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## 2017-2018 Academic Year Colloquium Schedule

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### September 7, 2017

 Title: Ergodic Theory and Normal Numbers. Speaker: Drew D. Ash Adjunct Assistant Professor Mathematics and Computer Science Albion College Albion, Michigan Abstract: The purpose of this talk is to expose the audience to subfield of dynamical system called ergodic theory. To do so, we will consider the following question. How many numbers in $[0,1)$ are there when we look at their base-10 decimal expansion have the following property: The asymptotic (or expected) frequency of seeing the digit $d$, $d\in\{0,1,\dots,9\}$, is $1/10$? Can you even think of a number that has this property? We will show, using ergodic theory, that a surprising amount of numbers have this property! If time allows, we will discuss another interesting transformation called the Gauss map. The Gauss map has connections with continued fractions! Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### September 14, 2017

 Title: Planning for Graduate Study in Mathematics and Computer Science Speaker: David A. Reimann Professor Mathematics and Computer Science Albion College Albion, Michigan Abstract: 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. Location: Palenske 227 Time: 3:30 Citation Click for BibTeX citation Flyer Click for a printable flyer

### September 21, 2017

 Title: How to Become An Extremal Graph Theorist Speaker: Lauren Keough Assistant Professor Mathematics Grand Valley State University Allendale, Michigan Abstract: Graph theory is the study of relationships that come in pairs. There are many such relationships occurring naturally, think of matching medical students to residencies, friendship on social networks, or even pairing animals with the regions in which they live. From these relationships we can draw graphs. For example, for each person on a social network draw a dot, and draw a line segment between two dots if the people are "friends". Graph theory is, broadly, the study of these pictures with these dot lines. So, what could extremal graph theory be? Unfortunately extremal graph theory is not doing graph theory while snowboarding. Think of "extremal" more like you may have in Calculus 1 — perhaps you remember finding "local and absolute extrema." By the end of the talk you'll be able to ask and answer extremal questions and perhaps even know a new card trick. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### September 28, 2017

 Title: The Black-Scholes- Merton Equation and Option Pricing Speaker: Darren E. Mason Professor Mathematics and Computer Science Albion College Albion, Michigan Abstract: The Black-Scholes option pricing formula is a 1997 Nobel Prize winning result in economics (& mathematical finance) that provides a framework for rational pricing of a large class of stock options. In this talk we will discuss the basic idea of an option on an asset as well as the problem of fair valuation of such a financial object. Then, assuming that the stock price St follows a geometric Brownian motion, we will discuss a rough hedging argument that results in the Black-Scholes partial differential equation as a necessary condition for risk-free portfolio evolution. Using changes of coordinate systems and integrating factors, the Black-Scholes partial differential equation will be transformed into the classic heat (or diffusion) equation, for which a standard integral solution form is known. Finally, we will use this integral solution to derive the celebrated Black-Scholes option pricing formula. Some limitations of this model will also be discussed. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### October 19, 2017

 Title: Symmetry: A mathematical approach using group theory and linear algebra Speaker: David A. Reimann Professor Mathematics and Computer Science Albion College Albion, Michigan Abstract: Symmetric patterns are used in many situations to decorate an object with a repeating motif that is translated, rotated, or reflected without changing size. We will see examples of several symmetry types and look at these from the vantage point of group theory. In particular, we will study rosette patterns, frieze patterns, wallpaper patterns, and patterns on the sphere. We will then see how we can create all these pattern types with a unified framework based on the vectors and matrices of linear algebra. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### November 2, 2017

 Title: All Parabolas Through Three Non-collinear Points Speaker: Michael A. Jones Associate Editor Mathematical Reviews Ann Arbor, MI Abstract: There are an infinite number of parabolas through any three non-collinear points. In this talk, I'll explain how solving a system of three equations and three unknowns and applying rotation matrices can be used to find the parabolas. The parabolas form a one parameter family. Geometric intuition about when a parabola doesn't exist for three specific values of the parameter is verified by recognizing when the equation for the parabola is undefined. Looking at the family from a calculus perspective, one can find the parabola with the widest mouth through the three points. We will use Desmos online software to visualize all the parabolas for an example. This talk is based on an article of the same title that is co-authored with Stanley R. Huddy and is forthcoming in the July 2018 issue of The Mathematical Gazette. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### February 1, 2018

 Title: A Mechanism Design Approach to Allocating Travel Funds Speaker: Michael Jones Associate Editor Mathematical Reviews/American Mathematical Society Ann Arbor, MI Abstract: 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. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### February 8, 2018

 Title: Diagonally dominant random matrices: Physical questions, Mathematical challenges Speaker: Rajinder Mavi Postdoctoral Researcher Institute of Mathematical Physics and Department of Mathematics Michigan State University East Lansing, MI Abstract: 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. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### February 15, 2018

 Title: Using Geometry to do Number Theory Speaker: Mckenzie West Visiting Assistant Professor Math Department Kalamazoo College Kalamazoo, MI Abstract: 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. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### February 22, 2018

 Title: Drawing Graphs on Surfaces Speaker: Heather Jordon Associate Professor of Mathematics Mathematics & Computer Science Albion College Albion, MI Abstract: A graph G consists of two sets, a finite nonempty set V of vertices and a set E of edges, where each edge is an unordered pair of distinct vertices. When we draw a graph, we want edges to intersect only at vertices, called an embedding of the graph. It turns out that not every graph can be embedded in the plane but every graph can be embedded in 3-dimensional space (even with straight line segments for edges). In this talk, we will discuss drawing graphs on surfaces that are "in between" the plane and 3-dimensional space. These surfaces will be compact 2-manifolds, and may orientable or non-orientable. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### March 1, 2018

 Title: Graceful Colorings In Graphs Speaker: Alexis Byers PhD Candidate Mathematics Western Michigan University Kalamazoo, MI Abstract: 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. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### March 22, 2018

 Title: Coloring the $n$-smooth numbers with $n$ colors Speaker: Andrés Caicedo Associate Editor Mathematical Reviews Ann Arbor, MI Abstract: 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. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### March 29, 2018

 Title: Finding cycles in graphs Speaker: Michael Santana Assistant Professor of Mathematics Mathematics Grand Valley State University Allendale, MI Abstract: 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! Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer

### April 5, 2018

 Title: The $P^2+P'$ problem and George Polya Speaker: Stephanie Edwards Professor of Mathematics Mathematics Hope College Holland, MI Abstract: 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. Location: Palenske 227 Time: 3:30 PM Citation Click for BibTeX citation Flyer Click for a printable flyer