Date | Speaker | Title | Abstract | Notes |
---|---|---|---|---|

9/4 |
George Andrews Penn State |
Ramanujan, Fibonacci numbers, and Continued Fractions
or Why I took Zeckendorf's Theorem along on my last trip to Canada |
This talk focuses on the famous Indian genius, Ramanujan. One object will be to give some account of his meteoric rise and early death. We shall try to lead from some simple problems involving Fibonacci numbers to a discussion of some of Ramanujan's achievements including some things from his celebrated Lost Notebook. | Notes by Loren Anderson and Jason Green |

9/11 |
Yuri Suhov
Penn State / University of Cambridge, UK |
Introduction to Entropy | The entropy is a famous quantity which is used widely in Math, Physics, Biology, Economics, let alone Information Theory. The concept of entropy is also popular in culture: it inspired (and continues to inspire) poets, artists and musicians. I will introduce and discuss basic properties of entropy which are of interest in many applications. Some of them will be quite surprising. I will also tell some elegant stories involving entropy. No preliminary knowledge of probability theory is required, apart from common sense and first principles. | Notes by Andres Arroyo and Hilton Galyean |

9/18 |
John Roe Penn State |
Commutative implies accociative? |
By introducing the symbol i, with i^{2}=-1, one can pass from the field of real numbers to the larger field of complex numbers. In the 19th century various attempts were made to define still larger "generalized number" fields, such as the quaternions and octonions, but all of these sacrifice some of the familiar "laws" of arithmetic: the quaternions are no longer commutative, the octonions not even associative. Notice that the commutative law apparently "dies" first. Around 1940, Heinz Hopf made an investigation of generalized number systems that were commutative but not necessarily associative, and he found that the reals and the complexes are the only examples. In other words, the commutative law implies the associative law (in the context in which he was working). Hopfs methods are topological, and are closely related to developments in topology in the latter half of the 20th century. |
Notes by Bobby Lumpkin and Todd Fenstermacher |

9/25 |
Vishal Vasan Penn State |
The William Pritchard Fluid Mechanics Laboratory |
Fluid mechanics is a very old branch of mathematics. However it is not only the source of some of the most difficult problems in mathematics but also a very relevant area of research in the modern age. Penn State is one of the few Departments of Mathematics that houses a physical laboratory to conduct experiments in fluid mechanics. In this talk, I give a brief description of some of the experiments we perform in the lab, the physical questions being asked as well as the associated mathematics. Then we will walk down to the basement for a tour of the lab and its facilities. | |

10/2 |
Carina Curto Penn State |
Stimulus space geometry and topology from neural activity |
Neural activity data can be used to infer subsets of co-active neurons in a network. By considering neurons in the hippocampus that encode position information, I will show how these data can be used to infer topological and geometric features of the stimulus space the neurons are encoding. Our results rely on an unexpected application of the Nerve Lemma from algebraic topology. | |

10/9 |
No colloquium midterm exam discussion |
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10/16 |
Vitaly Bergelson Ohio State University |
Ramsey Theory and Dynamics |
We will start the talk with formulating and discussing some of the classical results of Ramsey theory, a branch of combinatorics which studies the structure of mathematical objects that is preserved under partitions. Next, we will show that some of these results can be naturally viewed as dynamical questions about the recurrence in topological and/or volume preserving systems. We will conclude with the discussion of some of the recent developments and open problems. | |

10/23 |
Simon Tavener Colorado State University |
Evolution of resistance to white pine blister rust in high-elevation pines |
Five-needle white pines play an important role in high-elevation ecosystems but are highly susceptible to white pine blister rust (WPBR) caused by a nonnative fungal pathogen. We construct a nonlinear, stage-structured infection model to investigate the effect of WPBR on the dynamics and stand structure of high-elevation five-needle white pines. Management decisions are by definition short-term perturbations that require analysis of transient behavior and we have developed a general software package to examine both transient and equilibrium sensitivities and elasticities. The presence in a population of a resistant genotype can modify both transient and equilibrium behaviors and suggest potential new control strategies. We extend our model to include a resistant allele at a single genetic locus and provide preliminary results. This work was conducted as part of an NSF sponsored undergraduate research program (FEScUE) at the intersection of mathematics and biology. | |

10/30 |
Greg Lawler University of Chicago |
Random walks: simple and self-avoiding |
Many phenomena are modeled by walkers that wander randomly. The case of complete randomness is well understood -- I will survey some of the key facts including the idea that the set of points visited by a random walker in any dimension (greater than one) is two. I will then discuss a much harder problem -- what happens when you do not allow the walker to return to points? Many of the interesting questions about this "self-avoiding walk" are still open mathematical problems. | Notes by John Hirdt Hongxu Wei and Xiaolan Yuan |

11/6 |
Thomas Tucker University of Rochester |
Solutions to polynomials in two variables |
You may remember the quadratic formula for finding solutions
to quadratic polynomials in one variable. It is natural to ask: are
there formulas like this for polynomials of higher degree? The
answer, roughly speaking, is yes. Going further, one might ask: what
about polynomials in more than one variable? Here, the answer is far
more complicated, and involves geometry in what may seem a surprising
way. One famous example of this type of polynomial equation is the
Fermat equation x^{n} + y^{n} = z^{n}. |
Notes by Zachary Wampler and Xi (Amanda) Wang |

11/13 |
Richard Schwartz Brown University |
Lengthening the edges of a tetrahedron |
I'll describe what happens to the volume of a tetrahedron when one lengthens some or all of the edges of the tetrahedron. The analysis involves the Cayley-Menger determinant, a computer algorithm for certifying that a polynomial is positive on a simplex, and a very pretty triangulation of the moduli space of "tetrahedron-like" labelings of the complete graph on 4 vertices. | Notes by Chen Cai and Karim Shikh Khalil |

11/20 |
Alberto Bressan Penn State |
PDE models of traffic flow | Daily traffic patterns are the result of a large number of individual decisions, where each driver chooses an optimal departure time and an optimal route to reach destination. From a mathematical perspective, traffic flow can be modeled by a family of conservation laws, describing the density of cars along each road. In addition, one can introduce a cost functional, accounting for the time that each driver spends on the road and a penalty for late arrival. In this talk I shall explain how to construct solutions of these PDEs, and discuss the existence of (i) globally optimal solutions, minimizing the sum of the costs to all drivers, and (ii) Nash equilibria, where no driver can lower his individual cost by changing his own departure time, or the route taken to destination. An intriguing mathematical problem is to understand the dynamic stability of Nash equilibria. In this direction, some numerical experiments and conjectures will be presented. |