Ohio State University Math Club

Past talks:

• November 20

Professor Warren Sinnott gives a talk On the Conjecture of Birch and Swinnerton-Dyer

Abstract: It is a basic problem in number theory to understand the integer solutions to polynomial equations. Given such an equation — for example, y^2 = x^3 – 2 — can we find integers x and y which satisfy this equation? Or rational solutions? If so, how many solutions are there? Can we find them all? Such questions have a long history: they are called Diophantine problems, after Diophantus, who lived in Alexandria in the 3rd century C.E. The interest of Diophantine problems was emphasized by Hilbert, who made finding a general method of solution one of his fundamental problems for mathematics in 20th century.

Equations that define so-called “elliptic curves” — such as y^2 = x^3 – 2 — lie at the boundary between what we know and what we don't know about these questions. We will talk about the history of Diophantine equations, and discuss the conjecture of Birch and Swinnerton-Dyer, one of the Clay Millennium problems, which would give an effective method of determining all the rational points on elliptic curves.

• October 24

Graduate School info session, by Professor Thomas Kerler

• October 9

Abstract: A theorem of Riemann says that if a series of real numbers converges conditionally then, using different rearrangements, it can be forced to converge to any pre-chosen real number. What happens if a series of vectors in a vector space converges conditionally, what may then be the set of its “sum-after-a-rearrangement”s? In the finite dimensional case, the answer is given by Levy-Steinitz’s theorem, whose proof I am going to demonstrate in this talk.

• October 2

Dr. Jim Fowler will be giving a talk titled Projective planes

Abstract: Given a field (a number system), we can build a "geometry" -- something with points and lines. For instance, starting with the field of real numbers, we consider ordered pairs of reals, and we get the Cartesian plane.

What if we instead started with the geometry? Could we, from the points and lines, recover a field? Sometimes, yes! The geometries we start with will be "projective geometries" where any two lines meet, maybe "at infinity." How do we then get a field? Commutativity of multiplication---among the other axioms for a field---are encoded as gloriously complicated diagrams of points and lines. Desargues' theorem and Pappus' theorem show up to save the day.

A reference is Hilbert's Grundlagen der Geometrie.

• September 19

Professor Dan Shapiro will give a talk about Scissors-Congruence

Abstract: Two plane polygons A and B are “scissors-congruent” if there is a way to use a finite number of straight cuts to separate A into pieces that can then be rearranged (using rigid motions and with only trivial overlaps) into a polygon congruent to B. Certainly if A and B are scissors-congruent then they have equal areas. We will outline a proof of the converse: Polygons with equal area are scissors-congruent. How can this result be generalized? We will mention a few different possibilities.

• August 28

Professor Vitaly Bergelson gives a lecture Some interesting Diophantine equations

• April 17

Professor Ronald Solomon will give a talk entitled Groups, graphs, and counting

Abstract: The mathematical concept of a “group” and an elementary theorem about groups, Lagrange’s Theorem, will be defined and stated. Then the discussion will turn towards a useful trick for counting in groups of even order before culminating with an interesting and tricky problem.

• March 6

Love mathematical puzzles? Took the Rasor-Bareis-Gordon competition and are dying to know the answers? Come experience Vitaly Bergelson discuss the problems on the RBG and their solutions!

• February 27

Professor Matthew Kahle will give a talk entitled The (3n+1)-Problem.

Abstract: Start with any natural number n, and apply the following rule: If n is even, replace n by n/2. If n is odd, replace n by 3n + 1. Now repeat the process. The Collatz conjecture is that no matter what number n you start with, you eventually hit 1. The problem has been around for 75 years, though, and no one has been able to prove it.

We will discuss what is known, and especially connections to various areas of math: number theory, ergodic theory, and fractal geometry.

• February 6

Professor Robert Perry from the Physics Department will give a talk Careful with that Infinity, Eugene

Abstract: First I'll discuss the impossibility of physical infinities and implications for anyone contemplating eternity. Then I'll turn to the role of infinity in the renormalization group and discuss why all of our theories are at best effective theories.

• January 23

Dr. Moshe Cohen gives a lecture about Domino tilings, perfect matchings on graphs, and the Alexander polynomial of a knot.

Abstract: The goal of this talk is to investigate how well-understood problems in combinatorics interact with this polynomial from knot theory.

