Professor Wesley Calvert of SIU speaks on Proofs

Professor Wesley Calvert spoke on the topic 'How is a Proof Like a Function' at the Department Colloquium on April 1. No really he did! The abstract follows:

Mathematics seems to have quite a lot to do with functions; see how many mathematics courses are, implicitly, or explicitly, about functions. We mathematicians are also rather fond of proofs.

Proofs are the gold standard of what it means to have conclusively solved a mathematical problem. More recently, mathematicians have also been very interested in computers.

These three things (just a few of our favorite things) — functions, proofs, and computer programs — are related in very deep ways. In a sense, we can think of them as different forms of the same objects. I’ll tell you how this is so.

Gordon Discusses The Continuum Hypothesis.

Professor Evgeny Gordon spoke in the department colloquium on February 18th and 25-th the abstract of the talk follows:

L.V. Kantorovich, who is mostly known for the discovery of Linear Programming, was also an outstanding expert in functional analysis. He introduced and investigated conditionally order
complete vector lattices that are known in Russian mathematical literature as Kantorovich's spaces (K-spaces).

He came to these spaces in early 30's guided by the idea to find vector spaces that have as many properties of the field of real numbers as possible. At that time this problem couldn't be written in formal mathematical terms.

In the middle of 60's P. Cohen proved the independence of CH. The method of forcing that he developed for this problem was later rewritten in terms of Boolean-valued models of set theory by D. Scott and R. Solovay. In the middle of 70's I proved that Boolean-valued models of the field of real numbers are exactly the universal K-spaces. This fact did not only give a rigorous mathematical formulation to the problem of Kantorovich, but also allowed to transfer many properties of real numbers to K-spaces. In particular, it allowed generalizing a lot of theorems about linear functionals on operators with the values in Kantorovich spaces.

Many results in this area were obtained by functional analysts from Novosibirsk.

Galperin Talks on the Dynamics of Continued fractions

Professor Gregory Galperin spoke on the dynamics of continued fractions in the departmental colloquium on March 4th. Professor Galperin discussed various results due to Gauss, Kuzman, Bykovskij and Arnold.

Galperin's Talk on Invisible Objects

Gregory Galperin recently gave a talk in the colloquium series about invisible objects. These objects are typically invisible in one direction in the sense that light flows around the object and not through it. We give here an example of such an invisible object.





Can you show that light from infinity (parallel rays) flow around the object as shown if the base angles of each isosceles triangle is 30 degrees?

Gordon to Speak on the Continuum Hypothesis

From the abstract:

In the middle of 60's P. Cohen proved the independence of CH. The method of forcing that he developed for this problem was later rewritten in terms of Boolean-valued models of set theory by D. Scott and R. Solovay. In the middle of 70's I proved that Boolean-valued models of the field of real numbers are exactly the universal K-spaces. This fact did not only give a rigorous mathematical formulation to the problem of Kantorovich, but also allowed to transfer many properties of real numbers to K-spaces. In particular, it allowed generalizing a lot of theorems about linear functionals on operators with the values in Kantorovich spaces.

Many results in this area were obtained by functional analysts from Novosibirsk. However, this method did not become popular among specialists in analysis, since it requires a deep knowledge of mathematical logic, especially of the axiomatic set theory.

Recently, I found an exposition of this method that seems to me quite accessible for non- specialists in mathematical logic. I will try to explain the basic ideas of this method (including the idea of the independence proof of CH) in two consecutive talks.

Galperin to Give Feb 4 Colloquium

Gregory Galperin is slated to give the Friday colloquium this week. His topic is

A Geometric Problem That Leads to the Billiard Law of Reflection

Abstract:I will show some examples of "invisible geometric objects." This means that light bends around an object, causing it to become as it were invisible. The Billiard Law of Reflection, which states that the angle of reflection equals the angle of incidence, will play a crucial role. The main examples will concern billiard reflection for ellipses and hyperbolas.

Maxwell's House

There once was the physicist Maxwell
whose theories of light didn't wax well
lo, equations he found
with a d little round
by checking his E & M facts well

Newton's Fall

There once was a fellow named Newton
who sat neath a pear tree bout noon-time
his head was then hit
by the fruit not the pit
which forced him to gravitate sublime

More grafitti

A Supertask Puzzle

A supertask is a countably infinite sequence of tasks to be performed in order, say

{T₁,T₂,T₃,...}

such that we first perform task T₁ and then task T₂ and so forth.

