Department of Mathematics

West Chester University

**Mathematics Information**

Office: Room 101

25 University Avenue

West Chester, PA 19383

Phone (610) 436-2440

Fax (610) 738-0578

Email: Department Chair

Each Thursday there will be a mathematics seminar (usually in UNA 120 from 3:15-4:15), while colloquium talks will normally be on a Wednesday (usually in UNA 158 from 3:15-4:15).

These seminars/colloquium talks may be by visiting speakers, WCU faculty, or WCU students, and are open to all interested students and faculty.

Send an e-mail to jmclaughl@wcupa.edu, if you would like to be on the e-mail list to receive advance notice of upcoming talks.

Previous Semesters: Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Summer 2006, Spring 2006,

**Thursday, September 15th, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, I**

In the 1920s, Hardy and Littlewood devised a powerful analytic method for counting solutions of Diophantine equations, and in particular for studying the number of representations of an integer as the sum of a fixed number of kth powers. The initial set-up is similar to that used by Hardy and Ramanujan to study the partition function, but the details are quite different and involve a fascinating interplay between number theory and analysis. In this first seminar, we aim to provide an outline of the strategy, describe a variety of applications, and mention the strongest results attainable by modern refinements of the method. Subject to audience interest, we will flesh out the arguments in the coming weeks, culminating with a proof of the asymptotic formula in Waring's problem. For the most part, the talk(s) will be self-contained and make use of only undergraduate-level analysis and number theory.

**Thursday, September 22nd, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, II**

This week we begin the proof of the Hardy-Littlewood asymptotic formula for the number of representations in Waring's problem. Our main goals will be to describe the dissection of the unit interval into major and minor arcs, outline a rough plan of attack for each subset, and develop a few preliminary results that will be needed for the analysis.

**Thursday, September 29th, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, III**

This week we describe the dissection into major and minor arcs and outline the overall strategy in some detail. We will then try to understand the basic idea behind Weyl differencing, which is fundamental to bounding the contribution from the minor arcs. Along the way, we will need to prove some auxiliary estimates involving geometric series and the divisor function.

**Wednesday, October 5th, 2011**

3:15 to 4:15PM UNA 155

**Fall 2011 Mathematics Colloquium
THANE PLAMBECK**

We'll describe two games for the Apple iPad that are based on recent results from the mathematics of combinatorial game theory, and how we developed them. Along the way, we'll take occasional forays into French film noir and the surprisingly intricate theory of "impartial" Tic-Tac-Toe (both players play "X", and the first person to make three-in-a-row loses).

No prior knowledge of any of these subjects will be assumed.

*Thane Plambeck is a software entrepreneur and mathematician. He grew up in Nebraska and graduated from the Stanford Computer Science PhD program in 1990. He lives in Palo Alto, California.*

**For further information e-mail** mfisher@wcupa.edu or sgupta@wcupa.edu

**Thursday, October 6th, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, IV**

Following last week's introduction to Weyl differencing for cubes, we describe the general result for kth powers and indicate how to deduce Weyl's inequality, a non-trivial upper bound for our generating function that depends explicitly on the nature of rational approximations to the argument. If time permits, we may also begin discussing Hua's Lemma, which shows that the generating function is not too large on average. With these two estimates in hand, it will be an easy task to dispose of the minor arcs.

**Wednesday, October 12th, 2011**

3:15 to 4:15PM UNA 155

**Fall 2011 Mathematics Colloquium
JOSEPH H. SILVERMAN ( Brown University)**

An elliptic curve, being a curve, is naturally a geometric object, but it is also an algebraic object, because its points have a natural group structure. Elliptic curves have long been a testing ground for number theorists, who are especially interested in the points on elliptic curves having coordinates that are rational numbers or integers. In this talk I will explain what an elliptic curve is, both geometrically and algebraically, describe important theorems of Mordell and Siegel that classify the rational and integral points on elliptic curves, and conclude with a fascinating conjecture of Serge Lang that relates these two fundamental results.

*Joseph Silverman is a Professor of Mathematics at Brown University, where he has taught for 23 years. He is the author of seven books, including two on elliptic curves that were awarded the AMS Steele prize in 1998, and of more than 100 research articles in number theory, arithmetic geometry, dynamical systems, and cryptography. He has been a Sloan Fellow and a Guggenheim fellow, and in 2010 he received an MAA Award (Northeastern Section) for Distinguished Teaching.*

**For further information e-mail** mfisher@wcupa.edu or sgupta@wcupa.edu

**Thursday, October 20th, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, V**

This week we use the Weyl differencing apparatus developed last time to prove Hua's Lemma, which states that the exponential sum over kth powers cannot be too large on average. This is accomplished by observing that the sum's even moments count solutions of auxiliary symmetric equations. We then combine this with Weyl's inequality to complete the analysis of the minor arcs, showing that their contribution has smaller order of magnitude than the expected main term.

**Thursday, November 3rd, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, VI**

We begin analyzing the major arcs by obtaining an approximation for our generating function that is valid near rational points with small denominator. We then indicate how this approximation leads to a factorization of the number of representations in Waring's problem as the product of a singular series related to the underlying congruences and a singular integral capturing the density of real solutions.

**Thursday, November 10th, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, VII**

So far we have expressed the number of representations in Waring's problem as the product of a singular series that captures solutions of the underlying congruences and a singular integral that represents the density of real solutions. This week's goal is to express the singular integral in terms of values of the gamma function. This topic requires no background in number theory, and the necessary facts from analysis will be reviewed as needed. The argument does *not* depend on techniques introduced in earlier seminars!

**Wednesday, November 16th, 2011**

3:20 to 4:10PM UNA 158

**Fall 2011 Mathematics Colloquium
Daniel Juncos
(Former student, West Chester University)**

The Nyman-Beurling Criterion paraphrases the Riemann Hypothesis as the closure problem in a Hilbert space. The simplest version of this, due to B'aez-Duarte, states that Riemann Hypothesis equivalent to one particular element in a Hilbert space being in the closure of a span of countably many other elements. We investigate this numerically and analytically. In particular, we establish new formulas for the inner products of the vectors involved.

**For further information e-mail** mfisher@wcupa.edu or sgupta@wcupa.edu

**Thursday, November 17th, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, VIII**

The one remaining task in our proof of the asymptotic formula in Waring's problem is to show that the singular series is bounded away from zero. This involves studying the various congruence conditions that arise from the representation of an integer as the sum of kth powers. This final stage of the argument is much less analytic than previous parts and makes use of some results from elementary number theory.

**Wednesday, November 30th, 2011**

3:20 to 4:10PM UNA 162

**Fall 2011 Mathematics Colloquium
Professor Lin Tan
(West Chester University)**

We will study the solution structure of the type of multiple-choice problems that students dislike most in such standardized test such as the SAT, and provide a test-tested method of selecting the answer, without even having to know the question. The prerequisite of this presentation is some undergraduate mathematics.

**For further information e-mail** mfisher@wcupa.edu or sgupta@wcupa.edu

**Thursday, December 1st, 2011**

3:15 - 4:15 pm in UNA 120

**Professor Scott Parsell** (West Chester University)

**Diophantine problems via the circle method, IX**

In this final installment of the series, we use elementary number theory to investigate congruences for sums of kth powers modulo powers of small primes. By obtaining a suitable lower bound for the density of solutions to these congruences, we establish the positivity of the singular series and hence complete the proof of the asymptotic formula in Waring's problem.

Note: Talks will be added to the schedule throughout the semester. Check back for updates.