Combinatorics -- the art of counting -- asks questions like "How many ways can we cover a checkerboard with dominoes?". Knot theory asks "How can we tell two knots apart?". A knot is a circle embedded in three dimensional space. The Alexander polynomial is one example of a knot invariant -- that is, if the Alexander polynomial of two knots are not the same, the knots must be different. This polynomial is the determinant of a matrix, and we'll construct this matrix using techniques from combinatorics.

This talk is accessible to anyone with a love of problem solving and an understanding of matrix determinants.

• November 14

Professor Barbara Keyfitz will give a talk entitled Stuck in Traffic.

Questions: What can modelling traffic flow tell us about conservation laws? And what can conservation laws tell us about traffic flow?

Abstract: A classic continuum model for the density of vehicular traffic on a one-way road leads to a scalar conservation law, whose solution exhibits all the properties that the theory of conservation laws predicts: Formation of discontinuities (shocks), Necessity of admissibility conditions for uniqueness, Interaction of waves.

The model also sheds light on some well-known traffic phenomena, such as congestion at poorly timed traffic lights.

In this talk, I would like to use the example to describe some of the mathematical aspects of the theory of conservation laws, and also to show, by means of a simple extension to two-way traffic, how a seemingly simple model can lead quickly to open problems.

• October 31

Professor Daniel Shapiro will give a talk entitled Walking the Dog which considers the following problem: I like to walk with my dog Phido on the path that goes around the nearby park. That dog is so well trained that he stays exactly one yard to my right at all times. Since I walk with the center of the park to my left, Phido’s path is somewhat longer than mine. How much longer is it?

• October 10

Professor Sergey Chmutov will talk about Quantum Mathematics

Abstract: In the nearest future all mathematics is going to be quantized. Ultimate goal of my presentation is to show how some areas of mathematics will look like. I will start from 1687, when mathematics was separated from philosophy by I. Newton. Through all these centuries, mathematics was inspired, motivated, and stimulated by physics. This process continues nowadays, and it will continue in the nearest future. The second half of my talk will be devoted to quantum calculus.

• September 27

Professor Thomas Kerler will conduct a Graduate School information session

• May 23

Professor Jean-Francois Lafont will give a talk Finite Blocking Property.

Abstract: An assassin and a target are at fixed locations in a square room with reflective sides. The assassin has a single shot laser gun, while the target has the possibility of hiring a finite number of bodyguards, who he can place as he sees fit. Can the target place the bodyguards in order to ensure his survival? This problem was posed in the Leningrad Math Olympiad in 1990. We will give an answer to this question, and discuss various generalizations (curved surfaces, higher dimensions, etc).

• April 25

Jim Fowler will be speaking on Square Complexes - specifically, how to form negatively curved surfaces out of squares (and he'll creatively use free groups and Dehn functions in the process!).

• April 4

Professor Ron Solomon gives a talk on Sporadic Objects

• February 29

Dr. Bill Mance gives a talk about fractals, Schmidt games, and sets of non-normal numbers.

• February 22

Rasor-Bareis-Gordon exam debriefing, with Professor Vitaly Bergelson.

• February 15

Brian Li gives a talk entitled Normal Numbers: A Notion of Randomness.

• February 8

Professor Matthew Kahle will give a talk about the Crossing Number Lemma and its applications.

Abstract. The crossing number of a graph is the minimal number of crossings over all planar drawings (i.e. for planar graphs the crossing is zero). We will see a surprising proof that one can get a very strong lower bound on crossing number in terms of the the number of vertices and edges.

• February 1

Chris Altomare will be delivering a talk Cantor's Paradise: Sizes of Infinity.

Abstract: Are there different sizes of infinity or does one size fit all? If they're different, what do those sizes look like? Is there a smallest? A biggest? Can we compute these sizes? Can we add and multiply them? Exponentiate? What happens when we do? How can adding, multiplying, and exponentiating sizes of infinity possibly tell us anything about computer programming?! This stuff seems fishy; are there paradoxes?

• January 11

Paul Apisa and Ted Dokos will deliver a talk about math REUs; they will describe some available summer programs, recount their own experiences, and provide information on where to find more information for those who are interested.

• November 2

Dr. Dave Carlson will be delivering a talk on the IDA and the mathematics of blast mechanics: Mitigation of Blast-induced Glass Spall Using Safety and Security Film.

