**(For updated locations, click here;
All locations are subject to change**)
Two Short Course proposals have beeen selected for presentation just before the Joint Mathematics Meetings begin. These Short Courses will take place on January 4 and 5, 2011 (Tuesday and Wednesday).
The cost to participate is the same for both courses. Advance registration fees are: member of the AMS or MAA, US$100; nonmembers are US$140; students, unemployed, or emeritus are US$48. These fees are in effect until December 15. If you choose to register at the meeting, the fees are US$134 for members of the AMS or MAA, US$170 for nonmembers, and US$69 for students, unemployed, or emeritus.
**Evolutionary Game Dynamics **
**Location: **Rhythms I, 2nd Floor, Sheraton
Organizer: Karl Sigmund, University of Vienna
Evolutionary game theory studies basic types of social interactions in populations of players. It is the ideal mathematical tool for methodological individualism, i.e., the reduction of social phenomena to the level of individual actions. Evolutionary game dynamics combines the strategic viewpoint of classical game theory (independent, rational players trying to outguess each other) with population dynamics (successful strategies increase their frequencies).
A substantial part of the appeal of evolutionary game theory comes from its highly diverse applications, such as social dilemmas, evolution of language, or mating behavior in animals. Moreover, its methods are becoming increasingly popular in computer science, engineering, algorithmic game theory, network analysis, machine learning, statistical procedures, and control theory. They help to design and control multi-agent systems, often with large numbers of agents (for instance, when routing drivers over highway networks, or data packets over the Internet). While traditionally these fields have used a top down approach, by directly controlling the behavior of each agent in the system, attention has recently turned to an indirect approach: allowing the agents to function independently, while providing incentives that lead them to behave in the desired way. Instead of the traditional assumption of equilibrium behavior, researchers opt increasingly for the evolutionary paradigm, and consider the dynamics of behavior in populations of agents employing simple, myopic decision rules.
The lectures offer a menu whose main course is based on deterministic and stochastic dynamics describing the evolution of frequencies of behavioral types, and whose side dishes consist of examples drawn from disciplines as diverse as microbiology, genetics, animal behavior, evolutionary psychology, route planning, e-auctions, common resources management, or micro-economics.
An introductory part of the course is devoted to a brief sketch of the origins of the field, and in particular to the examples that motivated evolutionary biologists to introduce a population dynamical viewpoint into game theory. This leads to some of the main concepts: evolutionary stability, replicator dynamics, invasion fitness, etc. Much of it can be explained by means of simple examples such as the Rock-Paper-Scissors game. It came as a surprise when childish games of that sort, intended only for the clarification of concepts, were found to actually lurk behind important classes of real-life social and biological interactions. The Ultimatum Game, the Prisoner's Dilemma, or the Stag-hunt Game have been used in hundreds of economic experiments, leading to fascinating insights into the role of fairness norms, moral emotions, unconscious motivations, and cultural differences. Behavioral economics promoted the design of efficient mechanisms for broadband auctions or globalized e-commerce, but also led to hot debates on the economic roles of punishment, inequity aversion, beliefs in supernatural agents, or concerns for reputation.
The transmission of successful strategies by genetic and cultural means results in a rich variety of stochastic processes and, in the limit of very large populations, deterministic adjustment dynamics including differential inclusions and reaction-diffusion equations. Some economists view these types of dynamics as basic tools for so-called equilibrium refinement and equilibrium selection concepts. (Indeed, most games have so many equilibria that it is hard to select the "right one"). However, evolutionary games have also permitted the move away from the equilibrium-centered viewpoint. Today, we understand that it is often premature to assume that behavior converges to an equilibrium. In particular, an evolutionarily stable strategy need not be reachable. Limit phenomena such as periodic or heteroclinic cycles, or chaotic attractors, may be considered, perhaps not as "solutions of the game", but as predictions of play. On the other hand, large classes of games leading to global convergence are presently much better understood.
The team for this AMS course consists of **Ross Cressman**, Wilfried Laurier University; **Josef Hofbauer**, University of Vienna; **Sabin Lessard**, Université de Montréal; **Bill Sandholm**, University of Wisconsin, **Karl Sigmund**, University of Vienna; and **Sylvain Sorin**, Université Pierre et Marie Curie, Paris. These speakers have substantially contributed to the field. They will provide a thoroughly up-to-date introduction to evolutionary games for mathematicians interested in the bottom-up analysis of social behavior.** **
**Relevant Literature**
Cressman, R., *Evolutionary Dynamics and Extensive Form Games*, MIT Press, Cambridge, MA, 2003.
Hofbauer, J. and Sigmund, K., *Evolutionary game dynamics*, Bull. AMS 40 (2003), 479-519
Lessard, S. and Ladret, V., *The probability of fixation of a single mutant in an exchangeable selection model*, Journal of Math Biology 54 (2007), 721-744.
Sandholm, W. H., *Population Games and Evolutionary Dynamics*, Harvard, MIT Press, 2010.
