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Ohio State University Math Club |
General Announcement |
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Do you love to do math puzzles? Did you participate in the Razor-Bareis-Gordon competition? Come experience Professor Vitaly Bergelson discuss the problems on the RBG and their solutions. |
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Professor Daniel Shapiro gives a talk entitled Triangular reptiles Abstract:A plane figure A is an n-reptile if it can be exactly covered by n coungruent tiles, each similar to A. It is not hard to see that any triangle is a 4-reptile. For which n is every triangle an n-reptile? For which n does there exist a triangle which is an n-reptile? We will outline ideas used to answer those questions. |
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Professor Ghaith Hiary gives a talk entitled False Conjectures About the Primes Abstract: Prime numbers provide an excellent example for the general study of the value of computations in fundamental research, providing many tales of success and caution. We will discuss some conjectures about primes that have been believed to be true for a long time, and were supported by extensive numerics and heuristics, yet were shown to be false. |
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Professor Joseph Migler will present a talk entitled How Do You Exponentiate a Linear Transformation? Abstract: Certain functions like the exponential function can be applied to matrices. We will discuss this process, known as the functional calculus, and we will extend it to infinite dimensional settings. We will discuss recent applications as well as the behavior of traces and determinants under functional calculus. |
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Professor John Holmes gives a talk entitled What is the half derivative of x? Abstract: In 1695 L'Hôpital asked Leibniz what the notation for the nth derivative of a function would mean if n was replaced with a fraction, say 1/2, or an irrational number. In 19th sentury this question was explored by a variety of early analysts who proposed and studied several definitions. We will look at some of the proposed solutions to the question, as well as the modern approach using Pseudo-differential operators. |
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Professor David Sivakoff gives a talk about The Mass Trasport Principle Abstract: Imagine that for all pairs x and y in the cubic integer lattice, a random amount of "mass" is sent from x to y in such a way that the distribution of mass sent is invariant under translations of the lattice. The mass transport principle states that the expected total mass entering x equals the expected total leaving x, that is, mass is conserved locally on average. This simple idea has far-reaching consequences in probability theory and statistical physics. |
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Dr. Jim Fowler gives a talk about word2vec: Representing English Words as Vectors Abstract: This approachable talk discusses word2vec, a 2013 method by Mikolov et al. to produce "word embeddings", where English words are placed in a high-dimensional vector space. Starting with a large corpus of text, words that are surrounded by common words in the corpus are, through an efficient algorithm, placed near each other in the vector space. Students who have seen linear algebra can use this talk as an introduction to how linear algebra gets applied in machine learning and neural network. |
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Desmond Coles gives a talk about Tropical Curves and their Structure Abstract: A tropical curve is given by a set of "vanishing" points for a tropical polynomial, but what do tropical curves look like, geometrically and combinatorically? We will examine some structures which can't appear, as well as possible metrics on the edges of a tropical curve, and another geometric object arising from these metrics. |
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Professor Gary Kennedy gives a talk entitled What is Tropical Geometry? Abstract: Tropical Geometry is a piecewise-linear version of algebraic geometry. The core idea may seem bizzare: in tropical arithmetic, use addition in place o multiplication, and use min in place of addition. We'll apply this to polynomials and then study the reulting geometry. |
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Professor Vitaly Bergelson presents A Potpourri of Open Problems in Number Theory |
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Professor David Goldberg (Purdue University) gives a talk entitled How many ways can I tile my floor? Abstract: Aestetic appreciation of symmetry permeates human culture, and is evident across all civilizations. Natural mathematical questions arise in connection with tilings. In particular, what are the possible symmetries of a periodic tiling? It is surprising that the collection of possible symmetries is finite. We will describe what is meant by a tiling of the plane, how one classifies such tilings, why there are only finitely many classes, and what they are. |