Abstract: Explosive devices have become the weapon of choice for many domestic and international terrorists. In many bombing events emerge, while buildings and other structures emerge with limited structural damage, persons working or living in those structures are maimed or killed by flying glass, or spall, from glazings that fail and are unprotected.

This presentation will consider the potential uses (and users) of polyester safety and security film with the focus on blast mitigation, the basics of blast mechanics, and some test criteria and metrics defined the General Services Administration, as well as results of range testing on protected and unprotected glazings.

A patented ‘Next Generation’ safety film structure will also be presented for which laboratory and range testing has shown a marked increase in impact strength over current products.

• October 26

Professor Chris Miller gives a talk entitled P = NP?

• April 27

Derivation and transcendence: integrability in elementary terms.
Clark Butler will be delivering a talk on differential rings, transcendental extensions of function fields, and integrability in elementary terms. His goal is to give an accessible proof that exp(x^2) and sin(x)/x do not have elementary primitives.

• April 20

Dr. Jim Fowler gives a talk about Desargues' Theorem and projective planes

• March 2

Rasor-Bareis-Gordon exam debriefing, with Professor Vitaly Bergelson

• January 26

Ted Dokos will give a talk Hex and Brouwer's fixed point theorem.

• January 12

Chris Altomare will give a talk Whose Curve Is It Anyway? Improv Problem Posing.

Abstract: Solve it, prove it, find a counterexample ... to what? Too often, the honest answer is that students are working on the theorem they saw in the book, that the teacher mentioned in class, and so on. While important, a mathematician can't just rely on problems someone else gave. Someone made that problem in the first place.

The focus of this talk is on the much neglected art of problem POSING. How do you come up with a problem in the first place? What if it's too hard? What if it's too easy? If a conjecture turns out to be false, can it still be turned into something? How? How do I know how hard it is if I thought of it myself? How do I know if it's already known?

Specific problem POSING heuristics will be given during the talk. Many example problems will be given as well ... improvised on the spot to demonstrate there is just enough science to this art to do it fairly systematically, on the spot.

• November 17

Robert Behal, a lawyer, will be speaking about the interplay between mathematics and law.

• November 10

Dr. Magestro, a professor of finance, joins us to discuss his career path. Starting as a physics and math undergrad, he went to get his PhD in Physics. Later he pursued a carrer in the finance world. Come learn how a math degree can be applied in finance.

• November 3

Paul Apisa talks about Hindman's theorem.

• October 20

Jeff Lindquist will give a talk entitled Ford Circles.

• October 6

Professor Vitaly Bergelson will give a talk Homage to Catalan numbers.

• June 2

Professor G. A. Edgar will give a lecture entitled A counerexample(?) to the Fundamental Theorem of Calculus.

• May 26

Dr. Chris Altomare will give a talk The real line is a proof?.

Abstract: Yes, the real line is a proof. We've all proved facts ABOUT the real line, but how can the real line BE a proof? Does considering it as such have important mathematical consequences? Can seemingly disparate mathematical objects be put on the same footing with an abstract definition of a proof? Do these ideas allow one to state conjectures unifying "Laver's Theorem" and the "Graph Minor Theorem"?

• May 5

Logan Axon was speaking about Random Fractals.

Abstract: You've probably seen fractals before, and you'd probably know one if you saw one. But exactly what properties of a set earn it the name "fractal"? We'll look at some examples of recursively constructed fractals and their properties. We'll then introduce the idea of fractal dimension and figure out what it means to say that a set has non-integer dimension. Finally, we'll introduce an element of randomness into our constructions of fractals and look at the result.

• April 28

Jeff Lindquist, Jack Cheng, and Theodore Dokos talk about Conway Polynomials and Virtual Links.

• March 10

Dr. Bart Snapp gives a talk Moonlighting in Applied Mathematics.

Abstract: The primary focus of my research involves homological conjectures in commutative ring theory. In this talk I'm going to discuss solving a problem involving lasers, robots, and steel. How did I end up doing this? I'll talk about that too.

