# Upcoming Colloquia and Seminars

### Math Department Colloquium

Friday, Sept. 4, 18 and 25, 4:00-5:00 p.m. in DU 348
Speaker: Biswa Datta, NIU
Title:   Computational and Optimization Methods for Quadratic Inverse Eigenvalue Problems Arising in Mechanical Vibration and Structural Dynamics: Linking Mathematics to Industry

Abstract: The Quadratic Eigenvalue Problem is to find eigenvalues and eigenvectors a quadratic matrix pencil of the form P (λ) = M λ 2 + Cλ + K , where the matrices M, C, and K are square matrices. Unfor- tunately, The problem has not been widely studied because of the intrinsic difficulties with solving the problem in a numerically effective way. Indeed, the state-of-the-art computational techniques are capable of computing only a few extremal eigenvalues and eigenvectors, especially if the ma- trices are large and sparse, which is often the case in practical applications. The inverse quadratic eigenvalue problem, on the other hand, refers to constructing the matrices M, C, and K, given the complete or partial spectrum and the associated eigenvectors. The inverse quadratic eigenvalue problem is equally important and arises in a wide variety of engineering applications, including mechanical vibrations, aerospace engineering, design of space structures, structural dynamics, etc. Of special practical importance is to construct the coefficient matrices from the knowledge of only partial spectrum and the associated eigenvectors. The greatest computational challenge is to solve the partial quadratic inverse eigenvalue problem using the small number of eigenvalues and eigen- vectors which are all that are computable using the state-of-the-art techniques. Furthermore, computational techniques must be able to take advantage of the exploitable physical properties, such as the symmetry, positive definiteness, sparsity, etc., which are computational assets for solution of large and sparse problems. These talks will deal with two special quadratic inverse eigenvalue problems that arise in mechanical vibration and structural dynamics. The first one, Quadratic Partial Eigenvalue Assignment Problem(QPEVAP), arises in controlling dangerous vibrations in mechanical structures. Mathe- matically, the problem is to find two control feedback matrices such that a small amount of the eigenvalues of the associated quadratic eigenvalue problem, which are responsible for dangerous vibrations, are reassigned to suitably chosen ones while keeping the remaining large number of eigenvalues and eigenvectors unchanged. Additionally, for robust and economic control design, these feedback matrices must be found in such a way that they have the norms as small as possible and the condition number of the modified quadratic inverse problem is minimized. These considerations give rise to two nonlinear unconstrained optimization problems, known respectively, as Robust Quadratic Partial Eigenvalue Assignment Problem (RQPEVAP) and Minimum Norm Quadratic Partial Eigenvalue Assignment Problem (MNQPEVAP). The other one, the Finite Element Model Updating Problem (FEMUP), arising in the design and analysis of structural dynamics, refers to updating an analytical finite element model so that a set of measured eigenvalues and eigenvectors from a real-life structure are reproduced and the physical and structural properties of the original model are preserved. A properly updated model can be used in confidence for future designs and constructions. Another major application of FEMUP is the damage detections in structures. Solution of FEMUP also give rises to several constrained nonlinear optimization problems. I will give an overview of the recent developments on computational methods for these difficult nonlinear optimization problems and discuss directions of future research with some open problems for future research. The talk is interdisciplinary in nature and will be of interests to computational and applied mathematicians, and control and vibration engineers and optimization experts.

### 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.

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