Mathematics
Contact
Department Chair:
Susan Jane Colley

Department Email:


Phone: (440) 775-8388
Fax: (440) 775-6638

Location:
10 N. Professor St/King 205

Oberlin, OH, 44074

Spring-2013

Spring-2013
 

February 13 - Lecture
"A Short History of Length"
Professor Joel Langer -- Case Western Reserve University
4:30pm in King 239

   A handy old device called a waywiser–basically a wheel and axle mounted
on a handle–may be used to measure the length of a path, straight or curved.
It works well enough in practice––but does it also work in theory?
   The notion of length itself weaves like a thread through two millennia of
mathematical history:  From the Pythagorean theorem and constructions
by ruler and compass to the proof, in 1882, that Π is transcendental: from
euclidean to non-euclidean geometry and curved spacetime.  The theory of
length is yet a work in progress.
   In this talk I intend to lead a visual tour of mathematical thought from
ancient to modern times and to illustrate the diverse and far-reaching
consequences of seemingly simple ideas.  I hope to demonstrate that,
in the hands of Archimedes, Descartes or Gauss, a waywiser can be much
more than just a wheel on a stick.

 


 

February 14 - Student/Faculty Pizza Luncheon
Is Algebra Necessary?   A Discussion

12:15 in Wilder 112

Last July, Andrew Hacker wrote a controversial opinion piece in the  New York Times
suggesting that algebr be made an elective subject in school.  Take a look at the article
(see http://nyti.ms/OmuCla) and join us for pizza to discuss it and related matters.



March 7 - Lecture
Do Dogs Know Calculus?
Timothy Pennings and Elvis
4:30 pm in
Craig Lecture Hall

A standard calculus problem is to find the quickest path from a point on shore to a point in the lake,
given that running speed is greater than swimming speed. Elvis, my Welsh Corgi, has never had a
calculus course. But when we play “fetch” at Lake Michigan, he appears to choose paths close to
the calculus answer.  In this talk we reveal what was found when we experimentally tested this ability.
Elvis will be available for follow-up questions.

 


 

March 12 - Lecture
"Braids, Links and the Alexander Polynomial"
Professor Craig Jackson -- Ohio Wesleyan University
4:30pm in King 239

Braids groups are interesting objects that straddle the divide between algebra and topology. In particular,
braids are intimately related to knots and links via the operation of braid closure.
We discuss how topological invariants of knots and links, such as the Alexander polynomial, arise from
representations of braids as groups of matrices.

 



March 14 - Student/Faculty Pizza Luncheon
Mathematical Modeling of Climate
Jim Walsh
12:15 in Wilder 112

  Mathematics is playing an ever expanding role in the study of climate.  We present
a few of the simplest energy balance models of large-scale features of climate. We will
indicate the nature of a recently resolved mathematical problem which arose
in the study of a well known 1-demensional climate model.
 



March 20 -- Distinguished Visitor Lecture
"Playing Pool on Curved Surfaces and the Wrong Way to Add Fractions"
William Goldman -- University of Maryland
7:30 pm -- King 306


Mathematics is at once a science, an art and a language.  This gives mathematics a unique nature that makes it difficult to exactly pinpoint what this discipline really is.  In this lecture I will try to capture some of the excitement and fascination of mathematics, through examples relating the interplay between three fields of mathematics:  arithmetic (number theory), geometry and dynamical systems.


 

April 11 - Student/Faculty Pizza Luncheon
"
Can You Tie a Knot in 4-Space"
Jack Calcut -- Department of Mathematics
12:15 in King 203


Knotted ropes, power cords, and hoses commonly occur in 3-space.  What happens in 4-space?  5-space?
We'll untie loops in a 4-space, then we'll tie a sphere in a knot. 

 


April 18 -- Lecture
Recovering Information from Noisy Measurements
Alex Powell, Honors Examiner -- Vanderbilt University
4:30 pm in King 239

Estimation theory deals with the general problem of recovering information from
noisy measurements.  We will talk about estimation problems involving bounded
noise and will focus on a particular approach called consistent reconstruction.
Consistent reconstruction can be expressed as a linear programming problem and
has, for example, been used by electrical engineers for analog-to-digital conversion.
The mathematical analysis of consistent reconstruction turns out to be closely
related to questions in stochastic geometry that involve random polytopes and
random coverage problems.  We shall sketch some ideas in the error analysis of
consistent reconstruction and will also present alternative methods related to
randomized algorithms.
 


 

April 25 -- Lecture
Partition Dynamics: Avalanches and Bulgarian Solitaire
Brian Hopkins -- St. Peter's University
4:30 pm in King 239


We will explore the impact of ideas from dynamical systems on the classical topic of integer partitions.  Given an operation on partitions, the partitions of a fixed integer can be thought of as a finite dynamical system.  There are three natural questions: How many components are in the system?  Which partitions are fixed points or in cycles?  Which partitions are Garden of Eden states, with no predecessor?  We will see how combinatorial proofs and generating functions can help in analyzing Bulgarian Solitaire popularized by Martin Gardner and various sand pile model examples of "self-organized criticality" in theoretical physics.



