# Spring 2014

### Euclid's Fifth Postulate: The Drama of Ideas and the Drama of People

#### Dr. Oleg Gleizer, Chief Curriculum Developer, Lead Instructor<b/h4>

#### Los Angeles Math Circle at UCLA,

#### Tuesday, May 6, 2014 @ 2:30pm in AB 634

**Abstract:** The Great Theorem of Fermat, a proverbial symbol of mathematical complexity, stood open for 358 years. Conjectured by Pierre de Fermat in 1637, it was proven by Andrew Wiles with the help of his former student, Richard Taylor, in 1995. The Poincare Conjecture, another very hard math problem made famous outside of the scientific community by its conqueror Gregory Perelman's rejection of the $1,000,000 prize money, was proven in less than 100 years. It took humanity over 2,000 years to realize that the 5th Postulate of Euclid is indeed a postulate and cannot be derived from other axioms of Euclidean geometry. The breakthrough can only be compared to the Copernican Revolution in astronomy that replaced the geocentric model of the universe with the heliocentric one. Both discoveries broke millennia-long paradigms. The Copernican Revolution brought about the Newtonian mechanics and its box of tools now known as Calculus. The discovery of non-Euclidean geometry paved the way to the Riemannian revolution in geometry and to the Relativity Theory of Einstein.

At the beginning of the talk, we will compare the proofs of two theorems that give us the sum of the angles of a triangle in the Euclidean and elliptic planes. The first theorem is a staple of the middle school geometry. The second is presented in nearly any Intro to Geometry college course. However, while proving the second theorem is as easy as peeling an orange, an accurate derivation of the statement of the first theorem from the axioms is laborious and highly non-trivial.

Depending on the amount of time for the lecture, we may also do some or all of the following.

- Give an elementary (not involving the machinery of calculus) proof to the fact that geodesic lines on a sphere are arcs of great circles.
- Measure the radius of the Earth with nothing more than a camel and a stick.
- See a planet with one pole.
- Find out the curvature of the Universe at large.
- Learn some amazing facts from the biographies of the people who have unraveled the 5th Postulate riddle, Nikolai Lobachevsky of Russia and the Hungarian Janos Bolyai.
- Find out why Carl Friedrich Gauss, the great mathematician who had discovered non-Euclidean geometries before Lobachevsky and Bolyai, decided not to publish his findings.

The talk will be accessible to undergraduate students as well as to Davidson Academy students with a sufficient background in elementary geometry.

### Arithmetic vs. Non-Arithmetic

#### Dr. Mark Norfleet

#### Mathemaatics & Statistics Dept,University of Nevada, Reno

#### Thursday, May 1, 2014 @ 2:30pm at AB 634

**Abstract:** The talk starts by discussing some basic ideas about arithmetic Fuchsian groups. With some examples of both arithmetic and non-arithmetic groups, we consider how these groups act on models of the hyperbolic plane---the upper half-plane and the hyperboloid in the three-dimensional version of Minkowski space-time. Then we will conclude by constructing Fuchsian groups with a property that the set of hyperbolic fixed points will contain a given (finite) collection of points in the boundary of the hyperbolic plane. This construction is used to make infinitely many non-arthmetic non-cocompact Fuchsian groups of finite covolume sitting in PSL(2, Q).

#### Analyzing Single-Molecule Microscopy Data with Stochastic Differential Equation Models

#### Dr. Christopher Calderon

#### Numerica Corporation http://www.mendeley.com/profiles/christopher-calderon/

#### Thursday, April 24, 2014 @2:30pm in AB 634

**Abstract:** Recent advances in microscopy have equipped scientists with tools capable of measuring the motion of individual molecules (proteins, DNA, RNA, etc.) inside of living cells with high spatial and temporal resolution. Now, numerous labs are routinely producing high volumes of data characterizing molecular motion since quantitative information about the motion of individual molecules in the cell allows scientists to address many open problems in biophysics, chemistry, and molecular biology. However, statistical techniques capable of efficiently extracting the wealth of molecular information detectable by modern microscopes are under-developed. Reliably extracting scientific information from super-resolution measurements poses exciting challenges at the interface of several disciplines. In this talk, I will discuss estimation and other time series inference techniques associated with calibrating stochastic differential equation (SDE) models from measurements made by contemporary optical microscopes. The primary focus of the talk will be centered on discussing the utility of using hypothesis testing techniques to check the consistency of physically inspired model against experimental data. This task faces several technical challenges including complex measurement noise (on top of "thermal fluctuations" or "process noise"), non-stationarity of the time series, and unresolved latent features substantially affecting the dynamics of different experimental trajectories. Examples of these issues (and why life and physical scientists are interested in addressing these problems) will be provided through various examples taken from Refs. [1-3].

[1] Calderon, Phys. Rev. E (2013).

[2] Calderon, Thompson, Casolari, Paffenroth, Moerner, J. Phys. Chem. B (2013).

[3] Calderon, Weiss, Moerner, (submitted; http://arxiv.org/abs/1312.6742

### Adiabatic and Stable Adiabatic Times

#### Dr. Kyle Bradford(http://bradfordmathematics.tumblr.com)

#### University of Nevada, Reno, Mathematics & Statistics Department

#### Thursday, April 17, 2014 @ 2:30pm in AB 634

**Abstract:** This talk will detail the stability of Markov chains. One measure of stability of a time homogeneous Markov chain is a mixing time. I will define similar measures for special types of time inhomogeneous Markov chains called the adiabatic and stable adiabatic times. I will discuss the use of these Markov chains and I will discuss how the adiabatic and stable adiabatic times relate to mixing times. This talk is an exploration of linear algebra, analysis and probability.

