# 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. 28, 11:30-12:20 p.m. in Watson 110 | |

Speaker: | Scott Rexford | |

Topic: |
The spectrum of certain elliptic operators and Weyl's asymptotic law
| |

Abstract: | A description the eigenvalue problem for the Laplace-Dirichlet operator will be given, motivated by some simple, tractable examples. An overview of a rigorous proof of existence of a basis of eigenvectors in the Sobolev space H^1 will be given, the formulas for the asymptotic bounds on the eigenvalues will be discussed, along with applicability to more general elliptic operators in the H^1 setting. Hard formulas for the asymptotic bounds will be derived for the tractable examples. Finally, an application to the buckling spectrum of a beam will be discussed. |