Student Analysis Seminar: Winter 2010

Thursdays, 5:10-6:00pm, Room 3088, East Hall
University of Michigan Department of Mathematics



Winter 2010 Schedule

Date SpeakerTitle
Thu, 1/7/2010 n/aOrganizational Meeting
Thu, 1/14/2010 Tim FergusonA Continuous, Nowhere-Differentiable Function
Thu, 1/21/2010 Austin ShapiroEntropy
Thu, 1/28/2010 Several analysis students (listed in abstract)What sort of Math do I & My Advisor Do? A collection of 10-minute talks.
Thu, 2/4/2010Joe RobertsHyperbolic Conservation Laws and Vanishing Viscosity
Thu, 2/11/2010Steven FloresQuantifying Roughness: Fractal Dimension
Thu, 2/18/2010Alex MuellerInfinitesimal Adventures
Thu, 2/25/2010Joe RobertsComplex Analysis and Fluid Dynamics
Thu, 3/4/2010n/aSpring Break
Thu, 3/11/2010Zach ScherrErgodicity in Number Theory
Thu, 3/18/2010no talkno talk because of Terry Tao's lectures
Thu, 3/25/2010Alex Muellerp Adic Analysis and the Kubota-Leopoldt p Adic Zeta Function
Thu, 4/1/2010Nate TotzRigorously Justifying a Modulation Approximation of the sine-Gordon Equation
Thu, 4/8/2010Rafe KinseyAn Introduction to Brownian Motion
Thu, 4/15/2010Ross KravitzIto Calculus
Thu, 4/22/2010Hieu (Tom) NgoMultiplicative Spectrum and Integral Equations

This fills up the speakers for this semester. If people are around for the summer and want to do something (perhaps more informal), we could do that. Otherwise, stay posted for the fall. (We'll also check back to see if this is still the best time, or whether we want to shift it.)

Speakers: please email Rafe with your talk title and abstract by the Wednesday afternoon eight days before the talk, so that it can get on the weekly schedule.

Suggested Topics

This is a list of topics people expressed interest in hearing about (people who've expressed interest in parentheses--email Rafe to add yourself to the list) [possible volunteer to give talk in brackets], topics scheduled are struck through. Volunteers to give such talks encouraged! Please email Rafe with more suggestions.

Abstracts


Organizational Meeting (1/7/10): We're going to restart the student analysis seminar. Come to the organizational meeting to discuss how exactly we'll run it and what topics we'll go over. If you can't make the meeting, email rkinsey (at) umich (dot) edu for more information. First years are encouraged to come! Room 3088.

Entropy, Austin Shapiro (1/21/10): The (Shannon) entropy of a random variable is a measure of uncertainty regarding its value. I will illustrate this point of view by way of a series of mathematical vignettes. Along the way, you will see how entropy can be applied to problems in efficient communication, evolution of random processes, asymptotic problems in combinatorics, and the strategy of guessing games. This talk will be very accessible to beginners.

What Sort of Math do I & My Advisor Do? (1/28/10) Five students (Ross Kravitz [Bayraktar], Joe Roberts [Elling], Crystal Zeager [Fornaess], Nate Totz [Wu], and Qian Yin [Bonk]) describe what sort of math they work on, what sort of math their advisor does, and what their advisor is like to work with. First and second years very encouraged to attend!

Hyperbolic Conservation Laws and Vanishing Viscosity, Joe Roberts (2/4/10): It is possible for solutions to nonlinear hyperbolic conservation laws to lose regularity in finite time. In this case, it is necessary to define weak solutions. However, such weak solutions are not unique and in order to obtain uniqueness one must introduce an entropy condition or the notion of a vanishing viscosity solution. I'll go through some basic, computable examples to demonstrate the concepts, as well as briefly describe some of what is known about systems. I plan to sketch a construction due to Foy that shows that sufficiently weak entropy admissible shocks can be obtained as limits of viscous traveling waves.

Quantifying Roughness: Fractal Dimension, Steven Flores (2/11/10): Most subsets of $R^n$ encountered in traditional mathematics have a characteristic length scale so that when they examined much closer than that scale, "rough" features smooth out and classical techniques can be used to study them. Fractals can be thought of as sets that exhibit no such length scale. They have fine structure on all scales, and to look at them closer does not smooth out their roughness. Once thought of as pathological, fractals have become a critical tool for modeling real world phenomena. Thus, it is interesting to study their local and global properties. One global property is a measure of their roughness, called "fractal dimension." There is no one way to define fractal dimension, and different definitions such as Hausdorff, box, and packing dimensions are used. In my talk, I will discuss these different versions of fractal dimension, their interrelations, their invariance properties, how they can be used to characterize fractals, and techniques for computing them illustrated through examples.

