion, counting.)  The two appear to have little to do with each other.  Is there hope that that the two cultures can ever come together on common ground, or it is a case of never the twain shall meet?  

    In this talk, we’ll explore the notion of a generating function, a tool which can be used to transcend the superficial differences between the discrete and the continuous.  With specific examples, we’ll demonstrate several different ways of wielding that tool, including with the enhancement of a recent result (published in 2008) derived from a humble beginning – the familiar chain rule from calculus.

    Generating functions can be applied to topics as diverse as the Fibonacci sequence, Bernoulli and Euler numbers, counting restricted compositions of integers, and computing the Wiener index of a family of graphs.  Will they bring a lasting peace between the analysts and the combinatorialists, or will mathematics remain a house divided?  That remains to be seen, but in the meantime, come join us as we generate some fun along the way!

     

February 17, 2012
Speaker: Jon Bannon; Siena College, NY
Title: A Panoply of Pigeonhole Principle Pranks, with a Practical Application
Abstract:

    Who knew that being finite could be so much fun? Come and find out why in any group of six people there are either three mutual friends or three mutual strangers. Find out why we can prove that there are two people on Earth with the same number of hairs on their heads. Find out why, if we color every point in the plane either red or blue, there will always be a pair of points 1 mile apart that have the same color. Heck, I'll give you the practical application now, to get it out of the way: If you have a drawer full of white and black socks and you want to pick a pair of socks without turning on the light, you just need to pick out three socks.

     

January 27, 2012
Speaker: M. B Henry ; Siena College, NY
Title: The Infinitude of the Primes: The beauty of many proofs
Abstract:

    Mathematicians derive immense pleasure from discovering elegant proofs of existing theorems. We are, after all, artists of reason. During this talk, we will investigate as many different proofs of the infinitude of the primes as time allows. We will begin with Euclid's beautifully clever proof involving a few basic facts from arithmetic. From there we will venture into proofs involving counting arguments, calculus, infinite series, and group theory. Each proof is truly elegant and each gives us new insight into the complexity of the prime numbers.

     

SPRING 2011

    April 13, 2011
    Speaker: Lauren Peloso ; Siena College, NY
    Title: Queueing Theory - Mathematical Applications in Our Traffic System
    Abstract:
      Simulation is often a cheap and effective way to find a preferable solution to optimize a task. In using ProModel to simulate traffic flow for various arrival times and creating models for both the traffic light and stop sign systems, we examined the stop sign threshold. In studying the Poisson distributed traffic flow, we were able to determine which system would provide the most exits from the intersection over a set time. This was all performed with the intention of finding a traffic-reducing way to route vehicles, while utilizing our technological resources to study the data. While our focus was a specific Albany intersection, the simulation has been created so that, with only minor adjustment to variables and arrival rates, this model can be applied to any four-way intersection.
    Speakers: Bridget DeBardelaben, Megan DeRudder, Joe Fava ; Siena College, NY
    Title: Investigating Benford's Law in Financial Documents
    Abstract:
      Benford's Law is a probability distribution that says that the first digit of random numbers in real life data should have 1's show up more frequently then 9's. Benford's Law has been used to detect fraud in voting registration. We have investigated whether Benford's Law can be applied to balance sheets of Dow Jones Industrial Average companies.


 

    April 05, 2011
    Speaker: Satyan Devadoss ; Williams College, MA
    Title: Robot Motions and Collisions
    Abstract:
      What is the space of all possible ways robots can move in a room? What happens when we place obstacles in their path? We not only look at the important subject of robot motions but look at the ideas behind robot collisions. This leads to worlds of polyhedra, tilings, string theory, and phylogenetic trees. During this adventure, we visually learn what multiplication is really about as well as learning to draw in higher dimensions.


 

    February 17, 2011
    Speaker: Marie Snipes ; Kenyon College, OH
    Title: Minimal Surfaces, Soap Films and Flat Chains
    Abstract:
      If a piece of wire is twisted into some shape and then dipped into soapy water, a soap film forms. This film is a surface of minimal area whose boundary is the bent wire. Using soap films as inspiration, we will discuss the question of how to define a surface. The discussion will lead to a description of flat chains, a class of geometric objects that can be thought of as generalized surfaces. Opportunities to play with bubbles will be provided.


 

    February 14, 2011
    Speaker: Michael Bradley Henry ; The University of Texas at Austin, TX
    Title: A Roller Coaster Ride Through Knot Theory
    Abstract:
      In the 19th century, it was postulated that very tiny knotted circles formed fundamental building blocks within the structure of the universe. The discovery of the atom would eventually pull many scientists away from such knots, but mathematicians have persisted with their attempts to understand these beautiful and complex objects. In the last 25 years, a new type of knot has become popular. These are called Legendrian knots and they also originated as an attempt to understand our physical world.
      The track of a roller coaster can be thought of as a knotted circle. Riding the coaster gives you a close-up view of the complexity of the knot. A Legendrian knot is a roller coaster where the cars may move in two additional directions. The cars may spin like the cups on the Tea Cup ride and twist around the track as they move forward. The resulting three motions (forward/backward, spinning, and twisting) create a ride to test even the strongest stomach. In this talk, we will explore knots and roller coasters and, with any luck, avoid giving ourselves vertigo.


 

    February 10, 2011
    Speaker: Keir Lockridge ; Wake Forest University, NC
    Title: The Euler Characteristic
    Abstract:
      In the mid-18th century, Euler noticed an amazing fact concerning the number of vertices v, edges e, and faces f of any convex polyhedron: v - e + f = 2. (If you have a soccer ball, you can verify this identity!) This observation was foundational for the discipline of topology, and it underlies a topological proof of the classification of Platonic solids. In this talk, I will introduce a few simple surfaces (e.g., the sphere, torus, and Möbius strip) and discuss the role played by the Euler characteristic in their classification.