• March 3

Possibly interested in graduate school in mathematics? In addition to an inside view of what a good application should look and advice concerning which program to choose, Professor Thomas Kerler (our math department's Vice-Chair of Graduate Studies) will tell us, "from the point of view of a graduate recruiter from a large research university," what a sophomore or junior interested in math graduate school should keep in mind now and in the upcoming years. For more information about the talk, see the attached flier.

• February 24

Assistant Dean Teri Roberts of the College of Public Health will be speaking about graduate programs in Biostatistics and Epidemiology.

• February 17

Movie Dimensions (http://www.dimensions-math.org/Dim_E.htm) will be shown. Film produced by: Jos Leys (Graphics and animations), Atienne Ghys (Scenario and mathematics), Auralien Alvarez (Realisation and post-production). It's a movie explaining how we can view the 4th dimension, complex numbers and may other things. You can watch the movie on-line or come watch it on Wednesday February 17th at 5:30pm in EA060 on big screen.

• January 13

Jim Fowler gives a talk entitled Can you divide a square into three triangles of equal area?

• December 2

Professor Vitaly Bergelson will talk about REU's: What are they, which ones ar any good, and how to go about applying them. PLUS:: Get the scoop on the change to semesters in 2012 and how it will affect the math curriculum.

• November 4

Dr. Bart Snapp: The Hailstone problem. In this talk we will discuss a polynomial analogue of the Collatz Conjecture.

• October 21

Professor G. A. Edgar gives a talk Fractional iteration of series and transseries

• October 7

Dr. Bart Snapp is giving a talk on geometry produced by using a scoring metric from Dungeons and Dragons in place of the usual Euclidean distance metric.

• May 27

Phil Kilanowski is talking about Brownian Motion

Pizza and soda will be served.

• May 20

Chris Altomare gives a talk Graph Structure Theory

Abstract: "This bird's eye view of graph structure theory will touch on some of the gems of the field The Oracle touched on, including Kuratowski's Theorem, which gives the two "reasons" for the plane, and the Minor Theorem, which says each surface has a finite list of "reasons".

• April 22

Eric Katz (OSU alumnus) gives a talk on Tropical Geometry

Abstract: Tropical mathematics is what happens when one replaces the usual operations of plus and times with the operations of min and plus. Tropical mathematics is a sort of shadow of classical mathematics and has a piecewise-linear character. Much of mathematics has a tropical analog that captures a lot of subtle behavior. In this talk, we will give an introduction to tropical mathematics and discuss tropical algebriac geometry which studies solutions of systems of tropical equations.

• April 8

Jeff Lanz is speaking about financial mathematics.

• February 25

Radical Pi is meeting to discuss solutions to the Rasor-Bareis-Gordon examination, REU's, and undergraduate summer research opportunities in mathematics at OSU.

• February 4

Professor Alan Saalfeld, from School of Earth Sciences and Department of Computer Science and Engineering, talked about Discrete Barycentric Coordinates

Abstract: A coordinate system delivered from a plane graph embedding's combinatorial structure may be modified in many different ways to produce other plane graph drawings of graph embeddings that have the same combinatorial structure. In this talk on applications, a follow-up to my November 12th talk on theory, I will discuss some ways to modify coordinates to produce useful cartographic and computer graphics results. I will go over the main theoretical results of the earlier talk for anyone who did not attend that talk.

• November 12

Graph drawing in color - plain graphs don't have to be plain! A talk by Professor Alan Saalfeld.

• October 29

A talk by Rob Denomme: A first look at Number Theory.
What are numbers and what makes them so strange?

• October 15

Professor Vitaly Bergelson presents: The Hausdorff-Banach-Tarski paradox

• May 21

Bill Mance gives a talk What are Normal Numbers?

• April 2

Professor Miller gives a talk about the P vs. NP problem

Did you ever wonder how hard Minesweeper and Tetris really are? Did you know you could win one million dollars for solving a math problem? Come hear how to mathematically modal algorithms and discover problems that are truly hard!

• February 27

Professor Bergelson discusses the solutions of the Raisor-Bareis-Gordon contest problems and provides information on summer research opportunities for undergraduates.

• February 20

"N is a Number", a movie about Paul Erdos. From Amazon.com description of the movie: "Erdos was the most prolific mathematician who ever lived and universally revered among his peers. He became a wandering genius who eschewed the traditional trappings of success to dedicate his life to inventing new problems and searching for their solutions".