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Professor Elliot Paquette gives a talk about The Longest Increasing Subseqeunce Problem. Abstract: An increasing subsequence of a permutation π of 1,...,n is an indexed subset of those numbers so that they are increasing under π. Computing the average length of the longest increasing subsequences over all permutations of 1,...,n was a problem first mentioned by Ulam. Over time, the problem was connected to many different fields of math. We'll show some of these connections and give a taste of how the problem was eventually resolved. |
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Do you love to do math puzzles? Did you participate in the Razor-Bareis-Gordon competition? If so, join Professor Vitaly Bergelson on March 8 at 5 pm to discuss the solutions to problems on the exams. |
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Dr. John Johnson gives a talk about Szemerédi's combinatoral proof of Roth's theorem Abstract: In 1936, Erdös and Turán posed the following puzzle: "Every integer is connected toa power source and is either on or off. If only 0.00000001% are turned on, can one successively skip between three integers that are on?" Roth was able to give an affirmative answer to this puzzle in 1952, and in 1969, Szemerédi generalized from 3 to 4 skip. We'll explore Szemerédi's proof and outline how the result forms an ongoing area of mathematical research. |
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Professor Ovidiu Costin gives a talk entitled Evaluating Divergent Series Abstract: A good number of interesting problems in mathematics and physics often admit formal solutions as power series with zero radius of convergence. Can one somehow recover the actual solutions from those divergent series? the answer is yes, in striking generality, using the relatively new tools of "resurgent functions". |
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Professor Eric Katz talks about Chromatic Polynomials Abstract: The chromatic polynomial of a graph is a polynomial that counts the number of proper colorings of the graph. This talk will be about why such polynomial should exist, while introducing some generalizations, and discussing some recently resolved conjectures. |
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Professor Rachel Karpman gives a talk entitled From networks to Matrices. Abstract: A matrix is totally positive if all of its minor determinants are positive. Matrices with this property have important applications in mathematical physics. In this talk, we will give a way to construct totally positive matrices, using a remarkable connections between matrices and networks. |
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A talk entitled Continued fractions and the great mathematical planetarium, by Professor Joseph Vandehey Abstract: In 1680, Christiaan Huygens began work on the first ever mathematical planetarium, which would have all the planets rotating around the sun at the right speed. The finished product was a marvel of engineering for its time, but building it required Huygens to dive deep into a new area of mathematics: the area of continued fractions. |
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Professor Vitaly Bergelson will give a talk about Ramsey Theory |
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A meeting with Professor Vitaly Bergelson devoted to the 2016 Rasor-Bareis-Gordon Olympiad. Abstract: Love mathematical puzzles? Took the Rasor-Bareis-Gordon competition and are dying to know the answers? Come experience Professor Vitaly Bergelson discuss the problems on the RBG and their solutions! |
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Dr. Charles Baker gives a talk entitled Farmer Ted Goes Natural Abstract: We all know the problem from calculus of minimizing the perimeter of a rectangle given a fixed area; so if the area is 190 square meters, the optimal perimeter is 4 * √(190) square meters. But have you ever actually tried to order √(190) square meters of fencing? Following the eponymous paper by Prof. G. Martin (then U. Toronto, now U. British Columbia), we investigate the optimization among rectangles of integer length, but — to make it more interesting than "factorize the number and take the 'closest-to-square' factorization" — allow the area to dip below the given value if it gives us a better area-to-perimeter ratio. We call the integers representing areas with perimeters giving the best area-to-semiperimeter ratio "almost-squares," and derive a) which integers are almost-squares, b) the first two terms of the asymptotic for the counting-function for the almost-squares, and c) the existence of algorithms for computing both "Is this number an almost-square" and "Count the number of almost-squares up to this number" in polynomial time, time permitting. |
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Professor Bart Snapp will talk about How To Find Potential "Research" Problems Abstract: Where do problems in mathematics come from and how can we generate more? Come to this talk and we'll see if we can make more problems than we can handle. |
