The Mth 507 graduate seminar consists in part of talks given by OSU mathematics graduate students who are Summer GTAs and by invited speakers. The talks normally deal with the students' mathematics or mathematics education research, interesting byways, mathematics history, applications of mathematics, or recreational mathematics. Academic credit (3 hr P/N) is normally available only to graduate mathematics students.
The titles in the table below are links to the corresponding abstracts. The names are email links for your convenience.
Ryan Hass
June 28
Title: Noise Removal and Medical Images
Noise and Computer Tomography (CT) go hand in hand because measured data is prone to errors. A CT reconstruction algorithm can then amplify the noise further because local changes in the measured data has global effects in the CT reconstruction. We wish to de-noise a CT reconstruction and not degrade the quality of the reconstruction. How then can we reduce the effect of noise, while preserving the features in the reconstruction?
Corina Constantinescu
July 3
Title: On the Asymptotic Decay of the Probability of Ruin on a Sparre
Andersen model with investments
This talk considers one of the classical problems in the actuarial mathematics literature, the collective risk model. The claim number process N(t) is assumed to be a renewal process, the resulting model being referred as the Sparre Andersen risk model. The inter-claim times form a sequence of independent identical distributed random variables with distribution Erlang(n). The additional non-classical feature is that the company invests in a risky asset with returns modeled by a diffusion. The analysis is focused on the probability of ruin, if the company's initial capital is very large. As an example the case of an Erlang(2) inter-arrival times distribution and a geometric Brownian motion for the returns from investments is considered and the asymptotic decay of the probability of ruin is investigated.
Brian Dietel
July 5
Title: An Introduction to the Markoff and Lagrange Spectra
The Markoff and Lagrange Spectra are two related sets in \Re that occur in numerous contexts, including rational approximation, binary quadratic forms, and the modular group. Several elegant results concerning these sets have been proved since Markoff first considered them in a 1879 paper. However, there remains many open questions including the unicity conjecture for Markoff numbers and much is still unknown about the region of the spectra greater than 3 and less than Freiman's constant (\approx 4.5278).
Adam Wunderlich
July 10
Title: The Katsevich Formula for Cone-Beam Computed Tomography
The topic for this talk is a method for fully 3-D computed tomography (CT). In 2002, Katsevich disovered a new inversion formula for helical cone-beam CT. This formula is special beacuse it is both theoretically exact and it may be implemented using a filtered back-projection algorithm. The formula will be described with some numerical results.
Bryan Tardiff
July 12
Title: Markov Processes as Random Walks on Finite Networks
A finite-state Markov process can be seen as a random walk on basic network or graph. Certain processes are naturally adaptable to such techniques, and for situations where it can be used, common computations for Markov processes, such as finding limiting distributions, become greatly simplified. We will observe one situation where the use of a network graph is particularly useful, and then discuss how we might adapt these techniques to more general Markov processes.
Daniel Rockwell
July 17
Title: Mandelbrot Percolation on the Unit Square
A specific case of percolation on the unit square. The subdivision of the unit square in a "random cantor set" construction leads to interesting properties of this percolation.
Jason Schmurr
July 19
Title: Using algebra and topology to play triangular billiards
In attempt to understand the dynamics of a point bouncing around inside a triangle, we will end up looking at some compact Riemann surfaces and subgroups of PSL(2,R) that act as certain symmetry groups of the surfaces. While we're at it, we will discuss special covers of such surfaces.
Aaron Wangberg
July 24
Title: "If this lab doesn't help my grade, I ain't doing it!" -- A
laboratory introduction to gradient vectors in multivariable calculus
One aim of the Bridge Project, designed by Tevian Dray and Corinne Manogue, is to approach vector calculus from a geometric view-point, thereby giving students a geometric foundation for ideas and concepts taught throughout the course. Although the Bridge Project aligns itself with Math 255 at Oregon State, many of the concepts are applicable to multivariable calculus. I'll share one of the activities I've created to help Math 254 students understand the geometric meaning of gradient vectors.
Bent Petersen
July 26
Title: Fourier Transform
I will present an informal discussion of the Fourier transform. Fourier series were studied as early as 1750 by Daniel Bernoulli. The Fourier transform was introduced by Fourier in his 1811 paper on the propogation of heat by a limiting argument starting from Fourier series. The same argument occurs in many engineering textbooks today! He further elaborated the theory of the Fourier transform in his major work on heat in 1822. Since that time the Fourier transform has been applied in many areas -- almost periodic functions, operational calculus, quantum field theory, partial differential equations, and so forth. Since 1811 many people have extended the Fourier transform. The culmination of much of this work is Schwartz's beautiful theory of Fourier transforms of temperate (tempered) distributions developed during 1945--1950. This theory include most of the earlier extensions. A periodic distributions u is automatically temperate and it turns out the inverse transform of the Fourier transform of u is just the Fourier series representation of u! We have come complete circle. I will talk until I get tired, or you do, so I am unlikely to cover everything mentioned above. -- PDF Notes
Tat Hatase
July 31
Title: The Klein Unit Disk Model for Hyperbolic Geometry.
People are usually familiar with Poincaré's Unit Disk model and the upper halfplane model for hyperbolic geometry, but not the other one that was created around the same time. This talk will introduce the less known model, and show that the metrics defined on those models are equivalent by providing isometries going between the models. We will also discuss a certain kind of isometry on hyperbolic geometry.
Lance Burger
August 02
Title: Cramer's Paradox
A brief history of the origins of Linear Algebra and a discussion of Cramer's Paradox. The resolution of Cramer's Paradox, an interaction between geometry and algebra, allowed Euler to reason out some of the first notions of linear dependence.
Kyle Hickmann
August 07
Title: An Introduction to Microlocal Analysis in 50 minutes
Classically, microlocal analysis was motivated by finding parametrices, local solutions, for linear PDE and the Cauchy problem. This construction gives rise to the definition of Pseudodifferential operators (PDOs) and Fourier Integral Operators (FIOs). These operators propagate singularities of distributions in quantifiable ways. For this reason I will define the wavefront set of a distribution and state a theorem about the wavefront set of an FIO. This leads to defining equivalence classes of FIOs modulo smoothing operators by specifying certain conic manifolds instead of the phase function and principal symbol.
Dave Wing
August 09
Title: Inverse Limit Spaces and Near Homeomorphisms
In 1959, Morton Brown analytically proved a Theorem giving conditions for the Inverse Limit Space of an Inverse Limit Sequence of compact metric spaces to be homeomorphic to each of its factors. These conditions are general in the sense that the factors are only required to be "approximately homeomorphic". As a consequence, it follows that any two "approximately homeomorphic" compact metric spaces are homeomorphic. These results are interesting and useful in their own right. However, in 1986, Frederic Ancel gave a slick topological approach to these results, yielding elegance and topological insight. In this talk, we explore these ideas and some applications.
Renée Nolan
August 14
Title: The Directional Hardy-Littlewood Maximal Function
I will be discussing my paper, which develops the boundedness of the directional Hardy-Littlewood maximal function in the function space Lp( Lq(Sn-1), Rn ) under certain restrictions placed on p and q.
Bent Petersen
August 16
Title: Pizza and beverage of your choice!
I will provide pizza and appropriate beverage at American Dream Pizza on Monroe to thank you for your talks and to help you gather your strength for grading. We will meet at the usual time, 2:00 PM, at the Dream, or, if you prefer, met me at my office at 1:50 and walk over to the Dream with me.
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