• February 6

Professor Mike Davis gives a talk Nonpositively curved spaces

• January 16

Professor Edgar gives a lecture about Transseries

The differential field of transseries was discovered independently in various parts of mathematics: asymptotic analysis, model theory, computer algebra, surreal numbers. Some feel it was surprisingly recent for something so natural. Roots of the subject go back to Écalle working in asymptotic analysis, Dahn and Göring working in model theory, Geddes & Gonnet working in computer algebra, Kruskal working in surreal numbers. They arrived at eerily similar mathematical structures.

• November 14

Adam Hammet presents: Algorithm on steroids: How, for a small fee, randomizing can enhance performance.

• October 17

Professor Dan Shapiro gives a talk Barycentric coordinates. (This will be a quite elementary discussion of using coordinates in triangle geometry.)

• October 3

Rob Denomme: Elliptic curve primality tests, Alfred Rossi: Virtual links.

• May 16

Professor Edgar gives a talk A Counterexample(?) to the Fundamental Theorem of Calculus

• April 25

Professor Overman gives a talk The Logistic map

• April 18

Leonhard Euler's Birthday party. Posters: 1, 2, 3, 4, 5

• February 14

Jeremy Voltz presents Integer Partitions

Abstract: A partition of a positive integer n is a way of writing n as a sum of positive integers. The question is, how many partitions of a given integer are there? Does it depend on the properties of the number, as in even, odd, etc...? We will discuss these topics using Ferrer's Diagrams and generating functions.

• January 10

Undergraduate students presenting topics in Group Theory
Students of H590 will be presenting results they have explored during the honors algebra sequence.

• Min Ro: Cristallographic Groups
• Sam Fotis: Platonic Solids
• Rob Denomme: Free groups

• November 8

Professor Daniel Shapiro discussed whether Polygons Have Ears

Abstract: A polygon in the plane that has no self-crossings separates the plane into three parts: inside, on, and outside the curve. Those statements seem clear, and are often used as first steps in proving more advanced results (like Euler's formula). However the proofs are tricky. We will discuss the history of the problem, mention the "Jordan Curve Theorem", and prove the existence of a triangulation by showing that every polygon has an "ear".

• October 11

OSU students research experiences
Ohio State's undergraduates discuss their research in the fields available to young mathematicians.

• Michael Chmutov: Invariants of plane curves.
• Jeremy Voltz: Thistlewaite's theorem for virtual links.
• Deepak Bal: Algorithmic combinatorics on words.

• May 24

Finite Rotation Groups and Plato's Solids, by Professor Radu Stancu

Abstract: One of the mysteries that arise when we study the finite sets of rotations of the 3-dimensional space is that, besides the classical sets of rotations and symmetries of a regular $n$-gon, we have three exotic finite sets (no more, no less!). These last three sets correspond to the rotation sets of three Plato's solids: the regular tetrahedron, octahedron and icosahedron. We will try to pass from miracle to mathematical reality, by analyzing the finite rotation groups that appear in the space.

• April 26

Professor Andrzej Derdzinski gives a talk Special relativity in the Minkowski spacetime

Abstract: This talk is a presentation of the Minkowski spacetime, introduced by Hermann Minkowski in 1908 to provide a geometric model of Einstein's special theory of relativity. The Minkowski spacetime is a four-dimensional, time-oriented affine Lorentzian space; all these notions will be rigorously defined. Various geometric objects associated with the Minkowski spacetime and having physical interpretations will be discussed: examples are world lines and observers. We will also take a look at physical phenomena such as the twin paradox and mass-energy equivalence. The only mathematical tool needed is elementary linear algebra.

• March 8

Quandles: Illustrating the relationship between topology and algebra, a lecture by Professor Alissa S. Crans

Abstract: While it may sound surprising at first, algebra and topology have a very close relationship! One way to demonstrate this connection is through the language of quandles. After examining examples of quandles, we will illustrate their connection to knot theory, and in particular, to the three Reidemeister moves. We will see that we can obtain an action of the braid group on quandles and explore the method which enables us to associate a quandle to a given knot.