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Professor Krystal Taylor gives a talk Counting problems: the lattice and beyond Abstract: How many points with integer coordinates are contained in a disk centered at the origin of radius t>0 ? It is intuitive that the answer is approximated by the area of the disk. What is not so clear is a bound on the error. In the beginning of the 19th century, it was conjectured by Hardy that the error is bounded by a constant multiple of √t. This problem is still unsolved, but much progress and mathematics has been built up around it. We will look at some simple but beautiful techniques for estimating the error. We will also make some connections between this problem and different areas of math. In addition, we will look at a simple but interesting fact about the so-called "fat" Cantor set, C, which is constructed by removing the middle fourth from the unit interval, and then removing the middle 1/2n from each subsequent interval at stage n. |
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Professor Crichton Ogle presents a talk entitled Zero-determinant strategies and the Prisoner's Dilemma Abstract: The Prisoner's Dilemma (PD) - a term coined by A.W. Tucker in 1950 - is a well-known "game" in which the so-called Nash equilibrium differs from what is possible under (unstable) cooperation (hence the dilemma). The Iterated Prisoner's Dilemma (IPD), in which the game is played many times, has been one of the most intensely researched iterative games in the 70+ year history of game theory (with over 109,000 listings on Google Scholar). In 2012, W. Press and F. Dyson discovered a new set of strategies for IPD, called Zero Determinant Strategies, which extend to more general types of non-cooperative games. This talk should be accessible to all students who have taken Linear Algebra; in particular, no prior knowledge of game theory is assumed. After giving a brief introduction to two-person games and strategies for playing those games, I will go over the results of the Press-Dyson paper and try to explain why their work is of such interest. |
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Professor Jim Fowler gives a talk about Cutting squares into equal-area triangles Abstract: An important feature of Radical Pi and other pizza-and-math events is the "equidecomposition" of pizzas into equal-area triangle-shaped pieces. Suppose, however, that our pizzas were square. For which integers n can we divide our square pizzas into n triangles having equal area? For even n, this is easy, but what about for odd n? Somewhat surprisingly, it cannot be done. More surprisingly, a proof of this fact uses a 2-adic valuation on the reals! Disregarding the culinary question of using 2-adic valuations as a topping, we'll prove that a square does not admit an equidecomposition into an odd number of triangles. We will discuss some amusing generalizations to other polygons. In the meantime, we may accidentally prove Brouwer's fixed point theorem (as Sperner's lemma is at the heart of both these equidecomposition questions and the combinatorial proof of Brouwer's fixed point theorem). |
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Professor Vitaly Bergelson gives a talk entitled Formulari of Beautiful Formulas Abstract: One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than we originally put in to them. ---- Heinrich Hertz I think that mathematics is very much like poetry. I think that what makes a good poem – a great poem – is that there is a large amount of thought expressed in very few words. In this sense formulas are like poems. ---- Lipman Bers |
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Professor James Cogdell gives a talk about Zeta Function
Abstract: Zeta functions have become ubiquitous in number theory. They mix analysis (calculus) with arithmetic (such as primes). They are as mysterious as they are powerful. The most notorious example is the Riemann zeta function which looks harmless enough. But it and its compatriots, the L-functions, help us answer the following types of questions:
Professor Liz Vivas will talk about Non-Euclidean Geometry
Abstract: Euclid's book Elements (the longest running textbook in history) starts by stating five postulates. For at least a thousand years, geometers were troubled by the complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. This long standing controversy was only settled in the 19th century, giving rise to many modern concepts like curvature. In this talk we will state the postulates, review some history and describe what non-Euclidean geometry looks like!
A meeting with Professor Vitaly Bergelson devoted to the 2015 Rasor-Bareis-Gordon Olympiad
Abstract: Did you participate in the 2015 Rasor-Bareis-Gordon mathematics competition? Would you like to know the solutions? Attend this upcoming meeting of Radical Pi to hear Professor Bergelson discuss the problems and their solutions.
Professor David Simmons will be talking about Fractal sets, Measures, and Dimensions
Abstract: The theory of dynamical systems provides many interesting examples of “fractal” subsets of Rn which are “pathological” from a calculus viewpoint but exhibit many interesting symmetries. The question of how to understand the geometry of these sets leads to notions such as Hausdorff dimension and conformal measures. In this talk I will explain these notions and relate them to concrete examples.