 

    February 07, 2011
    Speaker: Louis Deaett ; University of Victoria, Canada
    Title: Communication Complexity and Linear Algebra
    Abstract:
      Suppose you and a friend are each given a number. Working together, you have to find the answer to a yes-or-no question about your two numbers. (Simple example: Are they equal?) The problem is that you are not allowed to show one another what your numbers are! Instead, you can only answer a series of yes-or-no questions back and forth until one of you knows the answer. And one more thing: You have to decide what the questions are going to be before you even get the two numbers! Can you design a sequence of questions that will always allow you to solve the problem? What is the absolute shortest sequence of questions you can come up with? This is the essence of communication complexity. We will discuss how these ideas are formally addressed in computational complexity theory and show how they can be related to some simple concepts from linear algebra.


 

    February 01, 2011
    Speaker: Mohammad Javaheri ; Trinity College, CT
    Title: Dynamics of Linear Fractional Transformations
    Abstract:
      Start with a positive rational number x, and repeat the following operation: if x>1, then replace x by (x-1)/3; if x<1, then replace x by 2x/(1-x). It is conjectured that any such iteration ends in 1 (for example, starting with 6, we have the orbit: 6, 5/3, 2/3, 4, 1). In this talk, we study this conjecture, some other related conjectures, and connections to quite a few mathematical fields such as number theory, functional analysis, and dynamical systems.


 

FALL 2010

    December 03, 2010
    Speaker: Joseph D’Avanzo; Siena College, NY
    Title: Knots, Surfaces and k-Polygrams
    Abstract:
      Take a piece of string, entangle it with itself in any manner without breaking it, then fuse the two loose ends together. The resulting closed loop is what mathematicians call, a knot. A knot, by definition, is an embedding of the unit circle into 3D. The standard way to view a knot on a piece of paper one creates a knot-diagram, which is a projection of a knot onto a plane, such that depth is preserved in the crossings. However, I will be discussing a new way of representing a knot as a planer diagram by drawing the knot on an orientable surface in 3D, and then unfolding the surface while keeping the information about the knot intact.
    Speaker: Maureen Jeffery; Siena College, NY
    Title: Noncommutative polynomials and Finite von Neumann algebras
    Abstract:
      von Neumann algebras are the right environment for doing Quantum Mechanics if you want your experiments to include “yes or no” questions. In 1976, in a groundbreaking paper on the classification of von Neumann algebras Alain Connes conjectured that all von Neumann algebras could be studied using finite matrices in a certain delicate way. This conjecture, when properly stated in detail, is known now as the Connes Embedding Conjecture. We consider the following equivalent form of Connes' conjecture due to our coauthor, Don Hadwin: Connes' Embedding Conjecture (CEC) is false if and only if there exists of a selfadjoint noncommutative polynomial p(t1,t2) in the universal unital C*-algebra A and positive, invertible contractions x1, x2 in a finite von Neumann algebra M with trace t such that (1) Tr_{k}(p(A1,A2))=0 for every positive integer k and all positive definite contractions A1,A2 in M_{k}(C), and (2) t(p(x1,x2))<0. We prove that if the real parts of all coefficients but the constant coefficient of a selfadjoint polynomial p in A have the same sign, then such a p cannot disprove CEC if the degree of p is less than 6, and that if at least two of these signs differ, the degree of p is 2, the coefficient of one of the t_{i}² is nonnegative and the real part of the coefficient of t1t2 is zero then such a p disproves CEC only if either the coefficient of the corresponding linear term ti is nonnegative or both of the coefficients of t1 and t2 are negative.


 

    November 05, 2010
    Speaker: Allison Pacelli; Williams College, MA
    Title: Algebraic Number Theory & Fermat's Last Theorem
    Abstract:
      The origins of algebraic number theory stem from attempts to solve Fermat's Last Theorem. In 1847, the French mathematician Lame claimed to have a proof of the famous theorem, but his proof was incorrect due to some mistaken assumptions about prime factorization in certain types of number systems. Although the world was forced to wait almost 150 years for a correct proof of Fermat's Last Theorem, the mathematics underlying Lame's mistakes led to this important branch of mathematics that is still heavily studied today. In this talk, we'll discuss the origins and some of the main ideas in the field.


 

    October 22, 2010
    Speaker: James Gatewood; Siena College, NY
    Title: A Mathematical Model of Network Communication
    Abstract:
      The behavior of a communication network can be modeled as a flow of traffic units along links connected by nodes. We derive a node/link network model and connect it to a fluid-like model of traffic flow. The discrete node/link model emphasizes packet queuing and the flow of packets from spatial point to spatial point. The model assumes that packets reside in buffers at each node, and are classified by their destination and the length of time they have resided in the buffer. An algorithm was created for packets to exit the buffer at each node according to their age and travel to the next node along a predetermined path to their destination. This algorithm calculates the rate at which packets distribute themselves to the next link in the route to their destination, assumes a source of packets originating at the node, and subtracts packets whose destination is that particular node. The continuum model derived from this discrete flow model leads to a flow continuity equation. The continuity equation describes the density of packets as a function of time and space, so that we are able to predict changes in global flow patterns and optimal paths in the network. Solutions to the flow equations in one dimension show that if the sources are too strong or the flow is restricted, the packet density grows at the nearest upstream node. When the source strength is reduced, or when flow is restored, the buffered packets flow at capacity until the density has been reduced.


 

    September 30, 2010
    Speakers: Colin Adams and Thomas Garrity; Williams College, MA
    Title: Derivative vs. Integral: The Final Slapdown
    Abstract:

      Ever since Newton and Leibniz, the derivative and the integral have been locked in mortal combat, doing whatever it takes to try to prove which is the better, and in the process tearing equations asunder and leaving broken and shattered math symbols in their wake. At this event, we determine once and for all who will be crowned the victor, derivative or integral. And mathematics can then revert once again to the bucolic garden of Eden, where students frolic with equations in peace and harmony.

       

SPRING 2010

    March 26, 2010
    Speaker: Gary Nan Tie ; The Travelers Companies, MN
    Title: A needle in a haystack
    Abstract:
      The optimality of square root biased sampling is shown to be a sharp instance of the Cauchy-Schwarz inequality.

