Math 127 | Trigonometry | Fall 2018 | |

## Course ScheduleLegend
Class Topics Reading Assignments Assignments Due College and Departmental Calendar Items
Mathematics and Computer Science Colloquium
A Mechanism Design Approach to Allocating Travel Funds
Michael Jones Associate Editor Mathematical Reviews/American Mathematical Society Ann Arbor, MI In mathematics and other disciplines, faculty members are required to give professional talks at conferences on their research or teaching. When I was at Montclair State University in New Jersey, the financial requests for travel exceeded the amount the School of Science and Mathematics had budgeted, which meant that only a percentage of travel was covered. Because faculty were exploiting the method used to distribute limited travel funds among the faculty, the associate dean asked me to construct a new method. In this talk, I'll explain the old method to award travel funds and how faculty were misrepresenting their financial needs to get a higher percentage of their travel paid for. Then, I'll explain the new method. The new method views allocating travel funds as a game. The method constructs a game in which it is each of the faculty member's best interest to reveal truthfully their financial needs. Thus, being truthful is a Nash equilibrium of the game. The method has the added benefit that it encouraged faculty to be conservative in their spending so that they get a higher percentage of their travel paid for. The process of constructing such a game is called mechanism design. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Diagonally dominant random matrices: Physical questions, Mathematical challenges
Rajinder Mavi Postdoctoral Researcher Institute of Mathematical Physics and Department of Mathematics Michigan State University East Lansing, MI A remarkable phenomenon in quantum physics is that impurities in solid state materials will inhibit mobile quantities, such as electrons, spin orientations, and even information. The consequences range from the familiar to the remarkable: copper wires with impurities of aluminum or silicon have higher resistivity, inhibiting the wire's conductance, a more surprising effect is that, left to their own devices, strongly disordered materials do not reach their thermodynamic equilibrium! This phenomenon is known as Anderson localization and it is a fundamental part of the theory of solid state physics. In the future, we might find applications for disordered materials exhibiting such behavior playing an important role in the construction of quantum computer microchips. A simple mathematical model exhibiting a physically relevant approximation to a disordered material is a diagonally dominant random matrix. A diagonally dominant random matrix is a random diagonal matrix perturbed by a symmetric, non-random, sparse matrix. In quantum mechanics, one is typically interested in the properties of the eigenbasis, i.e. the eigenvectors and eigenvalues of the matrix. If the system is one dimensional, or if the perturbation is small, the eigenbasis is similar to the unperturbed matrix. That is to say, most of the `mass' of most eigenvectors is at a single entry of the vector. Although this may seem unremarkable, the difficulty is showing this is true for fixed perturbation strength with probability one, regardless of the size of the matrix. We will also compare eigenbases of diagonally dominant random matrices to to eigenbases of `traditional' random matrices which have i.i.d. random variables at all entries of the matrix. In the later case, the mass of each eigenvector is more or less equally distributed over all entries of the vector. We will then examine an interesting interpolation between diagonally dominant random matrices and traditional random matrices. Finally, we will discuss some recent results and current questions in the field of diagonally dominant random matrices today. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Using Geometry to do Number Theory
Mckenzie West Visiting Assistant Professor Kalamazoo College Kalamazoo, MI Polynomial equations and their solutions form a cornerstone of mathematics. Solutions with rational coordinates are particularly intriguing; a fantastic surprise is the great difficulty of determining the mere existence of a rational solution to a given equation (let alone the complete set). We will discuss this problem in two cases, diagonal cubic surfaces, \[ax^3+by^3+cz^3+d=0,\] and degree 2 del Pezzo surfaces, \[ax^4+by^4+cx^2y^2+d=z^2.\] A surprising and successful modern approach, the Brauer--Manin obstruction, employs tools from linear algebra, geometry and non-commutative algebra. I will discuss a collection of interesting and motivating examples with simultaneous historical and modern interest, and also explain some of the tools and techniques that form the backbone of my research program. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Drawing Graphs on Surfaces