Professor Jeffery McNeal gives a talk Creating More Convergent Series
Abstract: Given an infinite sequence of real numbers a1, a2, . . ., when does their sum ∑k=1∞ak make sense? I.e., when is ∑k=1∞ak convergent? Many tests for convergence are studied in beginning analysis courses, as well as some structure theorems about infinite series. For instance, a theorem of Weierstrass – absolutely convergent series sum to the same number regardless of the order that the terms a1, a2, . . . are added – is part of our standard curriculum. However the following question is usually not asked: is there a once-and-for-all rearrangement of the positive integers so that adding sequences according to that ordering causes more series to converge? The somewhat surprising – but completely elementary – answer is "Yes"! We'll discuss this fact, after reviewing more standard convergence results, in this talk.
Professor Zbigniew Fiedorowicz will talk about Completeness, Reality, Unreality, Surreality
Abstract: The real numbers are obtained from the rational numbers by filling in the gaps in between, a process called completion. One can further extend the real numbers to the complex numbers, but the complex numbers can not be ordered. Is it possible to extend the real numbers to ordered fields? What do these fields look like? Are there any gaps in these fields? If so, can we fill in these gaps? Is calculus possible in these fields? I will discuss these questions and try to make my talk as accessible as possible to anyone with a good background in calculus.
Daniel Glasscock gives a talk about Foundational Problems in Additive Combinatorics
Abstract: In 1929, a twenty four year old Soviet mathematician named Lev Schnirelmann proved that every integer greater than 1 can be written as the sum of at most 800,000 prime numbers! Important in his work was not the number 800,000; indeed, just a few years later, fellow Soviet Ivan Vinogradov with advanced techniques from analytic number theory reduced this number to 3. Instead, it was the soft combinatorial approach Schnirelmann employed that persisted. Though the term "additive combinatorics" was coined much later, ideas stemming from Schnirelmann and others around his time laid the groundwork for this rapidly developing field.
In this talk, I will present many of the foundational problems and their solutions in the relatively short history of additive combinatorics. Emphasis will be placed on digesting statements and generating a feeling for the nature of the subject. Time permitting, we will discuss the fruitful intersections between additive combinatorics and other fields.
Professor Daniel Shapiro will talk about Products of Sums of Squares
Abstract: Suppose K is a field. For concreteness, think of K as the field of rational functions (quotients of polynomials) with real coefficients. For positive integer n, let DK(n) be the set of nonzero elements of K that can be expressed as a sum of n squares in K. The sets DK(1) and DK(2) are closed under multiplication because of the formulas: a2b2 = (ab)2 and (a2 + b2)(c2 + d2) = (ac − bd)2 + (ad + bc)2. A 4-square identity (due originally to Euler) shows that DK(4) is also closed under multiplication. Can you find a field K for which DK(3) is not closed? Theorem. DK(2m) is closed under multiplication for every m. We will outline the proof using an inductive argument and some elementary matrix theory. There are many ways to generalize those ideas, and we will mention some of them.
Professor Vitaly Bergelson gives a talk entitled The curse of Dimensionality
Professor Sergey Chmutov will give a talk about the Catastrophe theory
Abstract: The catastrophe theory appeared in mid 1960's in works of the French mathematician Rene Thom (1923-2002). Magazines reported about a revolution in mathematics comparable with Newton's invention of calculus. The aim of the theory was to model discontinuous change of a system as a response to a local smooth change of its parameters. Thom claimed that there are 7 typical elementary catastrophes. As a philosophical theory it has many funny non mathematical speculations. The rigorous mathematical theory started with a theorem of H. Whitney about the stable singularities of maps of one plane to another. Later on this approach was developed by V. Arnold to a big beautiful mathematical theory known as singularity theory and having many mysterious connections with various parts of mathematics and physics. I will review the beginning of this theory, describe 7 elementary catastrophes, and discuss some of its applications.
Rasor-Bareis-Gordon exam solutions by Professor Vitaly Bergelson
Professor David Sivakoff will give a talk Oriented Percolation and the Contact Process on Z
Abstract: In probability theory, a stochastic interacting particle system is a process on a graph (classically a lattice, like Zd) in which each vertex assumes one of two states (occupied or vacant), and randomly updates its state in continuous time based on the states of its neighbors. One of the most well-known stochastic interacting particle systems is the contact process, in which an occupied vertex becomes vacant at rate 1, and a vacant vertex becomes occupied at rate b > 0 times the number of its neighbors that are occupied. Taking our graph to be the integer lattice, Z, and starting with only the origin occupied, a simple analysis shows that if b is too small, then all vertices will eventually become vacant with probability one. A more difficult question is, if the origin is initially the only occupied vertex, can we take b large enough so that there is a positive probability that the occupied vertices never go extinct? We will show that the answer is “Yes” by exploiting a relationship between this process and another well-known model in probability theory called oriented site percolation, and much of the talk will be spent analyzing this model using the famous “contour argument.”
