SCU Mathematics Colloquium Series Schedule
Title: Alternating Sign Matrices
Abstract: I will tell the story of the Alternating Sign Matrix Conjecture, a tale that began in 1980 with a conjecture inspired by Charles Dodgson's (aka Lewis Carroll) algorithm for evaluating determinants. The conjecture was proven in 1996 when Greg Kuperberg realized that the square ice model from statistical mechanics held the key to the solution. During this 16 years, the search for a solution led to many remarkable discoveries. As we follow the twists and turns of this story, we will see how mathematical discovery really works.
Title: John von Neumann's Analysis of Gaussian Elimination and the Invention of Modern Computing
Abstract: Just when modern computers were being invented (those digital, electronic, and programmable), John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did concerning the accuracy of mechanized Gaussian elimination, but the paper was about more than that. Von Neumann and Goldstine described what they surmised would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them. This talk takes a broad view of the evolution of computing before computers and of von Neumann's efforts to involve mathematicians in building the machines, and then briefly describes the highlights of von Neumann and Goldstine's results in their paper.
Title: Chocolate Key Cryptography: A Delicious Way to Send Secret Messages
Abstract: Sharing a secret with someone you don't trust, message authentication, message integrity, user identification -- nowadays, these are all made possible by a branch of cryptography called Public Key Cryptography. The mathematics behind much of public key cryptography was once praised for its great uselessness -- and we'll talk about that. Finally, Chocolate Key Cryptography is a way to describe a certain public key cryptosystem, a way that is easy to learn, fun, and delicious.
Career Night/ Pi Mu Epsilon Initiation
Title: Linear Crypanalysis of Two Round 16 Step MD5 Over the Rationals
Abstract: Cryptographic hash functions are widely used as a method of generating a digest, or fingerprint, from an arbitrary length message. They are ubiquitously deployed as a method for password storage, data integrity verification, and online payment systems. We will present a method to attack the one-way property of a reduced-step MD5, a commonly used hash function, using linear cryptanalysis over the rational numbers.
Title: When is a Finitely Presented Group Linear?: A Glimpse of Geometric Group Theory
The algebraic objects known as groups appear naturally in mathematics in many different ways. For example, a set of invertible matrices that is closed under multiplication and inverse forms a linear group; and groups also arise naturally in the form of finite presentations, that is, sets of rules like < a,t | t^(-1) a^2 t = a^3 >. In this talk, we discuss the question: When is a finitely presented group isomorphic to (i.e., algebraically the same as) a linear group? The answers we provide in some special cases come from puzzle pieces, tilings, and geometry. Specifically, we explain why the geometry of the above innocuous-looking but non-linear group is not as "nice" as the geometry of the seemingly complicated but linear group < a,b,t | t^(-1)a^(-1)b^(-1) abt = a^(1000)b^(1000)a^(1000)b^(1000) >. The talk will be of greater interest to listeners who already know what a group is, but for those who are not familiar with groups, the particular examples discussed at the beginning of the talk should be enough to get by. No other background will be assumed.
Unless noted otherwise, talks will be at 4:00pm Tuesdays. Room O'Connor 204. There will be refreshments before all talks in O'Connor 31 starting around 3:45pm.