As an example one might consider the case of Zeno's Paradox where an arrow is shot at some target. The arrows first task is to complete the journey half way to the target. The second task is to complete the remaining journey half way to the target and so on.

Now consider the case of Jon Perez Laraudogoitia's beautiful supertask: There are a sequence of unit point mass balls stationary at x=1/2,x=1/4,x=1/8 and so on. A unit point mass at x=1 and time t=0 is moving with a velocity of 1 unit per second to the left (i.e. toward x=1/2).

We assume that there is an elastic collision at x=1/2 and so the moving ball is left stationary at x=1/2 and the ball at x=1/2 moves with velocity 1 unit per second in the direction of the ball at x=1/4. It is clear that at t=1 all the balls have been hit by the preceding ball.

Assuming that all the collisions are elastic and behave in exactly the same manner as the first collision, is there a ball that passes through x=0?

Sage Advice to be Given Jan. 14th

Patrick Coulton will give the first colloquium talk of the spring semester Friday January 14th. The topic will be "An introduction to using Sage".

Sage is a computer algebra system which is free and open source.
The talk will center on applications to the college classroom.

See the department website for more details.

A Philosophical Conudrum

If a tree falls into a black hole does it make a sound?

News from the Mathematics Castle

The new semester (spring) has begun. Classes started January 10th.
Andrew Mertz and family provided treats in the department lounge
to help start us off on the right foot.

Peter Andrews, Sylvia Carlisle and William Green attended the
joint AMS/MAA conference in New Orleans.

Peter was particularly disappointed in the beautiful weather
since the weather afforded no opportunity to play hockey.

The department will host a low dimensional topology conference on
March 26th. See the department website for more details.

Semester Update (News from Old Main)

The semester has flown by and like the proverbial bee we have been busy.
The Department has been sadden with the loss of our one time Chair
Alphonso (a.k.a Dr. D) DiPietro and Llyod Koontz.

We are now in the midst of Final Exams and we look forward to a
little break from the classroom routine.

The department held its holiday party on Friday December 10th after classes. Some 7 loaves of bread were consummed as well as a variety of treats.

The Comerford's hosted a holiday party for the math majors this past Saturday
December 11th. The highlight of the party was a game of departmental trivial pursuit.

The colloquium committee hosted several talks this semester including two lectures from outside the department ending with a delightful talk on boids and other animation curiousities by Professors Van Cleave and Mertz.

Once again the departments majors participated in the Putnam Exam December 4th. The exam preparation is headed by Keith Walcott.

Fall Semester

Well, the fall semester is upon us which leaves us wondering what happened to all those lovely 100 degree August days with 90 per cent humidity. The temperatures are down and the humidity too and classes have begun. Some old faces are missing and that makes us a little sadder, but there are new faces and new challenges. Good luck to all and have a good semester.

Phil Huling of SLU to Give April 23rd Talk

Homeomorphic topological spaces have isomorphic fundamental groups. An obvious questionc is then: when must spaces with isomorphic fundamental groups be homeomorphic? That is, when is the fundamental group a complete invariant? Flat conformal deformation theory investigates this question in the case of hyperbolic orbifolds and further asks if we can describe what happens when the fundamental group fails to be a complete invariant. We will look at what is known about these questions and then we will discuss my recent work with cofinite Coxeter groups and the lattices that contain them. In particular, we are able to develop tools which give the deformation spaces of the reflective Bianchi groups.

Sylvia Carlisle to give April 16th Colloquium

A metric space X is a real tree if between any two points in X there is a unique arc, and that arc is a geodesic segment. An isometry on a real tree is hyperbolic if it has no fixed points. In this talk, I will discuss the continuous theory of real trees with hyperbolic isometries, and discuss the model companion for that theory. As time allows I will talk about some properties of this model companion.

Colloquium April 9th by Martz and Parwani

Andrew Mertz and Kamlesh Parwani showed how lottery number distribution could be analyzed using Mathematica. Frequency distributions showed definite anamolies regarding the random nature of the numbers selected in lotteries. But it is generally the case that the odds of winnning are so low that these anomalies can not lead to effective strategies for winning money.

On the other hand, using Mathematica to analyze stock market data, Mertz and Parwani showed that there are low risk stategies that can work.