• February 22

Professor Alan Saalfeld of the Geodetic Science Department gives a lecture Plus-Minus Paths

• February 1

Professor Jim Brown gives a talk The congruent number problem and elliptic curves

Abstract: Many problems in number theory are so simple to state that a middle school student can understand them. The problem we will discuss is the congruent number problem. Given an integer n, we seek to determine when there exists a right triangle with rational sides that has area n. This easy to state problem turns out to depend on the Birch Swinnerton-Dyer conjecture about elliptic curves, one of the Clay Mathematics Institute's million dollar Millenium problems. We will discuss the problem itself, as well as provide a brief introduction to elliptic curves. We will conclude by giving a sense of how the conjecture of Birch and Swinnerton-Dyer determines which integers are congruent numbers.

• January 18

Problems on the plane, a lecture by Donny Seelig and Adam Chawansky

• November 30

Professor Satyan Devadoss gives a lecture Spaces of Trees

• October 26

Michael (CAP) Khoury, A Surreal Introduction to Numbers and Games

Abstract. In this talk, I will discuss the system of "surreal numbers" introduced by John Conway, which turn out to be an ideal setting for, among other things, making sense of the mathematical discussions you probably had when you were six, viz.
"Am not!"
"Are too!"
"Am not am not!"
"Are too infinity!"
"Am not infinity plus one!"
"Are too infinity squared!"
and so on ad nauseum. The surreal number line turns to be much richer than the real line, including (lots of) numbers further to the right than all the real numbers, and (lots of) numbers which, though positive, are less than all the positive real numbers.

In this talk, which is suitable for any undergraduate, we will take a short vacation from the real and complex numbers to explore the "exotic" surreal landscape. We will deal with Conway's definition, as well as very briefly with general games, focusing on the specific game of Hackenbush. We'll pay particular attention to the almost unbelievable elegance with which such a rich mathematical object comes from such a simple construction. If time permits, we will also give a slick construction of the real numbers from the surreal numbers which is totally independent of the usual construction from the rationals.

• October 12

Non-Standard Digits, by Professor Daniel Shapiro.

• May 4

Professor Henry Glover gives a lecture on Hamilton cycles in Cayley graphs

• April 20

Professor Dan Shapiro gives a lecture The four numbers game

• April 6

Professor Gerald A. Edgar gives a lecture on Hausdorff dimension

• February 9

Professor Ulrich Gerlach gives a lecture Space, Time, and Quantum Mechanics

Summary: We trace the fundamental physical ideas which led Einstein to his well known theory of space, time, and gravitation. We shall compare his line of reasoning and the use of the physical and mathematical ideas available at his time, with the line of reasoning he probably would have pursued if he had known and appreciated quantum mechanics during the creative part of his life. This comparison will focus on "the happiest thought of his life" (in 1907), which was the platform that launched him toward his theory of gravitation eight years later. We will present a simple quantum mechanical extension of his happiest thought for the purpose of grasping gravitation and quantum mechanics from a single point of view.

• January 19

Professors Peter March and Vitaly Bergelson speak on Research Opportunities for Undergraduates (at OSU and around the country).

• November 10

Professor Tadeusz Januszkiewicz gives a lecture Position spaces of Penduli

• November 3

Scott McKinley: Some Thoughts on Modeling Randomness in Continuous Times

• October 27

Professor Dan Shapiro gave a lecture entitled Walk the Dog

• October 20

Professor Saleh Tanveer gave a lecture on Mathematics of Bubbles

• June 2

Professor Andrew McIntyre: Sums of kth powers, Riemann zeta and regularization.

I will discuss the problem of evaluating 1^k+2^k+...+n^k for positive integer k, in particular the asymptotics, before describing the beautiful solution of Euler. Then I will describe Euler's evaluation of 1^k+2^k+... for even negative k, and its relation to the Riemann zeta function. Finally I will describe curious "regularized" sums like 1+2+3+...=-1/12, and mention that they can be used to get actual real-life numbers in physics.

• May 26

Professor Zbigniew Fiedorowicz: Classification of Surfaces and 3-Dimensional Manifolds.

For a long time people thought that the earth was flat, and indeed it does look flat from a myopic point of view. What other geometric shapes could give rise to a similar delusion? In this talk I will discuss the classification of surfaces, shapes having this property, from the point of view of topology, which is a very basic foundational branch of geometry. I will also briefly discuss the analogous three dimensional problem, including the recent apparent breakthrough by Grigori Perelman.

• April 21

Professor Warren Sinnott: Irrationality.