 

    March 12, 2010
    Speaker: Leona Sparaco ; Smith College, MA
    Title: Hypergraphs: what they are, why we study them, and why they are really cool.
    Abstract:
      A hypergraph is a generalization of a graph, where the edges are subsets of a given set of vertices. Computer scientists have been interested in them since the 1960s as they have many important applications. Several beautiful results about hypergraphs can be obtained using nothing more than basic proof techniques and some ingenuity. In this talk, I will present some of the results our research team discovered last semester.

 

    March 05, 2010
    Speaker: Darren Lim ; Siena College, NY
    Title: Where have all the lefties gone?
    Abstract:
      Modern statistics show that left-handed people have a lower life expectancy that right handed people. But in the face of these findings, left-handed people still form around 13% of our population. Why haven't they disappeared from the population altogether? In this talk, we will discuss the findings of two scientists: G.H. Hardy and Wilhelm Weinberg who, at the turn of the century, linked the binomial distribution to principles of population genetics governing recessive traits.

 

    February 18, 2010
    Speaker: Kim Plofker ; Union College, NY
    Title: Mathematics in Pre-modern India
    Abstract:
      India is widely but vaguely famous as the source of many important mathematical concepts, such as the decimal place-value numerals and the zero. What was the content and practice of mathematics in early India really like, and what led its practitioners to develop these and other fundamental ideas? This talk explores the history of Indian mathematics in general and its contacts with other mathematical traditions.

 

    January 29, 2010
    Speaker: Emelie Kenney ; Siena College, NY
    Title: "Preserving a Nation: Mathematics Education in Poland"
    Abstract:
      To provide context for a discussion of mathematics education in Poland, we briefly describe relevant aspects of her 1000-year history, with emphasis on the central place of education and culture. Some highlights of that history include the establishment of the world’s first ministry of education, as well as the clandestine flying universities. We then discuss the Polish tradition of mathematics education research, focusing on the Krakow Experiment. After describing the current system of education, we outline the primary and secondary school mathematics curricula, and include a series of examples typical of problems featured in those curricula. We conclude with some calculus problems that a student might encounter in a university calculus course.

 

FALL 2009

    December 11, 2009
    Speakers: Lauren Cassidy and Brittany Parahus ; Siena College, NY
    Title: Mathematics, Finance, and Paying Off Your Student Loans! An actuaries’ perspective geared towards the undergraduate
    Abstract:
      The heart of financial analysis is the application of the geometric series. With this mathematical tool, analysts are able to construct time value functions for valuation of annuities and perpetuities. These financial mechanisms derived from the geometric series are used to determine the repayment methods for student loans.

 

    December 04, 2009
    Speaker: Edwin Rogers ; Siena College, NY
    Title: Undergraduate Chaos
    Abstract:
      “Chaos Theory” has attained a certain level of visibility in popular culture. But what is “Chaos Theory” really? In this talk I will provide some answers to this question with some examples from elementary dynamical systems theory, using ideas accessible to anyone who has studied undergraduate calculus.

 

    November 06, 2009
    Speaker: Mark Steinberger ; SUNY, Albany
    Title: Topological equivalence of matrices plus very nice stipends for graduate school.
    Abstract:
      Let A and B be n x n matrices with real coefficients. Suppose in addition that some positive power of A and B is the identity matrix. We say that A and B are topologically similar (written A ~_t B ) if there is a continuous function f from R^n onto R^n, which is one-to-one with continuous inverse, such that f(Ax)=Bf(x) for all x in R^n. If f were a linear transformation, induced by the matrix P, then P is invertible and PAP^(-1)=B. In this case, A and B are linearly similar (written A ~ B ) which means that A and B differ by a change of basis. In 1935, de Rham conjectured that A ~_t B implies A ~ B. He was wrong. But the counterexamples are interesting, because if the function f were differentiable with differentiable inverse then A and B would be linearly similar. So the counterexamples display the difference between continuous and differentiable functions. Plus, we will describe new funding opportunities for the doctoral program in mathematics at the University at Albany.

 

    October 30, 2009
    Speaker: Scott Diehl ; Siena College, NY
    Title: "What Can Mathematics Teach Us About Dating?"
    Abstract:
      We take a mathematical approach to analyzing the game of dating and marriage. We pose and then answer the following questions: Is there a set of dating rules that society can follow so that all resulting marriages are relatively happy? Does "traditional" dating akin to the 1950's (where boys propose and girls choose) favor one gender over another? How can we apply these insights to our own personal dating life? The answers may surprise you!

 

    October 14, 2009
    Speaker: Joseph D’Avanzo ; Siena College, NY
    Title: Zeta[3] via hyperbolic functions
    Abstract:
      In 1736 Euler proved that the infinite sum of squares 1+1/2^2 + 1/3^2 + ... + 1/n^(2 ) + ... equals (Pi)^2/6. I will show you a modern way to prove this equality. What about the similar sum of cubes 1+1/2^3 + 1/3^3 + ... + 1/n^(3 ) + ... ? This question is an old mystery and we still do not know the "exact" value of this sum. I will present a plane region with area of (Pi)^2/6, and I will also present a 3D region, which has volume equal to the sum of cubes and discuss the similarities between these regions.
    Speaker: Nydia Negron; Siena College, NY
    Title: A Divisibility of Certain Integer Valued Functions
    Abstract:
      The greatest common divisor of a finite set of numbers is the largest number that will divide into every number in the finite set without a reminder. We will prove that if we have the outputs of certain integer valued functions (namely, polynomials and linear combinations of exponential functions) that the greatest common divisor could be found in finite time. For example, gcd{6,10,16,24, …} = gcd{x^2+x+4 | x is natural} = gcd{6,10) =2

 

    October 1, 2009
    Speaker: Michael Radin ; Rochester Institute of Technology, NY
    Title: Convergence of solutions and existence of multiple periodic solutions of a Non-Autonomous Rational Difference Equation
    Abstract:
      We investigate the convergence and the periodic nature of the positive solutions of a second order non-autonomous rational difference equation. We will analyze the similarities and differences when the periodic sequence is of even and odd orders. Also, we will examine how the relationship between the terms and the different arrangements of the terms can change the periods of the solutions of the difference equation.

