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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

### 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

### 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

### 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

### 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