There is a story that when Pythagoras discovered the existence of numbers that were not rational, he was so disturbed by the discovery that he tried to keep it a secret. In this talk we will discuss some of the early history of irrational numbers, including a curious passage in Plato's Theaetetus:

 "....Theodorus was telling us about square roots of 3 and of 5, and showed us that these were not rational; and he continued on with each case in turn until he got to the square root of 17, where for some reason he got stuck."
Why 17? And why didn't Theodorus start with 2? We will talk about the ways in which the Greeks may have shown that numbers were irrational, and why they cared.
• April 14

Professor Gary Kennedy: Bend, pinch, break & count

Maxim Kontsevich won the 1998 Fields Medal in part for his work on counting rational curves in the plane. In particular he gave a simple recursive formula which generates the sequence 1, 1, 12, 620, 87304, etc. The first term is the number of lines through two specified points; the second is the number of conics through five specified points; the third term 12 is the number of rational cubic curves through eight specified points; in general the nth term of the sequence is the number of rational plane curves of degree n through 3n-1 specified points. With the help of "four happy points", in this talk I will explain how the recursion works by calculating the third term of Kontsevich's sequence.

• February 25

Ronnie Pavlov: an Introduction to Symbolic Dynamics and the Perron-Frobenius Theorem.

In this talk, we plan to give a brief introduction to the topic of symbolic dynamics. As a starting point, we consider the following easily posable question: how many strings of 0's and 1's of length n are there which do not contain two 1's in a row? This problem may be solved with elementary methods, however one can see that generalizations might be much tougher: suppose instead we ask how many strings of 0's and 1's of length n there are in which any ten consecutive symbols contain between three and seven 1's? It turns out that all such questions may be solved by using linear algebra and a useful result known as the Perron-Frobenius theorem. Knowledge of some basic linear algebra will be helpful, but most concepts will be defined anyway.

• January 21

Professor Steven Miller:
Mathematics vs Monty Python: Primes and Elliptic Curves in Cryptography (and not the Bridgekeeper's Three Questions)

Cryptography has been defined as "The science of adversarial information protection". We will talk about several systems, ranging from simple ciphers to prime numbers to elliptic curves. Here's a nice problem to think about: The password to launch nuclear missiles is the triple (a,b,c), the coefficients of a quadratic ax² + bx + c. We have nine generals: ANY three generals must have enough information to launch the missiles, but NO set of two generals have enough info. What info should be given to each general? (Nice reference on ENIGMA: http://www.smecc.org/new_page_8.htm)

• October 12

Professor Alan Saalfeld of the Geodetic Science Department: What on Earth are Map Projections?

Map projections are differentiable transformations from one familiar mathematical surface (a sphere or ellipsoid) to another (a plane, cone, or cylinder). We will examine a variety of map projections and their properties. We will discover projections that preserve area ratios, projections that preserve local shapes and directions, and still other projections that preserve some, but not all, distance relations. Some of the projections we will introduce have had a long and useful history in navigation. Others will be intentionally contrived to illustrate extreme possibilities. All of these explicit examples of map projections will be used to illustrate and to help us try to understand important concepts in differential geometry of surfaces.

• October 29

Professor Sergei Duzhin: Mathematics of the Accordion

• October 15

Professor Thomas Kerler: Why is the Poincare Sphere not a sphere?

In the talk I will give some examples/understanding/visualizations of what a generalized 3-dim space can look like (aka a 3-manifold), and how one can distinguish these hard to imagine spaces. In large part I'll probably retell the story of Poincare, who founded algebraic topology by looking at exactly such questions. The aim of the talk is that there are 3-manifolds that cannot be distinguished from the 3-dim sphere using the combinatorics of "homology" or "Betti's numbers", but which are actually different - which follows from what Poincare (and, today, everybody else) called fundamental groups.

• October 1

Professor Steven Miller: How the Manhattan Project helped us understand primes

Often we want to understand the spacings between events. The events can range from being the energy levels of heavy nuclei to the waiting times at a bank to prime numbers. Amazingly, very different systems seem to be governed by the same, universal rules. We'll talk about how knowledge of nuclear spacings led to new insights into properties of primes.