 

SPRING 2009

    May 1, 2009
    Speakers: Brian McCourt, Allison Fazio, and Erin Ward ; Siena College, NY
    Title: Monkeys, Mods, and Remainders … Oh My
    Abstract:
      The Chinese Remainder Theorem was discovered in the early 300 century CE. Ancient mathematicians used it to solve problems with modular arithmetic. We will introduce the ancient and modern theorems with their corresponding proofs, discuss the implications in problem solving and solve some modular arithmetic problems.
    Speakers: Michael Cibiniak and Nicholas Noblett; Siena College, NY
    Title: A Prime Example
    Abstract:
      This talk will investigate primes, which can be written in the forms such as x^2+y^2, x^2+2y^2, and x^2+3y^2, for example 7 = 2^2+3•1^2. Specifically, it will be proved that there are infinitely many primes of the form 4n+1. The proof requires the use of Lagrange's lemma, Wilson's Theorem, and Fermat's Little Theorem, all of which are explained.

 

    April 20, 2009
    Speaker: Eric Heller ; Siena College, NY
    Title: The Genus of Regular Polyhedral Toroids
    Abstract:
      The familiar doughnut is an example of a torus of genus = 1. The number of holes a torus has is its genus. Imagine a polyhedron with a void in its center. Drill wholes through each face into the central void. Now you have a torus with many holes. What is its genus? We will answer this question by examining models and applying Euler’s Polyhedron Formula: V – E + F = 2 - 2g; V = number of vertices, E = number of edges, F = number of faces, g = the genus.

 

    April 8, 2009
    Speaker: Darren Lim ; Siena College, NY
    Title: Testing very large numbers for primality
    Abstract:
      In August and September of 2008, two very large numbers 2^(43112609) – 1 and 2^(37156667) – 1 were found to be prime numbers. These numbers, containing well over 10 million digits each, required special algorithms to prove their primality. This talk will take a look at prime numbers, with a special emphasis on Mersenne Primes. I will also discuss simple primality testing methods, including the Sieve of Eratosthenes and the Lucas-Lehmer test. Finally, I will discuss the GIMPS project, which has discovered the last 12 Mersenne Primes.

 

    April 1, 2009
    Speaker: Ryan Bakes ; Hudson Valley Community College, NY
    Title: Factoring Techniques
    Abstract:
      In this seminar, we will be discussing several techniques to factor general quadratics, binomials and some other special polynomials. The material is appropriate for mathematics educators who intend to teach at the high school or middle school level. If time permits, we will look into some theory on factoring polynomials in general.

 

    March 10, 2009
    Speaker: Robin Flatland ; Siena College, NY
    Title: Unfolding Polyhedra
    Abstract:
      An unfolding of a polyhedron is a set of cuts that can be applied to its surface so that it can be unfolded flat as a single piece without overlap. Often what is desired is an edge-unfolding in which the cuts are restricted to the polyhedron’s edges. For example, an edge-unfolding for a soccer ball shape is shown below. People have been interested in finding unfoldings since at least 1525 when painter and printmaker Albrecht Dürer published a book containing edge-unfoldings for many convex polyhedra. Today applications of polyhedra unfolding arise in manufacturing processes that construct three-dimensional objects by bending sheet metal. Very little, however, is known about what shape classes can and cannot be unfolded, with the biggest open question being whether or not all convex polyhedra have edge-unfoldings. In this talk I will give an introduction to polyhedra unfolding, discuss recent progress in unfolding orthogonal polyhedra, and present several easily accessible open problems.

 

    February 18, 2009
    Speakers: Nikolai Krylov and Edwin Rogers ; Siena College, NY
    Title: Dynamics of simple folds in a plane
    Abstract:
      Take a strip of paper whose two long edges are parallel, and fold in a crease intersecting these edges, creating two angles. Choose one edge and consider the angle between the crease and this edge. Now fold the opposite edge along the crease, creating a new crease that bisects the other angle. Fold again, this time using the newly created crease and the initial edge, creating a new angle along the chosen edge. It is well known that if this process is continued, the constructed angles along the chosen edge will approach a limiting value which is independent of the initial angle. In this talk, we explain why and generalize the result to when the edges are nonlinear or the strip lies in the hyperbolic plane.

 

FALL 2008

    November 21, 2008
    Speakers: Davide Cervone and William Zwicker; Union College, NY
    Title: Voting with Rubber Bands and Pulleys
    Abstract:
      Whoever gets the most votes wins, so voting is simple, right? Not quite. A variety of very different rules have been proposed for multicandidate voting. Some of these rules have a voter cast a ballot listing all candidates in descending order of preference. Rules that take account of a voter’s full ranking, rather than her first choice only, offer some advantages. Which rule we use certainly affects who wins the election, so the choice of rule is important. We’ll look at two rules: one dating back to the French revolution (or earlier) and another that was proposed very recently. Each of these rules can be defined in terms of machines that use weights and pulleys, or rubber bands, although that is not the way they were originally described.
      This mechanistic viewpoint gives us some insight into the different voting properties that might make us choose one rule over the other. That insight is enhanced when we play with the actual machines, in the form of interactive computer simulations of the rubber bands and the strings. If you'd like, bring your laptop to the talk, and we'll show you how to access the software on the web. This work represents collaboration between two members of the Union Mathematics Department: Davide Cervone (geometry and interactive web-based software) and William Zwicker (voting theory).

 

    November 06, 2008
    Speaker: Jonathan Brown; Dartmouth College, NH
    Title: Be careful what you vote for
    Abstract:
      Voting is a way to decide the preferences of a society, based on individual preferences. However, [as we've seen in recent elections] many people would contest the statement that the winner of an election is always the person who makes us happiest as a society. This brings up the question: how do we define who makes us "happiest as a society"? And if our elections don't always find this person, what would? Most democratic elections are held using some variety of a plurality vote, but there are other ways to hold elections and count votes. Are any of these other methods better at expressing the overall preference of a group of people, and can mathematics give us any insight into the best way to hold an election? In my talk we will discuss these questions and others in an attempt to find the best way to vote.