 
April 29 -- Honors Lecture
"Ramsey Numbers for Cycles: Finding Order in Randomness"
Claire Djang
4:30 pm in King 239

We begin with an introduction to Ramsey theory. A complete graph, denoted by Kn, consists of n vertices and has the property that every pair of vertices in the graph is connected by an edge. Consider coloring each edge of a Kn graph either red or blue. How large must n be in order to guarantee the existence of certain red or blue subgraphs, no matter how the graph’s edges are colored? Using simple proof techniques, deductive reasoning, and visual constructions, we determine exact answers to this question when the specified subgraphs are small. In particular, we focus on the case when the subgraphs under consideration are cycles, and illuminate parts of a relatively recent proof that solves this problem for pairs of cycles of any finite length.
 



May 1 -- Honors Lecture
"Intersections of Curves and Bézout's Theorem, with an Application to Classical Geometry"
Margaret Nichols
4:30 pm in King 239

In algebraic geometry, seemingly geometric problems can be solved using algebraic techniques. Some of the most basic geometric objects we can study are polynomial curves in the plane. Even in this familiar setting, simple questions to state can be difficult to answer. In this talk we address a natural first question: how many times do two curves intersect? Counting the intersections can quickly become complicated when we take into account singularities and multiple intersections. Despite this, Bézout's Theorem gives a beautifully concise answer. We develop the intersection number of two curves at a point, the key ingredient to Bézout's Theorem. This concept, although geometrically motivated, can be described in algebraic terms; it is this relationship that makes it such a powerful tool. Lastly, we illustrate the usefulness of the intersection number through quick and elegant proofs to two theorems from classical geometry: Pappus' Theorem and Pascal's Theorem.

 


 

May 2 - Student/Faculty Pizza Luncheon
"Donald in Mathmagic Land"

12:15 in King 203

 


May 2 -- Honors Lecture
"Incentive Theory and the Principal-Agent Problems"
James Foust
4:30 pm in King 239


How can an employee’s compensation be structured so that their incentives are in line with the incentives of their employer, even when the employer can’t perfectly observe the actions taken by the employee? This question lies at the heart of the problem of Moral Hazard, one type of principal-agent problem. How should a consumer decide whether or not to purchase a used car when they are unsure if it is a lemon? This question lies at the heart of the problem of Adverse Selection, another type of principal-agent problem. In this talk, I will address different solutions to these problems, and the circumstances under which each solution is optimal.


 

May 7 -- Honors Lecture
"Connected Sum and Infinite Swindles"
Patrick
Haggerty
4:30 pm in King 239


An infinite swindle is a proof technique that involves reassociating an infinite sum.  This is the technique used to evaluate telescoping series.  Infinite swindles are useful in many areas of mathematics for various definitions of "sum".  In topology, sum is often taken to be connected sum, which is roughly the operation of connecting two manifolds via a tube.  In this talk, we explore two examples of infinite swindles with connected sum.  Our first example involves knots; we will show that the connected sum of nontrivial knots cannot be unknotted.  Our main example is the Generalized Schoenflies Theorem.  The Classical Schoenflies Theorem says that a circle drawn on a sphere divides the sphere into two disks.  The Generalized Schoenflies Theorem essentially asserts the analogue in higher dimensions, although there are some potential complications such as horned spheres.  To prove the Generalized Schoenflies Theorem, we will use the infinite swindle technique of Barry Mazur - a technique which earned him the Veblen Prize in 1966.

 



May 9 -- Honors Lecture
Was the Earth Entirely Covered by Glaciers?  A Mathematical Investigation of "Snowball Earth".
Chris Rackauckas
4:30 pm in King 239


The geological and paleomagnetic record indicate that around 750 million and 580 millions years ago glaciers flowed into the ocean near the equator; as of yet we do not fully understand the nature of these glaciations. The well-known Snowball Earth Hypothesis states that the Earth was covered entirely by glaciers. However, this hypothesis has difficulty accounting for certain aspects of the biological evidence, such as the survival of photosynthetic eukaryotes. Thus the Jormungand Hypothesis was developed as an alternative to the Snowball Earth Hypothesis. In this talk I investigate previous models of the Jormungand state and look at the dynamics of the Hadley cells to develop a new model to represent the Jormungand Hypothesis. We end by solving for an analytical approximation to the model using a finite Legendre expansion and geometric singular perturbation theory. The resultant model gives a stable equilibrium point near the equator with strong hysteresis that satisfies the Jormungand Hypothesis.