SCU Mathematics/CS Colloquium Series Schedule
All talks will be in O'Connor 206, unless otherwise noted.
Title: A reciprocity formula for the growth series of right-angled Coxeter groups
Abstract: Right-angled Coxeter groups are a combinatorial generalization of the class of groups generated by reflections across hyperplanes that are either perpendicular or parallel. The standard growth series for such a group is the power series G(t) whose n-th coefficient is the number of group elements of length n (measured with respect to the standard generating set). The structure of a right-angled Coxeter group is completely encoded in a certain simplicial complex, and when this complex is a sphere, a theorem of Ruth Charney and Michael Davis says that the growth series satisfies the reciprocity formula: G( 1 / t )= +/- G(t). In this talk, we shall describe a certain noncommutative growth series for a right-angled Coxeter group defined with respect to a certain greedy normal form for elements of the group. We will show that this series also satisfies a reciprocity formula, implying not only the Charney-Davis formula, but also a host of other multivariate versions of reciprocity.
Title: Local Properties of Colored Graphs
Abstract: Most people have seen a tiling problem where you are given a set of tiles and asked to cover the plane with those tiles so that there are no holes. In this talk, we explore a similar problem except, instead of beginning with a set of tiles, we begin with a set of finite rooted graphs whose vertices are colored. We say that a graph is "tiled" by our set of graph tiles if every vertex in the graph looks like the root of one of our graph tiles. We will discuss the problem of determining whether there is a graph that can be tiled by a given set of our graph tiles. This talk should be accessible to anyone but knowledge of basic linear algebra will be useful.
Title:The Simple Mathematics of Utilitarianism, with Some Applications to Distributive Justice and Global Warming
Abstract: Utilitarianism is the ethical doctrine that actions and states of affairs should be judged according to their contribution to total happiness or satisfaction in society, as measured by the sum of individual utilities. The best distribution of society's resources is therefore the one that maximizes total utility. In mathematical terms, this comes down to a fairly straightforward constrained optimization problem. While frequently maligned as a philosophical doctrine, some version of utilitarianism underlies much of the normative (prescriptive) work done by economists.
In this presentation I will discuss the core assumptions of a mathematical version of utilitarianism and apply simple calculus to explore some properties of the utilitarian optimum. I will also show how utilitarianism, which seeks a societal optimum, is closely related to and can in fact be derived from a model of optimal individual decision making, given an assumption of impartiality. This leads to an interpretation of utilitarianism as a kind of social insurance. Finally, I will discuss recent debates among economists about how much present sacrifice should be made to mitigate the future impact of global warming, a decision that can be framed as an application of utilitarianism in an intergenerational setting. This allows us to clarify some important issues concerning time discounting.
Title: Conics: Searching for Beauty Reveals Deeper Truths
Note special day.
Abstract: Conic sections have occupied a central position in geometry since antiquity. In this talk we view conic sections from a perspective that makes the whole study more unified and more beautiful. Our principal guides--symmetry and eliminating exceptions --will lead us to the "natural habitat" of conics, the complex numbers. This will open our eyes to some startling facts that the traditional approach misses.
Title: Bidding Games
Abstract: What happens if you play your favorite two-player game, such as Connect-Four or chess, but instead of alternating moves you bid against your opponent for the right to move? For instance, suppose you and your opponent both start with one hundred chips. If you bid ten for the first move, and your opponent bids twelve, then your opponent gives you twelve chips and makes the first move. Now you have one hundred and twelve chips, your opponent has eighty-eight, and you bid for the second move. The goal is simply to win the game; chips have no value when the game is over. The basic theory of such games is simple and elegant, with a surprising relation to random-turn games, in which the right to move is determined by a coin flip.
Title: The Algebra of GPS
Abstract: This talk is centered on some of the algebraic underpinnings of the Global Positioning System (GPS). Just how does a GPS receiver isolate one particular satellite to use in its measurement protocol from among 24 choices? The answer involves error correcting codes, maximal length sequences, and a little bit of finite field theory. The talk will blend the flavors of linear and modern algebra over GF(2) with a smattering of combinatorics to reach the final piquant characterization of GPS identification codes, otherwise known as Gold codes. Along the way, the basic operations of a GPS receiver along with a brief snapshot of the underlying measurement protocol will be revealed.
If you have a disability and require a reasonable accommodation, please call or email Frank Farris 408-554-4430 or ffarris at scu dot edu .