 

    October 15, 2008
    Speaker: Erik Mulvaney; Siena College, NY
    Title: Putting different colored balls into boxes
    Abstract:
      Suppose we have 3 blue balls and 2 red balls and we want to count the number of ways of placing these balls into 3 boxes. Suppose we want to count the number of ways of writing 72 as a product of 3 of its divisors. How many ways are there to do each of these events? Actually, counting these events is equivalent. I will discuss why these two seeming different events are the same and present a formula for count m blue balls and n red balls into 3 boxes.

 

    October 07, 2008
    Speaker: Don Hadwin; The University of New Hampshire, NH
    Title: The Cantor set and space-filling curves
    Abstract:
      The Cantor set is a very dusty set. You construct it by taking a line segment, removing the middle third of the line segment, removing the middle thirds of the remaining two line segments, and so on ... ad infinitum. A space-filling curve is a curve in the square that wiggles around "fills space", i.e. for any point you pick, you can find a wiggle of the curve that passes as close as you like to the point. One thing that both of these objects have in common is that they both challenge our usual notion of dimension. We will discuss these objects further in my talk.

 

    September 30, 2008
    Speaker: Brad Lowry; School of Dentistry University of Southern California (USC), CA
    Title: Career Advice For CS/IT/Math Types from a CS/IT/Math Type Problem Solver
    Abstract:
      If you enjoy solving problems and are good at it then you may be interested in this talk. People need problem solvers. If you enjoy working with people then you might be interested in this talk. People need people who can understand and work with others. If you enjoy learning cool new things then you may be interested in this talk. People need people who can learn new things. How do you set yourself apart with your Siena degree? How can you make a life and a living doing things you like to do? How do you decide which tools to master? How can you survive in the low budget real world? How do you balance depth and breadth of skills and knowledge? In this talk you may learn some answers to these questions.

 

    September 18, 2008
    Speaker: Darren Lim; Siena College, NY
    Title: The Strategies and Tactics of Set and Setgame
    Abstract:
      This lecture will look at a popular card game called Set. The object of Set is to identify a group of three Set cards whose individual properties are all identical or all different. Setgame is an online version of the Set card game, where players try to find 6 3-card sets from a group of twelve cards while being timed. During the talk, I will present provably correct strategies for finding these 3-card collections, which involve aspects of computer science, mathematics and psychology.

 

SPRING 2008

    April 30, 2008
    Speakers: Jessica Hackett, Kellianne Kiely, Emery McDonald, Andrew Warner ; Siena College, NY
    Title: Knots Undressed
    Abstract:
      Ever wonder about the bare essentials of knots? Come for a mathematical explanation on what they are, what they can do, and what never changes about them. We’ll have your brain knotted up in amazement!

 

    April 30, 2008
    Speaker: Bianca Pier ; Siena College, NY
    Title: Do some biological enzymes understand topology? Examining the “link” between DNA supercoiling and knot theory
    Abstract:
      DNA must satisfy several requirements and undergo numerous, intricate processes in order to serve its function as the recipe for life. However, DNA cannot go at it alone. Several important enzymes play a role in maintaining and interpreting this message as well. It just so happens that many of these enzymes “understand” a division of topology called knot theory. These enzymes are responsible for knotting and unknotting, linking and unlinking, supercoiling and unsupercoiling the mess of DNA that is found in a living organism. Such a phenomenon encourages the collaboration of topologists and biologists toward understanding the function of these particular enzymes and opens the door to a new and exciting area of research which can produce knowledge about the mysteries of life.

 

    April 16, 2008
    Speaker: Vladimir Chernov ; Dartmouth College, NH
    Title: Linking Invariants of Fronts and Causality
    Abstract:
      A wave front at time t is a set of points that the light signal from a source has reached at time t. Arnold observed that one can associate a knot to a propagating front and that the knot type associated to a propagating front does not change during the propagation. Thus fronts associated to different knots can not change one into the other with time regardless of the geometrical shape of the ambient space. To study this relation he introduced his invariants of planar fronts. Low made similar observations relating linking to causality, i.e. to detecting if the information from one event could have reached the place of the other event, before the other event happened. We discuss the generalizations of Arnold's invariants to fronts on surfaces obtained independently by Inshakov and the speaker. We also discuss the generalized linking number constructed jointly with Rudyak and its applications to causality.

 

    April 09, 2008
    Speaker: Brandt Kronholm ; University of Albany, NY
    Title: Ramanujan-Like Congruence Properties of the restricted partition function p(n,m)
    Abstract:
      p(n) is the function that tells us how many ways we can add up positive whole numbers to total n. How many ways are there to add up positive integers to get 4? Let’s see, there is 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. So the answer is p(4) = 5. Or, you can just plug 4 in for n in the partition formula. But, would you believe that starting at 4 and counting by five (4, 9, 14, 19, 24, etc…) that the number of partitions is a multiple of 5? It’s true! Ninety years ago the great genius Ramanujan noticed (and proved) this and several other similar divisibility patterns in the partition function which can be stated as follows:
      p(5n+4) = 0 (mod 5)
      p(7n+5) = 0 (mod 7)
      p(11n+6) = 0 (mod 11).
      This talk will focus on families of Ramanujan-like congruences for the partition function p(n,m), the function which enumerates the partitions of a non-negative integer n into exactly m parts without ever using anything like the big scary equation. For example:
      p(60k, 5) = 0 (mod 5)
      p(420k, 7) = 0 (mod 7)
      p(420k + 406, 7) = 0 (mod 7)
      p(60k + 35, 5) = p(60k + 20, 5) (mod 5).

 

    April 02, 2008
    Speaker: Nicole Greaney; RPI, NY
    Title: Topologically Voronoi Nets
    Abstract:
      Voronoi diagrams provide a useful partition of space and have a wide variety of applications. In this talk, we will define what a Voronoi diagram is, its basic topological properties, and many of these applications. We will discuss some of the algorithms used to calculate Voronoi Diagrams and the efficiency of those algorithms. We will then discuss Voronoi Nets, which essentially "come from" Voronoi Diagrams, and see that for a finite set of sites, n, there exist a finite number of topologically distinct Voronoi Nets that can be generated by that set of sites. All nets have been found for sites n = 2..6. Thus, we will look at the catalog of distinct Voronoi Nets beginning with n = 7 sites.