Professor John Johnson will give a talk The Three Finite Sums Theorems of Schur, Folkman, and Hindman
Abstract: In 1916 Issai Schur published a paper that contains, as an important lemma in his proof of Fermat's last theorem modulo a large prime, one of the earliest results in Ramsey theory. This combinatorial lemma, in a simple form, asserts that there exists a positive integer N such that if every integer in {1, 2, ..., N} is colored either red or blue but not both, then there exist integers x and y such that x, y, and x + y are all monochromatic. Later Jon Folkman discovered a multi-dimensional generalization of Schur's lemma. Finally in 1974, Neil Hindman proved a more powerful generalization of both of these results; the proof of Hindman's theorem essentially requires the use of new tools. I'll survey these three finite sums theorems, compare their similarities and differences, and also give an outline for how each one is can be proved.
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.
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.
Graduate School info session, by Professor Thomas Kerler
Sasha Leibman will give a talk Rearrangements of conditionally convergent series of vectors.
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.
Dr. Jim Fowler will be giving a talk entitled 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.
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.
Professor Vitaly Bergelson gives a lecture Some interesting Diophantine equations
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.
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!
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.
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.
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.
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.
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?
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.
Professor Thomas Kerler will conduct a Graduate School information session
Professor Vitaly Bergelson gives a talk Patterns in primes
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).
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!).
Professor Ron Solomon gives a talk on Sporadic Objects
Dr. Bill Mance gives a talk about fractals, Schmidt games, and sets of non-normal numbers.
Rasor-Bareis-Gordon exam debriefing, with Professor Vitaly Bergelson.
Brian Li gives a talk entitled Normal Numbers: A Notion of Randomness.
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.
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?
Professor Warren Sinnott will speak about p-adic numbers.
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.
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.
Professor Chris Miller gives a talk entitled P = NP?
Professor Vitaly Bergelson gives a talk about Srinivasa Ramanujan and his beautiful formulas
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.
Dr. Jim Fowler gives a talk about Desargues' Theorem and projective planes
Rasor-Bareis-Gordon exam debriefing, with Professor Vitaly Bergelson
Ted Dokos will give a talk Hex and Brouwer's fixed point theorem.
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.
Robert Behal, a lawyer, will be speaking about the interplay between mathematics and law.
Adventures of an Academic in Real World Land
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.
Paul Apisa talks about Hindman's theorem.
Jeff Lindquist will give a talk entitled Ford Circles.
Professor Vitaly Bergelson will give a talk Homage to Catalan numbers.
Professor G. A. Edgar will give a lecture entitled A counerexample(?) to the Fundamental Theorem of Calculus.
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"?
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.
Jeff Lindquist, Jack Cheng, and Theodore Dokos talk about Conway Polynomials and Virtual Links.
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.
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.
Assistant Dean Teri Roberts of the College of Public Health will be speaking about graduate programs in Biostatistics and Epidemiology.
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.
Jim Fowler gives a talk entitled Can you divide a square into three triangles of equal area?
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.
Dr. Bart Snapp: The Hailstone problem. In this talk we will discuss a polynomial analogue of the Collatz Conjecture.
Professor G. A. Edgar gives a talk Fractional iteration of series and transseries
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.
Phil Kilanowski is talking about Brownian Motion
Pizza and soda will be served.
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".
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.
Jeff Lanz is speaking about financial mathematics.
Radical Pi is meeting to discuss solutions to the Rasor-Bareis-Gordon examination, REU's, and undergraduate summer research opportunities in mathematics at OSU.
Research in Normal Numbers - a talk about this exciting field and some new results by graduate student Bill Mance. In addition to learning about this topic, you will find out what it is like to be working towards a PhD in Mathematics.