Sigmund, K., *The Calculus of Selfishness*, Princeton University Press, 2010
Sorin, S, Viossat, Y. and Hofbauer, J. (2009): *Time average replicator and best reply dynamics*, Math. Operations Research 34 (2009), 263-269
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**Computational Topology**
**Location: **Rhythms II, 2nd Floor, Sheraton
**Organizer: Afra Zomorodian, **Dartmouth College
**Introduction**
The area of computational topology developed in response to topological impediments emerging from within geometric problems, such as reconstructing watertight surfaces in computer graphics. Topological problems, however, arise naturally in many areas of inquiry. In robotics, for instance, we need to capture the connectivity of the configuration space of a robot in order to plan optimal trajectories. In computational structural biology, optimal trajectories within the conformation space of a protein define its folding path. More recently, topological data analysis has emerged as a new paradigm for robust analysis of noisy, high-dimensional, heterogeneous, sampled data.
Like topology, computational topology is a large and diverse area. The aim of this short course is to introduce the audience to recent theoretical as well as practical developments of this field, starting with a theoretical grounding in algebraic topology and ending with analysis of real world data using fast software. Topics covered in the course may include:
1. Review of algebraic topology invariants, such as homotopy, homology, and the Euler characteristic, with an emphasis on algorithms and computation.
2. Theory for robust computation of features of spaces, such as persistent homology, multidimensional persistence
3. Data Structures for representing the underlying topology of sampled data, such as the witness complex.
4. Algorithms for computing robust invariants of spaces efficiently, such the persistence algorithm.
5. Software for analyzing sampled data, such as JPlex, mapper, and eucharis.
6. Applications of computational topology to real world data, such as in computer vision, biophysics, and sensor networks.
**Lectures**
We have the following confirmed speakers with titles and abstracts for their talks.
**1. Topological Data Analysis **
**Afra Zomorodian**, Dartmouth College
**Abstract:** Most real-world data may be modeled as a finite set of noisy points, embedded in some high-dimensional space. For analysis, we assume that the points are sampled from some underlying space, whose geometry and topology are lost during the sampling process. In the emerging area of topological data analysis, we focus on robustly cover the topology of the original space. The process has two steps: 1. representing the space with a combinatorial structure, and 2. computing topological invariants. In this talk, I will discuss several methods for each step, describing the theory as well as implementation and experiments with real-world data.
**2. ****Persistent Topology **
**Gunnar Carlsson**, Stanford University
**Abstract:** In recent years various algebraic constructions have been developed as a method for studying and "measuring" the shape of spaces only specified by finite samples. I will survey these developments, and give examples.
**3.** Topological Dynamics: Rigorous Numerics via Cubical Homology
**Marian Mrozek**, Jagiellonian University (Poland)
**Abstract:** In the talk I will present how certain existence problems in dynamical systems, in particular the existence and location of chaotic dynamics, may be rigorously solved with the help of a computer via topological tools. I will explain why cubical homology and efficient homology algorithms are crucial in such applications. I will also present CAPD::RedHom, a software package for cubical and simplicial homology based on reduction algorithms such as acyclic subspace construction, coreductions and discrete Morse theory.
**4. Euler Calculus for Sensor Data **
**Robert Ghrist**, University of Pennsylvania
**Abstract:** This talk covers an ingenious integral calculus based on Euler characteristic, stemming from work on sheaves due to MacPherson and Kashiwara in the 1970s, and connecting back further to classical integral geometry. I will emphasize its novel utility in data management, particularly in aggregation of redundant data and inverse problems over sensor networks. Applications to target counting and localization will be given. These applications motivate the investigation of a new branch of numerical integration theory.
**5.** Topology in Robotics: Planning with Uncertainty
**Michael Erdmann**, Carnegie Mellon University
**Abstract:** Programming robots to perform tasks involves dealing with uncertainty. Frequently one builds planners that anticipate multiple outcomes of desired actions. This talk shows how combinatorial topology informs the design and planning process by characterizing the capabilities of uncertain robotic systems topologically.
Planning problems, particularly for robotic manipulation tasks, may often be cast as discrete graph search problems. Motions in the graph are generally nondeterministic or stochastic as a result of control and sensing errors in the underlying physical system. The global capabilities of such a system may be modeled by the homotopy type of a "strategy complex". The simplices of this complex may be viewed as the eventually-convergent control laws (aka plans or strategies) possible on the graph.
One result that appears via this perspective is a controllability theorem: The robot can move anywhere in its state graph despite control uncertainty precisely when the graph's strategy complex is homotopic to a sphere of dimension two less than the number of states.
**6. **Optimization of Cycles and Bases
**Jeff Erickson, **University of Illinois at Urbana-Champaign
**Abstract:** Many applications of computational topology, such as simplification of three-dimensional surface models in computer graphics, require efficient descriptions of topological features. Erickson will discuss recent algorithms and hardness results for computing several such descriptions, including minimum-cost representatives in given homotopy or homology classes, optimal bases for fundamental groups and homology groups, and maximum flows and minimum cuts in surface graphs.
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Schedule
It is planned that each speaker will give a ninety-minute talk. Currently, there is one software session for examining three software programs described by the first three speakers (possibly, persistence, mapper, and CHomP.) The idea is to have some interaction with software on the first day, so there is time for discussion on the second day. The panel discussion currently scheduled at the end of the second day may also be turned into a software session.
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