MSRI-UP 2013: Algebraic Combinatorics

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Prof. Gibor Basri, University of California, Berkeley

NASA's Search for Earth-sized Planets

Four years ago a dedicated space telescope (“Kepler”) was launched to search for terrestrial planets around other stars, and even possibly discover a planet that might be like the Earth. The main purpose of the mission is to find out how common smaller planets are.  I explain how the mission works, and highlight some of its most amazing discoveries. Nearly 3000 potential planets have been found, including many in multiple planet systems. The most common planet may be something that we don’t have in our own Solar System: “super-Earths” which are 1.5-3 times as big as our planet. Some of these may be rocky, some may be “water worlds”, and some may be more like warm Neptunes. The Kepler mission is rapidly leading us to the conclusion that most stars have planets going around them, and the number of earth-sized planets in our Galaxy could easily be in the billions.

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Prof. Herbert Medina, Loyola Marymount University

Computing Pi Via New Polynomial Approximations to Arctangent:  a new contribution to (arguably) the oldest approximation problem

Computing Pi Via New Polynomial Approximations to Arctangent:  a new contribution to (arguably) the oldest approximation problemRational functions after integration can produce arctangent and therefore can be used to approximate \( \pi \). Using rational functions of the form $$\left\{\frac{{t^{km}\left(t-\beta \right)^{lm}}}{{1+t^2}}\right\}_{m\in{\mathbb{N}}}$$ for different values of \( k, l, \) and \( \beta \), we produce families of efficient polynomial approximations to arctangent, and hence, provide approximations to \( \pi \) via known arctangent values. Some of the polynomials produce rational approximations to \( \pi \) and others approximations that require only the computation of a single square root; moreover, they are orders of magnitude more accurate than Maclaurin polynomials. We analyze the efficiency of the approximations and provide algebraic and analytic properties of the sequences of polynomials. Finally, we turn the approximations of \( \pi \) into series including one that gives about 21 additional decimal digits of accuracy with each successive term.

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Prof. Talitha Washington, Howard University

A Glimpse of Dynamical Modeling in the Bioscienes

In mathematics, a differential equation is a tool which may be used to describe a quantity that changes with
respect to time. This talk explores how differential equations describe the spread of infectious diseases, hormone secretions of a cell, and calcium homeostasis.

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Prof. Alissa S. Crans, Loyola Marymount University & MSRI

A Surreptitious Sequence:  the Catalan Numbers

We are all familiar with Fibonacci’s famous sequence that begins 1, 1, 2, 3, 5, 8, …  as well as other popular sequences such as the perfect squares 1, 4, 9, 16, 25, … or the triangular numbers 1, 3, 6, 10, 15, … But what about the sequence 1, 1, 2, 5, 14, …?  These are the Catalan numbers, named after the Belgian mathematician Eugène Catalan (1814 – 1894), despite having been described by Leonhard Euler 100 years earlier.  It turns out these numbers take a variety of different guises as they provide the solution to numerous combinatorial problems!  After introducing this sequence, we will explore some of the many ways in which the Catalan numbers are hidden throughout mathematics.

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Prof. Mariel Vazquez, San Francisco State University

Random knots, viruses and DNA

DNA presents high levels of condensation in all organisms. We are interested in the problem of DNA packing
inside bacteriophage capsids. Bacteriophages are viruses that infect bacteria, and DNA extracted from bacteriophage P4 capsids is highly knotted. These knots can shed information on the packing reaction and DNA architecture inside the capsid. I here will overview a few research questions stemming from the DNA packing problem.

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"Zero Forcing and its Applications,"

Michael Young, Iowa State University

Zero forcing (also called graph infection) on a simple, undirected graph \[ G \] is based on the color-change rule: If each vertex of \[ G \]is colored either white or blue, and vertex \[ v \] is a blue vertex with only one white neighbor $w$, then change the color of \[ w \] to blue. A minimum zero forcing set is a set of blue vertices of minimum cardinality that can color the entire graph blue using the color change rule. In this talk will discuss the role of zero forcing in systems control, electrical engineer, and linear algebra. Even though various scientist have been using zero forcing, it wasn't until recently that it was realized they were all doing the same type of propagation. Zero forcing gets its name from the linear algebraists, who were using the propagation to force entries of a vector to be zero.

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"Compactifications in Algebraic Geometry,"

Pablo Solis, University of California at Berkeley

In this talk I'll give a basic introduction to the field of algebraic geometry and discuss the problem of compactification which has become a very active area of research today. I'll define what it means for something to be compact and show how compact and non compact things appear in algebraic geometry. I'll describe one problem algebraic geometry in detail which can be phrased as follows: how many degree 2 plane curves are tangent to 5 given given plane curves? This was a question posed by a mathematician named Steiner in 1846. The solution, which took almost 20 years to be found, is one of the earliest instances of how compactification can be used.

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"A (very) brief introduction to graphical models,"

April Harry, Purdue University

Probabilistic Graphical Models is a framework used in statistics and machine learning to represent complex dependencies between random variables. By combining concepts from graph theory and probability, graphical models allow us to leverage efficient algorithms from computer science for statistical decision making and inference. We explore one class of graphical models, Bayesian networks, through an example with three random variables. Also, we discuss some practical applications of graphical models in bioinformatics, speech and language processing, and atmospheric sciences.

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Workshops

"Using LaTeX,"

by Ivelisse Rubio, University of Puerto Rico

• 2013 MSRI-UP Latex Handout.pdf
• 2013 MSRI-UP Latex Handout.tex
  
• MSRI-UP Report Template.tex
  
• grafica.jpg