Wednesdays 4.30 - 5.30
Mariana is maintaining the current website.
A seminar where graduate students give expository talks to other students about the math they're working on. The goal is to get some idea about what we're doing in our various specialized areas, practice giving talks, and have fun :)
The talks should be accessible to all grad students - and you certainly don't need to be an expert in your area to give one. So let me, Mariana (Smit Vega Garcia), Kyle (Kloster), or Sohei (Yasuda) know if you are interested in speaking!
I wrote down some tips on preparing a talk for the Student Colloquium.
12/07/11 Yuanzhe Xi: Introduction to sparse matrix computation
Sparse or data sparse matrix computation lies in the heart of many science and engineering problems. For example, Google's web page ranking problem is the world's largest sparse matrix problem. Almost every month, Google has to find an eigenvector of a billions-by-billions sparse matrix to represent the connectivity of the web. Other applications can be found in quantum mechanics, optimization problems, acoustics, seismic imaging problems and so on. In order to solve those problems efficiently, we have to take advantage of the structure of those matrices. In this talk, I will briefly talk about the basic graph algorithms and related data structures to handle the fill-in in the factorization of sparse matrix and the Hierarchically semi-separable matrix (HSS matrix) framework for data sparse matrix. Some projects using these techniques will also be introduced.
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11/28/11 (Monday!, UNIV 119) Artur Jackson: The Elliptic Curve Discrete Logarithm (ECDLP) in the Presence of Weak Entropy Sources
In Elliptic Curve Cryptography (ECC) the main source of computational hardness is from the Elliptic Curve Discrete Logarithm Problem (ECDLP), i.e., inverting the function $d \mapsto [d]G$ for a public point $G$ where $d$ is an integer taken modulo the order $n$ of $G$. The complexity of the ECDLP for a large class of curves is thought to be roughly a square root of $n$. Due to this rather low upper bound on complexity, it can be seen that the integer $d$ if taken uniformly modulo the $n$ has in some sense ``too much entropy.'' I'll define all of the above more precisely, and show an original result which shows that we can take $d$ from a ``less random'' distribution, apply a transformation $f$, yielding a (composite) function $d \mapsto [f(d)]G$ which is just as hard to invert as the original function where $d$ is taken from the (maximally entropic) uniform distribution. The mathematics will be fairly simple, and I'll introduce some information theoretic concepts (e.g., Shannon Entropy, Computational Entropy) and some proof techniques which are used in modern theoretical cryptography and complexity theory.
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11/21/11 (Monday!, UNIV 119) Gabriel Sosa: Why Generic Initial Ideals?
When working over k[x], the polynomial ring in one variable over a field, the division of two polynomials is well defined due to Euclid’s Algorithm, and the fact that identifying leading terms is natural. The main problem when tackling the division of polynomials in several variables, (i.e. polynomials in k[x,y,z]), is how to identify leading terms. This obstacle can be overcome by defining a monomial order, (this is a total order on the set of monomials). The study of certain algebraic properties of ideals of polynomial rings, such as projective dimension and regularity, benefitted from concepts developed because of monomials orders. One of these concepts was that of generic initial ideals. In this talk, I will define monomial orders and generic initial ideals, discuss some of their properties and the difficulties that arise when trying to calculate them. As a motivation, I will describe projective dimension and regularity through examples and present an open question by Stillman.
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11/16/11 Vu Dinh: Mathematical Modeling of the T-cell Signaling Pathway
T-cells play an important role in cell-mediated immunity by recognizing and eliminating threats presented by invading pathogens and cancerous host cells. Studying of the T-cell signaling pathway is currently an active research area that has direct impact on medical science, especially on cancer treatments.In this talk, using T-cell as the main theme, I will give an overview of my current research on mathematical biology: the motivation of studying the T-cell signaling pathway, the link between biological models and mathematical models, and the framework on which different branches of applied mathematics help solving problems arise from systems biology. No fancy biological background required, no complicated applied math technical detail, the talk will be just a fun introduction to mathbio and should be accessible to everyone.