### Boundary Control Approach to Inverse Spectral and Dynamical Problems

#### Dr. Sergei Avdonin

#### University of Alaska, Fairbanks

#### Thursday, April 10, 2014 @ 2:30pm in AB 634

**Abstract:** The Boundary Control (BC) method is based on the deep connection between control theory for partial differential equations and inverse problems of mathematical physics and offers an interesting and powerful alternative to previous identification techniques based on spectral or scattering methods. This approach has several advantages: (i) it is applicable to a wide range of linear lumped and distributed systems and reconstruction situations; (ii) it is, in principle, dimension-independent; (iii) it lends itself to straight forward algorithmic implementations. Being originally proposed for solving the boundary inverse problem for the multidimensional wave equation, the BC method has been successfully applied to all main types of linear equations of mathematical physics. In this talk we discuss connections between the BC method and the classical Gelfand {Levitan and Krein theories, and the recently proposed Simon and Remling approaches. We also demonstrate how our approach can be extended to inverse problems for differential equations on graphs.

My web page address is:https://www.cs.uaf.edu/~avdonin/

### Autonomous Cars For City Traffic

#### Prof. Raul Rojas -- Freie Universtaat Berlin

#### Thursday, April 3, 2014 @ 2:30pm in MIKC 107

In my talk, I will describe the autonomous vehicles we have been developing in Berlin since 2006. Our car MadeInGermany" was certified for city and highway traffic and has been navigating the Berlin streets since 2011-- fully automatically. I will review the sensor architecture and the hierarchy of embedded processors that makes it possible to control the car using just a laptop. The car uses a combination of radars, laser scanners, video cameras, and a GPS navigation unit. The sensor architecture is redundant. An independent processor supervises the control chain and provides additional safety. We are now focusing on being able to drive using only computer vision in the future. With this objective in mind, we have developed our own stereoscopic cameras, which are now also being used in large utility trucks in Berlin. The automotive industry is also starting to develop systems for highway autopilots. I will mention some projects and will discuss the still existing legal barriers, as well as the timetable of the German automotive industry. http://www.autonomos.inf.fu-berlin.de/

### Middle Censoring

#### Prof. Sreenivasa Rao Jammalamadaka

#### Dept of Statistics & Applied Probability, University of California, Santa Barbara

#### Thursday, March 13, 2014 @ 2:30pm in AB 634

**Abstract: **In connection with survival analysis and related problem, there is considerable literature which treats data that is censored from the left, the right or both. In this talk, we consider situations where the data becomes unobservable if it falls inside a random interval in the middle, which we call middle-censoring. This happens in clinical trials and lifetime studies where a subject is temporarily absent or withdrawn from the study and the event of interest occurs during this period, so that the exact time of occurrence cannot be observed. This is also applicable in cases where some of the observations are so imprecise that they are stated as intervals, as for example the situation where peoples' willingness to pay for a natural resource. In this talk, we present a "self-consistent estimator" and the Nonparametric MLE for the survival function, study its large-sample properties, and demonstrate how well it works even in the presence of heavy censoring. Presence of covariates can also be dealt with both in a parametric and semi-parametric set-up.

### Actuarial Work in Property and Casualty Insurance

#### Bryan Ware and Jon Marshall, Actuaries

#### Employers Insurance Group Reno, Nevada

#### Thursday, March 6, 2014 @ 2:30pm in AB 634

**Abstract:** A brief introduction to Property and Casualty Insurance and the P&C actuarial profession. Discussion of Property & Casualty Insurance in general, the Casualty Actuarial Society and the actuarial profession, what actuaries do, work environment and compensation.

### Distinguishing Topologically and Smoothly Doubly Slice Knots

#### Jeff Meier

#### University of Texas, Austin

#### Thursday, Feb 13, 2014 @ 2:30pm in AB 634

The purpose of this talk is to show the existence of an infinite family of smoothly slice knots that are topologically doubly slice, but not smoothly doubly slice. After outlining what is known about doubly slice knots, I will present some sufficient conditions for a knot to be topologically doubly slice, and show how the knots in question satisfy these criteria.Then, I will outline how the correction terms coming from Heegaard Floer homology can be used to obstruct these knots from being smoothly doubly slice. Along the way, we will encounter some very interesting open questions related to the study of doubly slice knots.

### "History of Gaussian Elimination"

#### Sandia Laboratory, Lawrence Berkeley Nat'l Laboratory

#### Wednesday, Feb 5, 2014 @ 2:30pm in DMSC 103

**Abstract:** Gaussian elimination is universally known as "the" method for solving simultaneous linear equations. As Leonhard Euler wrote in 1765, it is the most natural way of proceeding ("der natürlichste Weg"). Because Gaussian elimination solves linear problems directly, it is an important technique in computational science and engineering, through which it makes continuing, albeit indirect, contributions to advancing knowledge and to human welfare.

What is natural depends on the context, so the algorithm has changed many times with the problems to be solved and with competing technology. The sole development in ancient times was in China. An independent origin in modern Europe has had three phases. First came the schoolbook lesson beginning with Isaac Newton, whose contribution has only recently been recognized. Next were methods for professional hand computers which began with Carl Friedrich Gauss. Lastly was the interpretation in matrix algebra by several authors including John von Neumann and Alan Turing. Gaussian elimination is living mathematics that has successfully mutated for hundreds of years to meet changing social needs. Perhaps the only certainty about future algorithms is their name.

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