Ohio State University Math Club

Upcoming events:

• Wednesday, October 15, 5pm, Undergraduate Math Lounge, Room 052

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

  Pizza and soda will be served.

Past talks:

Year 2007-2008

• 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.

Year 2006-2007

• 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.

Year 2005-2006

• 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.

• May 10

  Professor Satyan Devadoss gives a lecture Juggling Links

• 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

• November 9

  Professor Alissa Crans gives a lecture about Knots, Links and Braids

• 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.

Year 2004-2005

• 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 gave 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

Year 2003-2004

• 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 Sinnot: 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.

Year 2002-2003

• 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.

• February 4

  Professor Samir Mathur: "String Theory and Black Holes"

• 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”

• November 6

  Professor Alan Saalfeld, "Some Mathematics for Computer Mapping"

• 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

Year 2001-2002

• 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 20

  Professor Warren Sinnott speaks about "The p-adic integers"

• 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"