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11/09/11 Jamie Weigandt: Ranks of Elliptic Curves: Folkore, Conjectures, and Folklore Conjectures
An elliptic curve is a surprisingly simple mathematical object to call up. All one needs to do is chose two rational numbers A and B so that 4A^3 + 27B^2 is not equal to 0, and then write down the cubic plane curve, y^2 = x^3 + Ax + B. Once this is done, there are a myriad of questions one can ask about this curve. How many pairs of integers (x,y) satisfy this equation? How many pairs (x,y) of rational numbers? A satisfying answer to the later question would resolve some very ancient problems like: "Which integers can be the area of a right triangle with rational sides?" They also raise more modern questions like the Birch and Swinnerton-Dyer conjecture. In this talk I'll discuss these questions, and others related to the so-called Mordell-Weil ranks of elliptic curves.
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11/02/11 Andrew Homan: A Brief Introduction to Tomography
Broadly speaking, tomography is the study of techniques for indirectly observing the interior of a body based only on data coming from physical processes observed at the boundary of the body. For example, in a CT scan, one measures how x-rays propagate through the body, and the problem becomes how to reconstruct the density of the body inside. In this talk I'll describe some techniques people use to solve this sort of problem, and some of the ideas behind the basic theory.
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10/26/11 Taylor Hines: What is a noncommutative topological space?
As mathematicians, we usually think of 'noncommutativity' as gross and annoying. How much easier would our lives be if all matrices commuted? However, we have to deal with it everywhere. Walking a mile west and then a mile north will land you in a different place than walking a mile north and then a mile west. Studying for a test before taking it usually works out better than taking the test before you study. In the early 20th century, a few scientists noticed that measuring a particle's momentum and then its position gave different answers than measuring it's position first and then its momentum, a phenomenon at the root of the Heisenberg uncertainty principle.
For these reasons, mathematicians had to invent new objects in order to deal with the noncommutativity that could not be explained by classical physics. C*-algebras came into mathematical view (roughly) in the mid-20th century in an attempt to provide a framework for modern quantum mechanics. By now, however, C*-algebras are recognized as important mathematical objects in themselves, and C*-algebra theory has found its way into many other areas of mathematics.
In this talk, I will give an introduction to C*-algebras, in particular, I will hopefully demonstrate the intrinsic connections that C*-algebra theory has with the theory of topological spaces, and the reasons why C*-algebras can be reasonably thought of as "noncommutative spaces." I'll also try and explain how this noncommutativity idea spills over into other areas, to things like "noncommutative cohomology" (absolutely no knowledge of cohomology is assmed) or "noncommutative dimension theory" (whatever that means).
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10/19/11 Mariana Smit Vega Garcia: The intriguing and elusive Bernstein problem
The Bernstein problem is remarkably easy to explain: is it true that the only entire solutions of the minimal surface equation in the plane are of the form u(x) = (a,x) + b? This was proved to be true in 1915 by Bernstein himself. The question remained whether this result could be extended to higher dimensions. Several different proofs were given for the plane, but they didn't use methods that allowed for an extension... In the end, was a counterexample found? Was the Bernstein problem extended? Come to find out! :)
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10/12/11 Partha Solapurkar: Fun with Projective Geometry
Projective geometry arose as the study of geometric properties that are invariant under projections. Early development of projective geometry was fueled by the study of perspective drawing in fine arts. In nineteenth century it became a prominent research area in mathematics. Today it continues to be a major research area in the form of modern algebraic geometry. In this talk we'll meet points at infinity, projective spaces and assorted fun facts about complex projective curves. I will try to keep the talk elementary for the most part.
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10/05/11 Bobby Bridges: An Exciting and Fun Introduction to Composition Operators
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09/26/11 Vita Kala: Towards the Langlands Program
The Langlands Program is a central area of modern number theory. Number theory originated as the study of properties of integers and primes, yet gradually it changed into something much more abstract, using notions from many other parts of mathematics. I'll try to motivate and explain some of these changes. We'll meet p-adic numbers, Galois groups, and class field theory, finally arriving at the representation-theoretic conjectures of Langlands. Don't worry if you don't know what any of this means - I hope to be able to keep the talk understandable and entertaining :)
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