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.
Graph drawing in color - plain graphs don't have to be plain! A talk by Professor Alan Saalfeld.
A talk by Rob Denomme:
A first look at Number Theory.
What are numbers and what makes them so strange?
Bill Mance gives a talk What are Normal Numbers?
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!
Professor Bergelson discusses the solutions of the Raisor-Bareis-Gordon contest problems and provides information on summer research opportunities for undergraduates.
"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".
Professor Mike Davis gives a talk Nonpositively curved spaces
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.
Adam Hammet presents: Algorithm on steroids: How, for a small fee, randomizing can enhance performance.
Professor Dan Shapiro gives a talk Barycentric coordinates. (This will be a quite elementary discussion of using coordinates in triangle geometry.)
Rob Denomme: Elliptic curve primality tests, Alfred Rossi: Virtual links.
Professor Edgar gives a talk A Counterexample(?) to the Fundamental Theorem of Calculus
Professor Overman gives a talk The Logistic map
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.
Undergraduate students presenting topics in Group Theory
Students of H590 will be presenting results they have explored during the honors algebra sequence.
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".
OSU students research experiences
Ohio State's undergraduates discuss their research in the fields available to young mathematicians.
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.
Professor Satyan Devadoss gives a lecture Juggling Links
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.
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.
Professor Alan Saalfeld of the Geodetic Science Department gives a lecture Plus-Minus Paths
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.
Problems on the plane, a lecture by Donny Seelig and Adam Chawansky
Professor Satyan Devadoss gives a lecture Spaces of Trees
Professor Alissa Crans gives a lecture about Knots, Links and Braids
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.
Non-Standard Digits, by Professor Daniel Shapiro.
Professor Henry Glover gives a lecture on Hamilton cycles in Cayley graphs
Professor Dan Shapiro gives a lecture The four numbers game
Professor Gerald A. Edgar gives a lecture on Hausdorff dimension
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.
Professors Peter March and Vitaly Bergelson speak on Research Opportunities for Undergraduates (at OSU and around the country).
Professor Tadeusz Januszkiewicz gives a lecture Position spaces of Penduli
Scott McKinley: Some Thoughts on Modeling Randomness in Continuous Times
Professor Dan Shapiro gave a lecture entitled Walk the Dog
Professor Saleh Tanveer gave a lecture on Mathematics of Bubbles
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.
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.
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:
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"....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." |
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.
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.
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)
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.
Professor Sergei Duzhin: Mathematics of the Accordion
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.
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.
Professor Susan Goldstine: Polynomial Dynamics
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.
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!)
Math club was hosting a special movie session on the life and mathematics of the “Prince of Problem Solving” Paul Erdos.
Professor Susan Goldstine gave a talk Shoestring Arithmetic.
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.
Professor Samir Mathur: String Theory and Black Holes
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.
Professor Peter Brooksbank gave the talk “A guide to a successful marriage in an age of piracy”
Scott McKinley was talking about Two Laws of Large Numbers and the “Supreme Law of Unreason”
Professor Alan Saalfeld, Some Mathematics for Computer Mapping
Professor Susan Goldstine, Sunflowers and the least-rational number
A talk on the results when Topology, Quantum Mechanics and Superconductivity meet.
Professor Vitaly Bergelson speaks on Billiards
Professor Satyan Devadoss: The shape of the universe
Professor Susan Goldstine speaks on Hexaflexagons.
Professor Mark Evans is talking about when the product of two derivatives is a derivative.
Professor Bostwick Wyman talks about Actuarial Science.
Professor Sergei Chmutov: Knots.
Professor Warren Sinnott speaks about The p-adic integers.
Professor Gerald Edgar gives a talk on Fractal Dimension.
Dr. S. Brueggeman presents Solutions to (x+y+z)³=nxyz where x, y, z and n are positive integers.
Measuring liquids by Professor Dan Shapiro.
Roninie Pavlov gives a talk on Various Notions of Size in the Natural Numbers.
Professor Susan Goldstine gives a talk on The Discovery of Non-Euclidean Geometry.
Professor Vitaly Bergelson gives a talk on Continued Fractions Through the Ages.