 

    April 02, 2008
    Speaker: Csilla Szabo; RPI, NY
    Title: Do the Ends Justify the Lengths?
    Abstract:
      Actin polymers form the cytoskeleton of cells, giving cells structural support and allowing for cell motility. They are also instrumental in axon (nerve cell) growth, where they determine the directions taken by the axons as they grow. Models for the growth of actin polymers are based on adding and/or deleting actin monomers from the ends of the polymer chains. Consequently, it is straightforward to track the ends of these actin polymers. A natural question to ask is given the probabilistic distribution of the growing and decaying positions of the ends of the polymers, does one know the lengths? I will present a 1-dimensional model of actin polymerization, which will include diffusion of monomers and determination of length distribution based on the maximum entropy principle.

 

    March 12, 2008
    Speaker: Emelie Kenney; Siena College, NY
    Title: Babylonian Triples and Pairs of Equations: Origins of Algebra in Antiquity
    Abstract:
      Some historians of mathematics argue that algebra evolved from the ancient Babylonian practice of what is called numerical algebra, while others claim that Babylonian mathematics should more properly be considered geometric algebra. Which claim is more reasonable? Is Babylonian mathematics truly algebra? In this talk we address what may be considered the contributions of these early thinkers to the development of algebra, showing both numerical and geometric features of their mathematics. We focus especially on their problem-solving, highlighting what are called Babylonian triples and their relationship to Pythagorean triples. We conclude with some related, but very modern, problems, at least one of which is still unsolved.

 

    February 20, 2008
    Speaker: Edward Burger; Williams College, MA
    Title: Coincidence, Chaos, and All That Math Jazz
    Abstract:
      In this presentation, as we honor the new inductees into Pi Mu Epsilon, we celebrate the power of looking at the world through the lens of mathematical thinking. Here we will highlight some of the true life lessons that mathematics has to offer all of us--both within our math studies as well as far beyond into our everyday lives.

 

    February 06, 2008
    Speaker: Brenda Johnson; Union College, NY
    Title: Searching for Missing Links: the Evolution of Some Undergraduate Research Projects
    Abstract:
      Consider a computer system with six components, each of which has to be connected to each of the others by a cable or wire. The result is a mess, but how messy is it? The mathematical analog of this set up is called a spatial embedding of the complete graph on six vertices. We’ll look at a theorem that establishes the minimum degree of messiness in this situation, and the undergraduate research on knots and links in graphs that it has inspired.

 

FALL 2007

    December 06, 2007
    Speaker: Jason Irwin, Erin Keys, Brittany Lindhorst; Siena College, NY
    Title: Permutation Groups and Conjugacy Classes

    Speaker: Jacqueline Busching, Daryl Doty, Kelly Lawrence; Siena College, NY
    Title: Avoiding the Identity

    Speaker: Jennifer Roberts and Celeste Sisson; Siena College, NY
    Title: Groups as Unions of Their Proper Subgroups

    Speaker: Jeffrey Dujardin, Amy Gadziala, Erin Guldenstern, Nicholas Orlando ; Siena College, NY
    Title: Isomorphism, Equality, Apples and Trees

 

    November 30, 2007
    Speaker: Gove Effinger; Skidmore College, NY
    Title: Encryption, Asymmetry, and Prime Numbers
    Abstract:
      In the 1970's Diffie and Hellman had an idea for what they called "public-key" cryptography which hinged on the existence of functions which are asymmetric in the sense that they are easy to compute but hard to invert. Subsequently Rivest, Shamir, and Adelman introduced a specific implementation of this idea which exploits the asymmetry in multiplication versus factorization. We explore the elegant elementary mathematics which makes "RSA" encryption work and illustrate the algorithm using Mathematica.

 

    November 09, 2007
    Speaker: Alice Dean; Skidmore College, NY
    Title: Visibility Graphs: A story of faculty and student collaboration over 10+ years
    Abstract:
      Sometimes people say that mathematics and computer science are not well suited to collaborative research by faculty and undergraduate students. This talk contradicts that assumption, by describing two research projects in graph theory, one involving an undergraduate at Skidmore College, and the other involving an undergraduate at St. Michael's College. The questions studied concern geometric representations of graphs, using unit bars or squares in the plane for vertices and visibility for edges. The talk will look at several results, including in particular which trees can be represented in this way. Prof. Dean will also talk about current research that has grown out of these two projects with undergraduates.

 

    November 01, 2007
    Speaker: Junsheng Fang; University of New Hampshire, NH
    Title: Rational, Irrational, Algebraic and Transcendental Numbers
    Abstract:
      We define and give characterizations and examples of all of the types of numbers in the title. We prove, using only concepts from high school algebra, a Theorem of Liouville and use it to construct transcendental numbers. With the exception of an infinite series to prove e is irrational, we only use very elementary ideas.

 

    September 28, 2007
    Speaker: Emelie Kenney; Siena College, NY
    Title: Color my world: some open questions in graph coloring theory
    Abstract:
      Graph coloring theory is a subfield of combinatorics, or combinatorial analysis, which sounds exotic, but is simply about counting. Although graph coloring theory can be applied to scheduling and other problems, the beauty and elegance of its ideas make it worthwhile in its own right. In this talk, which assumes no familiarity with graph coloring theory and, indeed, very little mathematical knowledge in general, we present some open questions in the field that are deceptively easy to state, but devilishly difficult to solve.

 

    September 14, 2007
    Speaker: Kristin Farwell; Siena College, NY
    Title: Sudoku Variants and Other Logic Puzzles
    Abstract:
      Are you addicted to Sudokus? Are you sick of Sudokus and are looking for another puzzle to be addicted to? Well, you are in luck. There are numerous variations on Sudoku and logic puzzles. I will present different types of puzzles and various solution techniques on these variations.