Andrew Schultz of UIUC to Give Colloquium April 2nd

Linear algebra is often thought of as a gateway to the abstraction of modern mathematics, but it is also a discipline with a number of terrific real-world applications. In this talk we'll discuss some applications of linear algebra which we encounter in our day-to-day digital lives. Topics include Google's PageRank algorithm, image compression, noise filtering and - if time allows - mp3 compression.

53rd EIU Math. Ed. Conference set April 5th, 2010

Using Billiards in the classroom?

Using the equinox to find latitudes?

Social Justice and Dark Matter: What do they have in commmon?

Just how big is a trillion?

Interested? Attend the EIU Mathematics Education Conference and find out more

see

http://www.eiu.edu/~adulted/10MathConf.pdf

for more details.

Department to Host Geometry/Topology Day March 27:

The Eastern Illinois University Department of Mathematics and Computer Science will host a day of hour long talks in geometry and toplopogy on Saturday March 27th. The schedule follows:

9 am coffee

9:30 am Bruce Kitchens - IUPUI

The Dynamics of the Nash Map for 2 by 2 Games

11 am Kamlesh Parwani, EIU

Quasi-isometric foliations for partially hyperbolic diffeomorphisms

12 pm Lunch

2 pm Hong Kun Zhang U-Mass-Amherst

Spectral gap for the Markov operator of certain random billiards


3:30 pm Serge Tabachnikov, Penn St. U.

Pentagrama Myrificum, Old wine into new wineskins

Peter Andrews to Give March 26th Colloquium:

Peter Andrews will speak today on 'Well Behaved Matrices and Knotty Diagrams'. In particular, he will discuss how to compute the Khovanov invariant for various knots or not.

More Nerd Graffiti

Svetlana Butler to Give March 12th Colloquium

Abstract:

We would like to analyze what can be said about the large deviation behavior of martingales,approximate identities, and related operators. Given some sequence m_n of positive integers, and some sequence w_n of positive real numbers, and let the linear operators

T_n : L1(R) \to L1(R)

be either the dyadic Lebesgue derivatives or the dyadic martingale. We will prove positive and negative results concerning certain convergences related to these sequences and operators

Rick Anderson to Give March 5th Talk: Abstract follows

Mathematics educators have an ongoing concern with students' achievement and participation in mathematics courses. It is recognized that all groups of students do not have comparable access to mathematical opportunities or success in mathematics courses. Students in rural areas and small towns approximately one-fifth of US school-aged children are one such group whose mathematics achievement and participation have been tracked in recent decades. In this talk I will summarize the mathematics achievement of rural high school students in the US as it compares to students in non-rural areas. Then I will present results of rural high school students' mathematics course-taking drawn from data collected for the 2005 NAEP High School Transcript Study. I will conclude with a discussion of the implications of the results for mathematics teachers in rural areas.

Patrick Coulton to Give Feb 26th Talk on Colorings

Historians trace the origins of the map coloring problem to Francis Guthrie, a nineteenth century cartographer. Francis' brother, Fredrick, was a student of mathematics and so he naturally asked his brother whether it was known that only four colors were sufficient to color a geographic map of distinct regions. Fredrick passed the problem onto Augustus De Morgan who wondered whether a necessity for five colors could not be contrived.

It occurs that such a necessity and more exists on the torus as well as other well known surfaces such as the Klein bottle. We will investigate the coloring of maps on various surfaces and show that the topology of the surface plays an important role.

The colloquium will begin at around 4 pm in Old Main 2231.

Nerd Graffiti

Who am i?

a complex ol creature am i
O this negative radical guy
as i travel in fours
all out thru my pores
will flow my idenitie

Challenge of the Week

The Department sponsors a problem solving competition for undergraduates at the university with a new challenge each week or so. We will feature some of the old problems on this blog, but changed slightly. Here is an example

Suppose that we have two rational numbers whose sum is negative 9 times their product and the sum is also negative 16 times their quotient. Find the two numbers.

x+ y = -9(xy)

x+y = -16(x/y)

Find x and y.

Galperin to Continue with Talk on January 28th

Gregory Galperin will give a second lecture on the foundations of hyperbolic geometry today at 4 pm. This talk will concentrate on certain geometric constructions of Gauss, Lobachevsky and Bolya.