# NIU Department of Mathematical Sciences

# Upcoming Colloquia and Seminars

**January 20-23, 2015**

### Math Department Colloquium

**Friday, March 6**, 4:00-5:00 p.m. in DU 348

Speaker:

**Steven Doty**, Loyola Univ.

Title:

*Generalizations of Schur-Weyl duality*

**Abstract:**Classical Schur-Weyl duality (the name is a misnomer) is a double centralizer property for the commuting actions of a symmetric group and a general linear group on a tensor power of a vector space. Discovered by Issai Schur in a landmark 1927 paper, this duality relates the representation theories of general linear and symmetric groups. After explaining the classical picture, I will discuss a few of the many generalizations, starting with work of Richard Brauer on other classical groups and extending through more recent times. These generalizations have led to various diagram algebras, a class of finite dimensional algebras possessing a basis given by certain graphs, along with a graphical based multiplication. This leads to interesting problems in combinatorics. There are connections to mathematical physics, representation theory, and knot theory.

### Seminars

Algebra Seminar: | Wednesday, Mar. 4 , 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, Mar. 3, 11:00-11:50 p.m. in DuSable 464 | |

Speaker: | Ben Wallis | |

Topic: | A new family of operator ideals
| |

Abstract: | Using upper $\ell_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ with parameters $1\leq p\leq\infty$ and $1\leq\xi\leq\omega_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\leq p\leq\infty$ and, when $p\neq 1$, for infinitely many $1\leq\xi\leq\omega_1$. For the case $\xi=1$, there exists an ideal norm $\|\cdot\|_{(p,1)}$ on the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ under which it forms a Banach ideal. We also prove that each space $\mathcal{WD}_{\ell_p}^{(\infty,\omega_1)}(X,Y)$ is the intersection of the spaces $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}(X,Y)$ over all $1\leq\xi<\omega_1$. |

Applied Math Seminar: | Tuesday, Mar. 3, 11:30-12:20 p.m. in Watson 110 | |

Speaker: | Nathan Krislock | |

Topic: | Semidefinite Optimization and Combinatorial Optimization
| |

Abstract: | During the last two decades, semidefinite optimization has grown into a significant field of research with applications in many diverse areas such as graph theory, distance geometry, combinatorial optimization, low-rank matrix completion, and polynomial optimization. In combinatorial optimization, semidefinite optimization is used to compute high-quality bounds to many difficult (in fact, NP-hard) problems, such as Max-Cut and maximum k-cluster. This has led to the development of state-of-the-art branch-and-bound methods for solving such problems to optimality. |