# NIU Department of Mathematical Sciences

# Upcoming Colloquia and Seminars

### Math Department Colloquium

**Friday, April 24**, 4:00-5:00 p.m. in DU 348

Speaker:

**Juan Pablo Mejia-Ramos**, Rutgers Univ.

Title:

*On the presentation of proofs and the assessment of proof comprehension in undergraduate mathematics courses*

**Thursday, April 23**, 4:00-5:00 p.m. in DU 152

Speaker:

**Juan Pablo Mejia-Ramos**, Rutgers Univ.

Title:

*Reading and evaluating mathematical text: expert and novice approaches*

### Seminars

Algebra Seminar: | Wednesday, Apr. 22, 4:15-5:15 p.m. in DU 378 | |

Speaker: | Mike Geline | |

Topic: | Introduction to Representation Theory and Brauer's
Induction Theorem
| |

Abstract: | This seminar series will be aimed at graduate students, and will constitute an informal (but hopefully informative) course on representations of finite dimensional algebras with an emphasis on representations of finite groups. We will begin with a (hopefully brief) review of semisimplicity and the distinction between irreducible (versus absolutely irreducible) modules. We'll define characters early as well. Then we'll define the ubiquitous concept of induced representations (and induced characters) with the aim of proving a fascinating and powerful result known as Brauer's induction theorem (or if time permits one of its stronger variations). Brauer's theorem began life as a conjecture of Emil Artin, who anticipated an application to L-functions attached to representations of Galois groups. But it turned out to have many applications to group theory as well, and also quite surprisingly to certain questions in algebraic topology. |

Complex Analysis Seminar: | Tuesday, Apr. 21, 11:00-11:50 p.m. in DuSable 464 | |

Speaker: | Michael Geline | |

Topic: | Artin's L-functions
| |

Abstract: | Let K be a finite Galois extension of the rational numbers with Galois group G. In an effort to understand the decomposition of primes in the ring of algebraic integers of K, Artin attached a so-called "L-function" to each complex representation of G. These are functions of a complex variable. I will present the definition of these functions, discuss their basic properties, and explain how they led Artin to a conjecture about finite groups that ultimately became "Brauer's induction theorem." |

Applied Math Seminar: | Tuesday, Apr. 21, 11:30-12:20 p.m. in Watson 110 | |

Speaker: | James Benson | |

Topic: | Continuous time Markov models and Markov Chain Monte Carlo
methods for modeling ice propagation in tissues
| |

Abstract: | Propogation of ice in tissues is a stochastic process that for an individual cell depends on the number of frozen neighbors. The system is thus well modeled by a continuous time discrete state Markov process. For (very) small cell systems, the state transition matrix can be written and solved explicitly, but the number of states scales with at least 2^[number of cells]. Attempts have been made to use Markov chain Monte Carlo (MCMC) methods to estimate the behavior, but applications have been limited to at most 10^5 cells and are computationally expensive. In this talk I will present a brief backgound of the ice formation model and an example calculation for a 4-cell system. I then show a novel and useful state-space reduction technique that allows simple analytic solutions for system quantities of interest. Interestingly, this reduction technique requires Monte Carlo estimation of parameters related to the classical "packing problem" that generate a surprising quadratic relationship between complicated objects. Finally, I present possibilities for modern mathematical approaches to solving the associated large (10^6 cell) MCMC system and compare these results with the analytic solution. |