• May 28

Professor Susan Goldstine: Polynomial Dynamics

• May 14

We were watching the film Outside In. The movie illustrates an amazing mathematical discovery made in 1957: you can turn the surface of a sphere inside out without making a hole, if you think of the surface as being made of an elastic material that can pass through itself. Communicating how this process of eversion can be carried out has been a challenge to differential topologists ever since. Computer graphics helps to explain as well as present the visual elegance of this process.

• April 30

Professor Chris Miller was talking about Turing Machines, recursive algorithms, and more in relation to computibility.

Title: P versus NP

Suppose that we are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided us with a list of pairs of incompatible students, and requested that no pair from this list appear in our final choice. Now, it is easy to check if any proposed choice of one hundred students is satisfactory (that is, no pair from taken from the proposed list also appears on the list from the Dean's office), but the task of generating such a list from scratch seems to be so hard as to be completely impractical. Indeed, the total number of ways of choosing one hundred students from the four hundred applicants is greater than the number of atoms in the known universe. Hence, it is fair to say that there is no hope of anyone ever building a computer capable of solving the problem by brute force, that is, by checking every possible combination of 100 students. Of course, this apparent difficulty may only reflect our lack of ingenuity in coming up with an algorithm. Can we do better? Before we can answer this question, we need to know what we mean by "better" and, more importantly, what we mean by "algorithm"; I'll make all this precise, and state the "P versus NP" problem (the solution of which will earn the solver a \$1,000,000 prize!)

• April 23

Math club was hosting a special movie session on the life and mathematics of the “Prince of Problem Solving” Paul Erdos.

• March 5

Professor Susan Goldstine gave a talk Shoestring Arithmetic.

• February 12

Professor Alan Saalfeld of the Geodetic Science Department was speaking on the Piecewise Linear Functions.

Piecewise Linear? Think Triangles. Piecewise Linear Functions are a handy math tool to get to know. They turn up in many proofs of existence of approximating functions and approximating topological spaces. Because the "pieces" are, after all, only "linear," very little attention is paid to explicitly constructing piecewise linear functions. We will describe a quick and easy way to represent (and simultaneously "construct") piecewise linear functions on the plane. Our representation will allow us to build and test important alignment transformations used in computer cartography. Our representation will also permit us to state succinctly some intriguing open problems involving the existence of extensions of point maps to piecewise linear functions defined on the entire plane.

• January 29

Professor Peter March talked about undergraduate research opportunities available both at OSU and elsewhere in the country: Undergraduate Research Experiences

Do you like doing mathematics? Would you like to join a team of undergrads, grads and faculty working on a research project? Would you like to receive a stipend for doing it? Then you're in luck! There are a tremendous number of exciting opportunities for undergraduate research experiences both here at Ohio State and at universities, government laboratories and mathematics institutes around the country. I will describe many of these opportunities, especially ones connected with the National Science Foundation's VIGRE and REU programs. I'll also illustrate some of the mathematical projects out there, the financial support available and how to apply.

• January 15

Professor Peter Brooksbank gave the talk “A guide to a successful marriage in an age of piracy”

• November 20

Scott McKinley was talking about Two Laws of Large Numbers and the “Supreme Law of Unreason”

• October 23

Professor Susan Goldstine, Sunflowers and the least-rational number

• October 22

A talk on the results when Topology, Quantum Mechanics and Superconductivity meet.

• October 9

Professor Vitaly Bergelson speaks on Billiards

• May 22

Professor Satyan Devadoss: The shape of the universe

• May 15

Professor Susan Goldstine speaks on Hexaflexagons.

• May 8

Professor Mark Evans is talking about when the product of two derivatives is a derivative.

• May 1

Professor Bostwick Wyman talks about Actuarial Science.

• March 6

Professor Sergei Chmutov: Knots.

• February 6

Professor Gerald Edgar gives a talk on Fractal Dimension.

• February 4

Dr. S. Brueggeman presents Solutions to (x+y+z)³=nxyz where x, y, z and n are positive integers.

• January 23

Measuring liquids by Professor Dan Shapiro.

• November 19

Roninie Pavlov gives a talk on Various Notions of Size in the Natural Numbers.

• November 7

Professor Susan Goldstine gives a talk on The Discovery of Non-Euclidean Geometry.

• October 24

Professor Vitaly Bergelson gives a talk on Continued Fractions Through the Ages.

Last updated by leibman.1 on 11/21/13