Heather Jordon Associate Professor of Mathematics Mathematics & Computer Science Albion College Albion, MI A 3:30 PM All are welcome! Mathematics and Computer Science Colloquium
Graceful Colorings In Graphs
Alexis Byers PhD Candidate Mathematics Western Michigan University Kalamazoo, MI We describe how a 19th century problem on sets led to a 20th century problem on decompositions of graphs. This, in turn, resulted in a graph labeling problem which gracefully led to a 21st century concept on colorings of graphs. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Coloring the $n$-smooth numbers with $n$ colors
Andrés Caicedo Associate Editor Mathematical Reviews Ann Arbor, MI Fix a number $n$. Can we color the positive integers using precisely $n$ colors in such a way that for any $m$, the numbers $m, 2m, \ldots, nm$ all receive different colors? The question was posed by Péter Pach about 10 years ago. To this day it remains open in general, although some cases are known. I will present a survey of known results and some other problems it leads to. This is joint work with Pach and my former master's student Tommy Chartier. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Finding cycles in graphs
Michael Santana Assistant Professor of Mathematics Mathematics Grand Valley State University Allendale, MI In chess, a "knight" is a piece that has the ability to move two spaces vertically (or horizontally) and one space horizontally (or vertically). This unusual movement led to questioning whether or not it was possible for a knight to travel the entire board and end where it started, visiting all other spaces exactly once. This question turns out to be one of the earliest cases of the Hamiltonian cycle problem in graph theory (a notoriously difficult problem that has inspired many other cycle structure problems). In this talk, we'll see explore some of these cycle structure problems (some of which are very recent!), and see how doing research in mathematics can be like playing Jenga! Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
The $P^2+P'$ problem and George Polya
Stephanie Edwards Professor of Mathematics Hope College Holland, MI Many open problems in entire function theory, specifically, the distribution of zeros of real entire functions, can be tracked back to work by George Polya. One of these such problems was stated in a Polya and Szego text from the early 1900's: If $P$ is a real polynomial with only real zeros, find the number of non-real zeros of $P^2+P'$. If one removes the hypothesis that $P$ has only real zeros, the problem becomes quite difficult and was not solved until the 1980's. We will discuss a simple solution to the problem, look at natural questions that arise from the problem and discuss some open questions which have their roots in Polya. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Casino Carnival Games: Past, Present, and Future
Mark Bollman Professor of Mathematics Albion College Albion, MI Beyond the "big four" casino table games of baccarat, blackjack, craps, and roulette, over 1000 different games have been designed, proposed for casino use, and approved by the state of Nevada. In this presentation, we shall look at the math behind some of the games that have fallen by the wayside and at the mathematical issues that arise in designing a new game of chance. An opportunity to investigate the mathematics behind a new game proposal will be announced. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Planning for Graduate Study in Mathematics and Computer Science
David A. Reimann Professor Albion College Albion, Michigan A degree in mathematics or computer science is excellent preparation for graduate school in areas such as mathematics, statistics, computer science, engineering, finance, and law. Come learn about graduate school and options you will have to further your education after graduation. Palenske 2273:30 All are welcome! Mathematics and Computer Science Colloquium
Reducibility and Balanced Intransitive Dice
Michael Ivanitskiy ^{1} and Michael A. Jones^{2}^{1}University of Michigan; ^{2}Mathematical ReviewsAnn Arbor, MI We will review some results about balanced intransitive $n$-sided dice and what it means for a set of dice to be reducible based on a concatenation operation. Using data from the Online Encyclopedia of Integer Sequences, the lexicographical ordering of dice, and permutations, we are able to construct new integer sequences representing the number of $n$-sided reducible and irreducible dice. We define a notion of margin and explain how margins are effected by concatenation. We introduce a new splicing operation that generalizes concatenation and give conditions for when the resulting dice are balanced and irreducible. Finally, we construct new integer sequences for the number of fair, balanced dice and the largest margins for $n$-sided, balanced intransitive dice.