 

SPRING 2007

    May 07, 2007
    Speakers: Meghan Davey, Madeline Gialanella, and Jaclyn Iraci; Siena College, NY
    Title: "Mirror, Mirror on the Wall, Which Knot is Fairest of Them All?"
    Abstract:
      In this talk we will discuss what a knot is, what a mirror knot is, and how to deform knots to look like “different” knots. We will show how one could see if some knots are equivalent visually and mathematically. The talk will focus on whether all knots are equivalent to their mirror image. In particular, we will present a polynomial to distinguish one knot from another. The main ideas of the talk can be understood by any undergraduate student with exposure to math.

 

    May 04, 2007
    Speaker: Sue Hurley; Siena College, NY
    Title: Geometry meets Algebra; Vector Space and Field Properties of the Regular Polygon.
    Abstract:
      The diagonal lengths in a regular polygon generate a vector space (with rational scalars) that is also a field. This talk explains how the arithmetic of the field is apparent in the geometry of the polygon and in two families of related matrices.

 

    April 13, 2007
    Speaker: Alexander J. LaPoint; Siena College, NY
    Title: To Be or Not To Be? Let us Try and Answer the Question
    Abstract:
      Have you ever watched a play or movie and tried to guess what the characters will do next? By applying the mathematical techniques of game theory to the decisions characters make in the theatrical text we will try and find a mathematical structure within these decisions. Through the use of different types of game theory analysis, guessing what your favorite character’s next move will be might not be so hard.
    Speaker: Matthew Farrelly; Siena College, NY
    Title: The General Burnside Problem in Exponent 3
    Abstract:
      In 1902 William Burnside raised the question of whether a finitely generated group must be finite if each of its elements has order dividing a natural number n, called the exponent of the group. Along with this question Burnside provided cases in which it had an affirmative answer, namely for any group of exponent 2 or 3 and for all groups of exponent 4 that can be generated by 2 elements. This talk will handle the case of groups of exponent 3. Here is the exact question that I will be answering: Suppose a, b are elements in a group G and that every element can be written as a product of a, b, and their inverses and further assume for every element g in G, we know that g^3 = e. What is the order of G?

 

    March 30, 2007
    Speaker: Thomas Rousseau; Siena College, NY
    Title: An ocean front is not the beach
    Abstract:
      I will describe several interesting physical features found in the oceans, seas, and other estuarial objects. I will describe in detail the region of an ocean that is called front. There being no closed-form function that describes an oceanic front, I will develop a simple model that will be used to analyze the propagation of sound through the region that is called an oceanic front. The influence of the front on the travel time of a signal and the total field amplitude of the received sound will be discussed. The effect of a front the perceived azimuth of the source from the receiver will be discussed. Mathematics studied in Calculus II and Calculus III suffices to perform all calculations. I will mention the role of linear algebra and of number theory in the development of my model.

 

    March 09, 2007
    Speaker: Mingchu Gao; University of Illinois at Urbana-Champaign, IL
    Title: Mathematical models in daily life
    Abstract:
      I’d like to present mathematical models in daily life. I first introduce a strategy for mathematical modeling. 1. Set up a mathematical model. 2. Solve the mathematical problem. 3. Apply the mathematical answer(s) to the original question. Then I will present three examples: 1. Complementary coffee cups. Suppose there are two coffee cups, one convex and the other concave, such that their profile curves fit together. Question: Which cup can hold more coffee? 2. Suppose that a bug is climbing up the inner wall of a bowl while water is poured into the bowl at a constant speed. Question: How fast should the bug crawl upward to keep its feet dry? 3. How can we determine a mathematical model for traffic flow in order to prevent traffic jams?

 

    February 16, 2007
    Speaker: Jim Matthews; Siena College, NY
    Title: Cracking the case of the twenty cases problem
    Abstract:
      In this talk we will pose a version of a search problem for succinct data structures that appeared in a paper written by Peter Bro Miltersen and Anna Gál which won the best paper award at the 30th International Colloquium on Automata, Languages and Programming in 2003. We plan on discussing several solutions to the problem with a goal of determining an optimal strategy for success. We will make good use of material from undergraduate courses on computer programming, calculus and probability in solving and analyzing the Twenty Cases Problem and its more general solutions. Note: The main ideas of the talk are accessible to the general public and the searching solutions that will be discussed can be carried out by average middle school level students. The applications of the college level material to this problem are elegant and may be surprising.

 

    February 02, 2007
    Speaker: Darren Lim; Siena College, NY
    Title: The Educational Value of Boardgames
    Abstract:
      This talk will present a wide assortment of boardgames, and how they can be used within the context of a classroom. In particular, we will discuss the use of a particular board game as a device for introducing graph theory. By experiencing a hands-on demonstration of graphs, students can better understand the basic principles of graph theory and can better design algorithms and programs which manipulate graph data.

 

FALL 2006

    December 08, 2006
    Speaker: Alan Wiggins; Texas A&M University, TX
    Title: Constructing Numbers with Straightedge and Compass

 

    November 09, 2006
    Speaker: Peter N. Wong; Bates College, ME
    Title: The three B’s in fixed point theory
    Abstract:
      Through topological fixed point theory, the foundation of algebraic topology was laid during the first half of the twentieth century. In this talk, I will present some of the classical results and their (elementary) combinatorial analogs that were discovered by the early pioneers in the field.

 

    October 20, 2006
    Speaker: David Vella; Skidmore College, NY
    Title: The Higher Chain Rule and Composite Generating Functions
    Abstract:
      In this talk we pose the question of how to generalize the chain rule of calculus to higher derivatives. It turns out the question has a 150 year old answer which involves a sum over the partitions of an integer. We add a new twist however, by applying the formula in a novel way to create a sort of machine for generating and proving many identities involving Stirling numbers, Bell numbers, Bernoulli numbers and Euler numbers.

 

    October 17, 2006
    Speaker: David V. Feldman; University of New Hampshire, NH
    Title: Sonar and Radon Transforms
    Abstract:
      We introduce Sonar and Radon transforms, and establish new relations which connect them. The talk will be suitable for juniors and seniors.

