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Math Institutes Modern Mathematics Workshop October 08, 2008 - October 09, 2008

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Oct 08, 2008
Wednesday

02:00 PM - 03:00 PM		Using uncertainty to establish certainty Po-Shen Loh
Location -- Video -- Abstract "Leave nothing to chance." When accuracy is paramount, this maxim reflects the common belief that elaborate planning is essential for success. One would expect that in the rigorous environment of mathematical logic, such a statement would hold the status of doctrine. However, half a century ago, researchers discovered that to the contrary, artificially-introduced randomness could be used as a powerful tool to prove deterministic statements with absolute certainty. This revolutionized the field of discrete mathematics. Indeed, consider the following application. Suppose one needs to show the existence of a combinatorial arrangement with certain properties. Instead of exhibiting an (often intricate) satisfying construction, consider a random arrangement, sampled from a suitable probablility space. It is then enough to show that the random arrangement is suitable with probability strictly greater than zero. This alternative perspective allows one to apply results from the theory of probability, and in many cases it makes the problem substantially more tractable. In this talk, we will showcase this technique, now known as the Probabilistic Method, through examples and applications. Supplements -- Show Detail
02:00 PM - 03:00 PM		Using knot theory to understand DNA packing in viruses Javier Arsuaga
Location -- Video -- Abstract Understanding the basic principles that govern chromosome organization poses one of the main challenges in mathematical biology of the postgenomic era. In viruses chromosome organization varies greatly across different families and is highly dependent on the assembly pathway of the proteins that form the virus. In this talk I will show how knot theory can be used to understand the three dimensional organization of the bacteriophage genome.In the ealry eighties L. Liu and colleagues found the DNA molecules extracted from P4 bacteriophages where knotted with very high probability. The question of why these knots are formed, or whether they contain any information about the packing of DNA inside the viral capsid have remained open. We have developed experimental protocols (Trigueros et al. 2001) as well as computaitonal methods to address these questions. We have shown that the formation of knots inside the viruses is driven mainly by the effects of the confinement (Arsuaga et al 2002) and that the knot distribution reflects a chiral organization of the genome (Arsuaga et al. 2002, Arsuaga et al. 2005). Our current work aims at establishing a bridge between models of DNA packing in bacteriophages and the topological information drawn from bacteriophage P4. Supplements -- Show Detail
03:00 PM - 04:00 PM		Knot homologies and applications to low-dimensional topology Eli Grigsby
Location -- Video -- Abstract Roughly speaking, low dimensional topology is the study of 3-dimensional manifolds and the 4-dimensional cobordisms between them. Low dimensional topologists also like to study knots, i.e., smoothly imbedded circles in 3-manifolds (up to ambient isotopy), because of a classical theorem of Lickorish-Wallace: Every closed, connected, oriented (c.c.o.) 3-manifold can be obtained from the three-sphere (the simplest c.c.o. 3-maniofld) by doing surgery on a finite collection of knots. A corollary of the Lickorish-Wallace theorem is that any c.c.o. manifold is the boundary of some 4-manifold. Yet knots are remarkably tricky to study directly; it is difficult to tell, just by staring at pictures of two knots, whether they are the same or different. We confront this problem through the use of knot invariants, algebraic objects associated to knots that do not depend upon how the knots are drawn. I will discuss a couple of these: Khovanov homology and knot Floer homology, both inspired by ideas in physics. In the less than ten years since their introduction, they have generated a flurry of activity and a stunning array of applications. There are also intriguing connections between the two theories that have yet to be fully understood. Supplements -- Show Detail
03:00 PM - 04:00 PM		Right triangles and elliptic curves Karl Rubin (University of California, Irvine)
Location -- Video -- Abstract Which natural numbers occur as the area of a right triangle with three rational sides? This is a very old question and is still unsolved, although partial answers are known (for example, five is the smallest such natural number). In this talk we will discuss this problem and recent progress that has come about through its connections with other important open questions in number theory. Supplements -- Show Detail
04:30 PM - 05:30 PM		Applications of symplectic geometry to topology Lenhard Ng
Location -- Video -- Abstract In geometry and topology, there are certain structures that are very "rigid" (like Riemannian manifolds) and others that are very "flexible" (like topological manifolds). Symplectic and contact geometry lies in between these two extremes and incorporates some attractive features of both. One consequence is that symplectic techniques have recently been used, to great effect, to give combinatorial approaches to questions in topology that were previously only amenable to difficult gauge-theoretic and analytic techniques. I will introduce symplectic and contact structures and describe some recent developments linking them to the study of three-dimensional manifolds and knots. Supplements -- Show Detail
04:30 PM - 05:30 PM		Molecular interactions within the cell: Network, Scale, and Complexity Baltazar Aguda
Location -- Video -- Abstract Biological processes can be characterized by different degrees of complexity at microscopic (genes, molecules), mesoscopic (protein-DNA complexes) and macroscopic (cells, organisms) levels. Historically, all biological systems have been studied at different levels. However, an increasing amount of experimental results and theoretical studies suggest that a more comprehensive system approach would tackle better biological problems. It would require a collaboration and intensive exchange between experimental and theoretical researchers from physics, chemistry, biology, mathematics, computer science, and engineering.   The MBI’s 2009-2010 activities will focus on the following fundamental questions: What are the properties of biological networks? How do they function? How do genes come together to form networks, and how can we use bioinformatics to discover such networks? Can our understanding of the fundamental mathematics inform the design of those bioinformatics methods? How is information transferred in cells? What role can synthetic biology perform in aiding our understanding of real life processes? How can different subjects of biological systems interact together to create effective dynamic systems? Supplements -- Show Detail
05:30 PM - 06:30 PM		Combinatorial Matrix Theory: Origins and Applications Luz DeAlba
Location -- Video -- Abstract In her lecture, DeAlba will explore the timeline of how the concepts of determinant and matrix evolved from the 16th century to today. She will give brief accounts of the background and work of the most prominent mathematicians, whose work on these two concepts eventually let to Matrix Theory. Early mathematicians studied and solved physical problems, such as the motion of a wave, or the elasticity of a spring. Although much of their work contained matrix theory ideas, they did not recognize this fact, and it was not until the 19th century that the study of matrices became an essential part of mathematics. DeAlba will present applications of graph theory and recent developments that show the natural connection between Combinatorics and Matrix Theory. Supplements -- Show Detail
05:30 PM - 06:30 PM		Tropical Geometry Josephine Yu (Georgia Institute of Technology)
Location -- Video -- Abstract Tropical geometry is the geometry over the tropical semiring, which is the set of real numbers where the tropical addition is taking the minimum, and the tropical multiplication is the ordinary addition. As the ordinary linear and polynomial algebra give rise to convex geometry and algebraic geometry, tropical linear and polynomial algebra give rise to tropical convex and tropical algebraic geometry. Tropical algebraic varieties are polyhedral complexes that behave very much like usual algebraic varieties. For example, two tropical quadratic curves in the plane intersect at four points. Tropical geometry has connections to various branches of pure and applied mathematics such as enumerative geometry, computational algebra, dynamical systems, and statistical physics. In this talk, I will introduce tropical varieties and basic results about them. Supplements -- Show Detail