Infinitesimal Adventures, Alex Mueller (2/18/10): References to infinitesimal numbers were common in analysis in the 17th and 18th centuries. Questions regarding the rigor of this approach caused infinitesimal numbers to fall into disrepute, and they were later replaced with the epsilons and deltas familiar today. Infinitesimals languished in the mathematical rubbish bin until the 1960s, when logician Abraham Robinson saw a safe way to revive them.

Nonstandard analysis develops the familiar ideas of analysis using infinitesimal numbers rather than epsilons and deltas. This approach supplies proofs of familiar results which are sometimes more compact, more intuitive, or more weird and magical than their standard counterparts. The topic is also historically interesting, in that many basic concepts were originally formulated in the language of infinitesimals.

I intend to focus on analysis, and I will black box as much of the underlying mathematical logic as possible. My goal is to provide interesting nonstandard proofs of a number of familiar theorems from real analysis, and to give a sense of the alternative perspective infinitesimals provide.

Complex Analysis and Fluid Dynamics, Joe Roberts (2/25/10): The equations for ideal fluid flow are equivalent to the Cauchy-Riemann equations. The boundary conditions are preserved under conformal mapping, and so one can use a simple flow on a half plane and a conformal map to explicitly compute flows that would be difficult to obtain directly. In addition, basic properties of holomorphic functions can be used to make useful conclusions about general flows. I'll show some concrete examples using familiar conformal maps as well as discuss some general conclusions that can be drawn from complex analysis. I intend for it to be an accessible introduction to how basic complex analysis can be applied to fluid dynamics.

Ergodicity in number theory, Zach Scherr (3/11/10): In this talk I will discuss ergodic theory and its applications to number theory.

Rigorously Justifying a Modulation Approximation of the sine-Gordon Equation, Nate Totz (4/1/10):
When studying a nonlinear PDE, it is often useful to formally approximate a subclass of its solutions by a model equation whose solutions have known qualitative and quantitative properties. However, it can be difficult to show that the solutions to the model equation are indeed close (in a suitable sense) to solutions of the original equation.

In this talk I'll consider the example of justifying that small wave packet-like solutions to the sine-Gordon equation have amplitudes that are close to satisfying the nonlinear Schrodinger equation on its governing time scale. Along the way, we'll motivate some of the standard methods and tools of pure and applied PDE, including multiscale analysis, contraction mappings and semigroups to establish existence of solutions, and energy estimates with Sobolev embedding to bound the remainder.

This talk should be accessible to those who have no or little experience with PDE, but some familiarity with basic functional analysis and the Fourier transform will be assumed.

An Introduction to Brownian Motion, Rafe Kinsey (4/8/10):
I will give an introduction to Brownian motion, discussing where it comes from, how it's constructed, and what it's useful for. The talk should be relatively accessible, and will also serve as an introduction to Ross Kravitz's talk next week on stochastic calculus.

Ito Calculus, Ross Kravitz (4/15/10):
In the physical and social sciences, there are many phenomena which cannot effectively be modeled using deterministic methods, either because they are considered fundamentally random or because the underlying mechanisms are too complex to be completely understood. In these cases, instead of using a classical differential equation, one considers a model augmented by some random noise process. The most fundamental noise process is Brownian Motion, and so in order to analyze such models, one needs to develop a theory of integration for BM. Due to the wild oscillations of BM, classical Lebesgue integration doesn't work, and so one must develop a new integral, the Ito Integral. There are also some interesting applications in pure mathematics, and after defining this integral, I will try to briefly mention some connections of the theory to probability, PDE, and complex analysis.

Multiplicative Spectrum and Integral Equations, Hieu (Tom) Ngo (4/22/10):
Given an arbitrary multiplicative function $f(n)$ with values inside the unit circle, does it have a mean value (i.e. does $\lim_{x\to\infty} \sum_{n\leq x} f(n)$ exist)? Wirsing (1967) gave an affirmative answer for real-valued functions; his result is essentially as deep as the Prime Number Theorem (look at the M\"obius function)! Complex-valued functions are more exotic; for example the mean value of $f(n)=n^{i\alpha}$ for any nonzero real $\alpha$ does not exist. However, H\'alasz (1968) ingeniously realized that basically this is the only exception. A recent breakthrough of Granville and Soundararajan (2001) initiated a geometric study of the multiplicative spectrum, revealing that the spectrum of mean values is intimately related to the solution of a family of integral equations.

We shall start with some motivating probabilistic ideas behind the classical central results. Then we shall convey the insights and the analytic techniques underlying the beautiful work of Granville and Soundararajan. The talk is completely elementary in nature; virtually no background in analytic number theory is assumed.
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Last modified: Wed Apr 21 22:46:20 EDT 2010