3:30 PM All are welcome! Mathematics and Computer Science Colloquium
Multi-Lens Analysis of Office Dynamics and
Space Usage
Angela Morrison, '17 Graduate Student Mathematics Michigan State University East Lansing, Michigan Workplace optimization is critical for organizations to make the most of their real estate as well as help employees stay more productive at work. Steelcase Space Analytics equips organizations with tools and data needed to measure and improve the effectiveness of the workplace by applying their proprietary sensing capability. This project aims to analyze office space dynamics and usage by investigating correlations between and within sensing and survey datasets sourced from Steelcase's 2 West (2W) facility. The data consists of sensor output that describes how often spaces are in use, as well as survey data that reports how the 2W employees feel about using certain spaces. Clustering analysis was developed to study the hidden trends of the sensor data and generalized linear mixed model (GLLM) was constructed to investigate the correlations between the sensor data and space traits data. The results showed that the significant space traits indicated by the GLMM were also the popular ones from survey data analysis. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Spherical Panoramic Photographic Processing
David A. Reimann Professor Albion College Albion, Michigan Conventional cameras view a small solid angle, limiting the both the field of view and projective distortion. However, multiple individual pictures are need to have full spherical coverage. Cameras that can directly take spherical panoramic photos, such as the Ricoh Theta S, have become available as relatively inexpensive consumer products. Unlike a traditional camera, this camera has two hemispherical lenses, allowing it to see simultaneously in every direction around the camera. These cameras produce an equirectangular projection, where each latitude row has the same number of pixels, which has severe distortion at the poles. One approach to hardcopy display is to map the image onto the surface of a small polyhedron, such as a Platonic or Archimedean solid, which reduces the distortion. Using such polyhedra resembles the process artist Dick Termes uses for painting on a sphere, which he calls a Termesphere. These techniques force the viewer to see the world inside-out. This work maps the spherical photo to the inside of a large polyhedra to create a miniature pavilion which can be entered for a personal panoramic experience. Other interesting applications and issues will be discussed. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Solving an Easy Equation in an Interesting Way
Mckenzie West Visiting Assistant Professor Kalamazoo College Kalamazoo, Michigan In number theory, we are interested in studying the behavior of integers. For example, Fermat's Last Theorem classifies all triplets of integers that satisfy $x^n+y^n=z^n$. In this talk, we will look at a special collection of "integers", the $S$-units, and determine every solution to the equation $x+y=1$. We will draw from areas such as linear algebra, geometry, and computer algorithms in this talk as we cover my relation to the $S$-unit equation and its applications. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
Mathematics and Computer Science in Sports
Jon Lorenz Business Analyst Business Strategy and Analytics Detroit Lions Detroit, MI The sports industry is incredibly competitive. How do we find business advantages through computer science and mathematics? I will walk through my own personal path to working in the NFL, what kind of analysis and contributions I make on a daily basis, how math and CS are essential to best business practices, and what opportunities are arising with the Lions this winter. Palenske 2273:30 PM All are welcome! Mathematics and Computer Science Colloquium
The Geometry of Polynomials
Matt Boelkins Professor of Mathematics, Grand Valley State University Grand Valley State University Allendale, MI In the 3:30 PM All are welcome! Mathematics and Computer Science Colloquium
Random Walks and Ito Calculus with Applications to Finance
Darren E. Mason Professor Albion College Albion, MI Rational pricing of many financial products rely on mathematical modeling of random phenomena (e.g. the time evolution of a stock price). In this talk I will discuss the fundamental ideas of random walks, Brownian motion, and Ito Calculus, with applications actuarial science and finance. This presentation also serves as a nice introduction to what can be expected by students in the last portion of the topics course Math 389 being offered in Spring 2019. Palenske 2273:30 PM All are welcome!
Copyright © 2018, David A. Reimann. All rights reserved. |