 

    October 17, 2006
    Speaker: David V. Feldman; University of New Hampshire, NH
    Title: Computing with Continued Fractions
    Abstract:
      How can you add or multiply two continued fractions without passing to their decimal expansions first? A famous book claimed that this couldn’t be done, but it can, and the algorithm is simple enough for a freshman to understand.

 

    September 27, 2006
    Speaker: Mark Rosenberry; Siena College, NY
    Title: An Introduction to Quantum Cryptography
    Abstract:
      How secure is the credit card number you submit on a website? How secure will it be when quantum computation becomes a mature technology? This talk will explain the basis of public key cryptography in terms of modular arithmetic and the factoring of large numbers into component primes. It will also cover how the laws of quantum mechanics lead to a special kind of computational device, a quantum computer, that is capable of solving certain kinds of problems much faster than its classical analogue. Several possibilities for physical implementation of a quantum computer will be mentioned. Finally, a new way to generate secure transmissions using a quantum computer will highlighted.

 

SPRING 2006

    May 03, 2006
    Speakers: Jeffrey Avila, Linda Braham, Christine Kawczak, and Kristin Prochilo; Siena College, NY
    Title: Games on the Torus and Klein Bottle
    Abstract:
      We will illustrate topological differences between the torus and Klein bottle by studying various strategies of Tic-Tac-Toe and Chess on each surface.

 

    April 27, 2006
    Speakers: Kerry McKnight, Daniel Lomanto, Matthew Sayles, and David Smith; Siena College, NY
    Title: On the Cantor Set and Various Dimensions
    Abstract:
      We will define the Cantor set and outline a few fascinating properties of this basic fractal including the dimensional property of having length zero and the same number of points as the whole real line. We will also formally introduce the topological and fractal dimensions and illustrate these concepts with examples (e.g. Sierpinski triangle).

 

    April 24, 2006
    Speaker: Dierk Schleicher; International University Bremen, Germany
    Title: On Newton’s method for root finding and chaos, and how to really find roots of complex polynomials
    Abstract:
      Newton's method, as old as calculus itself, is in wide use in order to find roots of differentiable functions. However, it is surprisingly complicated to understand even when finding roots of polynomials. We will try to explain why, using a combination of mathematical theory and computer pictures. Even though this method is very old, it is still necessary and possible to prove new mathematical results about it. We will give some examples. We try to make this talk accessible to curious undergraduates.

 

    March 29, 2006
    Speaker: David Vella; Skidmore College, NY
    Title: Symmetry, Group Actions, & Euclidean Geometry
    Abstract:
      I begin with a simple theorem about quadrilaterals and observe that while it is easy to give both synthetic and analytic proofs of this theorem, neither approach explains what happens when we try to generalize the theorem to other polygons, nor what happens when we try to 'invert' the theorem. However, by paying attention to certain symmetry in the problem and taking a transformational (group-theoretic) approach, all of the generalizations and inversions become crystal clear.

 

    March 22, 2006
    Speaker: Sergey Solodukhin; International University Bremen, Germany
    Title: Mysteries of Black Holes
    Abstract:
      I will review the development which led to understanding that black holes behave like thermodynamical systems. They have mass, temperature and entropy and satisfy first and second laws of thermodynamics. Although classically black holes are absolutely dark objects, they radiate particles quantum mechanically and, thus, evaporate. It remains a great mystery what is the statistical origin of the huge entropy of a black hole and what is going to happen to the information that was ever put into a black hole if the black hole evaporates completely. I will discuss some recently suggested approaches which may give us answers to some of these questions.

 

    March 14, 2006
    Speaker: Ryan Decker; Siena College, NY
    Title: Uniqueness and the Doubling Operator
    Abstract:
      What types of functions f give us a unique solution to the differential equation f'(x)=2f(2x)? We show that if f is analytic the solution is unique, any finite degree of smoothness isn't enough to get the solution to be unique.
    Speaker: Andrey Taran; Siena College, NY
    Title: A Taste of Dynamical Systems
    Abstract:
      In this talk, we introduce basic ideas of discrete dynamical systems. In particular, we explore the period doubling route to chaos.

 

    March 01, 2006
    Speaker: Kristin Farwell; Siena College, NY
    Title: Finding the closed path of minimal length within a regular tetrahedron
    Abstract:
      “Indiana Junebug” was the most famous explorer of pyramids among the insects. He was always having wild adventures. However he did not always have adventures in pyramids. Unknown to most people, he actually started out exploring regular tetrahedrons when he was a small bug. The pyramids and tetrahedrons always had one only entrance so he always had go back to the starting point to finish his adventure. Inside this regular tetrahedron, he discovered that he could touch every face of the tetrahedron by starting at the top of the tetrahedron, going down to the bottom and then back up. So he used this same idea when he started exploring pyramids. Now Indiana Junebug brags to all of the ladybugs that he has touched every face of each pyramid. Is Indiana Junebug’s approach the shortest way of touching each face? What are some other options? Maybe you should touch each centroid? What about flying from the midpoint of one edge to the midpoint of the opposite edge? Come and find out more about the adventures of Indiana Junebug.

 

    February 10, 2006
    Speaker: Nikolai A. Krylov; Siena College, NY
    Title: What is a real number?
    Abstract:
      If we can trust legends, Pythagoreans believed that everything was expressible in terms of whole numbers. This belief was so strong that the discovery of line segments whose ratios are incommensurable is said to have caused a great shock in the community. Nowadays we are not surprised when somebody says that v2 is not a rational number. We know lots of other irrational numbers, but how to describe the set of all such numbers? Is it that simple? It turns out that many founders of modern mathematics, such as Bachmann, Bolzano, Cantor, Cauchy, and Weierstrass have worked on the foundations of the real number system, and it was finally Richard Dedekind (1831-1916) who constructed the set of all Real numbers from the Rational numbers with exemplary clarity using our geometrical intuition of the continuum. In this talk I will follow the basics of this construction to define the Real numbers as the Dedekind cuts.