Oct 09, 2008
Thursday

09:00 AM - 12:00 PM		<b> Session 1 </B>
Location -- Video -- Abstract -- Supplements -- Show Detail
09:00 AM - 12:00 PM		<b> Session 2 </b> <b> Undergraduate Mini Course </b>
Location -- Video -- Abstract -- Supplements -- Show Detail
09:00 AM - 12:00 PM		The State of the Planet: how mathematics can help Mary Lou Zeeman (Bowdoin College)
Location -- Video -- Abstract The mathematical community offers a wealth of largely untapped resources for addressing problems related to the state of our planet. For example, climate change research is full of mathematical challenges, ranging from continuous and stochastic dynamical systems, to inverse problems, data assimilation, efficient numerical numerical methods, quantifying uncertainty, and more. Global sustainability and resource management questions raise new classes of problems at the interface of mathematics, economics and computer science. Spread of disease and ecosystem preservation both take on new dynamic spatial components as species move poleward. Insight into tipping points can be provided by the language of bifurcation theory. This mini-course will describe how mathematics can play a role in understanding and addressing challenges faced by our planet. The mini-course is aimed at the undergraduate level and is open to all students regardless of their majors. Supplements -- Show Detail
09:30 AM - 10:30 AM		<b>Panel of all the Institute Reps</b>
Location -- Video -- Abstract -- Supplements -- Show Detail
10:30 AM - 11:00 AM		Coffee, Tea Break
Location -- Video -- Abstract -- Supplements -- Show Detail
11:00 AM - 12:00 PM		Statistical approaches for parameter estimation in climate models Gabriel Huerta
Location -- Video -- Abstract To quantify the uncertainties arising in climate prediction it is necessary to estimate a multidimensional probablility distribution. This is know as the calibration problem. The computational cost of evaluating such a probability distribution for a climate model is impractical using traditional methods such as Gibbs/Metropolis algorithms. This talk will describe an optimization based method that has been applied for non-linear problems in geophysics and that is currently in use to calibrate parameters of an atmospheric general circulation model (ACGM). Furthermore, we will also consider adaptive Monte Carlo based methods in the context of a climate model that is able to approximate the noise and response behavior of the AGCM. Comparisons and efficiency evaluations between approaches will be made. Another aspect of this talk is to overview the current role of spatical methods in providing emulators to climate model output and reducing computational burden. In particular we will discuss the use of Gaussian process (GP) in this context andon potential limitations and challenges for these methods. Supplements -- Show Detail

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