## Langlands Duality In Representation Theory And Gauge Theory

### [RG # 122] Langlands Duality In Representation Theory And Gauge Theory

September 1, 2010 - February 28, 2011

Organizer:

David Kazhdan (The Hebrew University)

September 1, 2010 - February 28, 2011

Organizer:

David Kazhdan (The Hebrew University)

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This research group will explore new and important developments in Category theory, Topology, Representation theory and Physics.

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March 1 - August 31, 2011

Organizers:

Michal Feldman (Tel Aviv University)

Noam Nisan (The Hebrew University)

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The last decade has seen the emergence and growth of a new interdisciplinary field of research often termed "Algorithmic Game Theory". This field lies at the crossroads of computer science, game theory, and economics; a combination which is necessary for addressing many of the challenges posed by the Internet. Not only is this field full of intellectual excitement internally, and not only has it already begun to intellectually influence the three parent disciplines, but it also has significant implications for the Internet, as evidenced by the large number of researchers in the field hired by Google, Yahoo, and Microsoft.

At the approximate age of ten years, it seems that the field of Algorithmic Game Theory is maturing. The goal of this group is to elucidate the main challenges of the field and attempt to chart the future course of the field for the next decade.

Some research topics that will be explored:

- Networks with contagious risk, the different aspects of how the evaluation of the Generalized Second Price mechanisms are used for selling ads on the Internet, and the understanding of the performance of simple auctions and modeling auctions used in practice (Eva Tardos)

- Interviewing in stable matching problems and cost-sharing mechanisms (Nicole Immorlica)

- Sketching valuation functions, the equilibria of simple market mechanisms, and optimal multi-item auctions (Noam Nisan)

- Auction design for agents with uncertain, private values (Anna Karlin)

- A general framework for computing optimal correlated equilibria in compact games, computing Nash equilibria of action-graph games via support enumeration, mechanical design and auctions, and computational equilibrium analysis of voting games (Kevin Leyton-Brown)

- Envy-free mechanisms for multiunit auctions with budgets, cost sharing games with capacitated network links, and game theoretic perspectives of the facility location problem (Michal Feldman)

- Bargaining in networks (Amos Fiat)

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March 1 - August 31, 2012

Organizer:

Elchanan Ben-Porath (The Hebrew University)

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The classical model in economic theory assumes that the economic agent is fully rational. In particular, it is assumed that the agent is aware of the set of actions that is available to him and has a correct model of the environment in which he is operating. In particular, he understands the relationship between his actions and outcomes. Any calculation or consideration that is relevant to achieve this complete understanding of the environment can be done without mistake, with no delay, and without cost. In addition, the agent has a complete and consistent preference over the set of possible outcomes and chooses the action that leads to the best outcome with respect to his preference.

This model is clearly unrealistic. A human agent is often unaware of actions, contingencies, and considerations that are relevant for the decision problem that he is facing. He often finds it difficult to form a preference (for example, to determine his trade-off between price and quality, or the trade-off between current pleasure and future welfare), and there are specific limits on his ability to process information (specifically, attention, memory, and thinking are bounded and costly). Economists have of course realized that people are subject to these limitations; however, until they were exposed to the research in cognitive psychology they did not have a concrete sense of the systematic deviations of human decision making from the rational model.

The research agenda of our group consists of two main components:

(1) Studying models of decision making that depart from the standard model and in particular take into account cognitive limitations and non-standard preferences.

(2) Studying the implications of bounded rationality in multi-person interactions, in particular, games and market economics.

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Organizers:

Matthias Staudacher (Humboldt-University, Berlin)

Romuald Janik (Jagiellonian University)

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The focus of the group is on a currently intensively-studied model in theoretical physics, which has been termed by some the "hydrogen atom of the 21st century". The basic idea and goal was to construct a mathematically exact solution of an, admittedly idealized, quantum field theory of the general type as occurs in the description of the forces between our universe's elementary particles, with the notable exception of the gravitational force.

Yang-Mills gauge theory is named for its inventors, Chen Ning Yang and Robert Mills. The word *gauge* refers to the fact that at the heart of these theories lies a certain built-in redundancy in its mathematical description very hard to eliminate, while apparently necessary in order to properly record and understand the rules of the game. The idealized system at the focus of our group is called N=4 super Yang-Mills gauge theory. It stipulates that in addition to our standard continuous ("bosonic") spacetime dimensions, certain hidden discrete ("fermionic") dimensions exist. The number N=4 refers to the fact that this model has four such curious symmetries.

The N=4 gauge model is the most beautiful and simplest Yang-Mills theory one can come up with, even though it certainly does not directly appear in nature. It is also a deeply mysterious model, and it has become clear in recent years that it possesses further hidden symmetries as well as seemingly contradictory, alternative descriptions, which promise to allow for a complete solution of the model, at least for certain quantities and in certain limits. This is precisely what we are setting out to achieve with our program at the IIAS.

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Organizer:

Amnon Aharony (Ben-Gurion University)

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Molecular electronics, one of the major fields in nanoscience, studies electronic devices based on single molecules, and on molecular networks connected to other electronic components. Its potential applications include sensors, displays, smart materials, molecular motors, logic and memory devices, molecular scale transistors and energy transduction devices. Besides being the next step in device miniaturization, molecules are able to bind to one another, recognize each other, assemble into larger structures, and exhibit dynamical stereochemistry. In addition to its technological potential, molecular electronics has raised many new fundamental questions, e.g. concerning the interactions of molecular systems with their environment and their functioning far from equilibrium. Also, fluctuations and noise constitute an important part of the physics of such microscopic systems. At the moment there already exist several ingenious experimental realizations of transport through molecular bridges. There also exist a variety of different theoretical tools (both in chemistry and in physics) to attack the above important issues.

This group will bring together physicists and chemists, experimentalists and theoreticians, senior and young scientists, aiming to understand existing experiments, to propose new experiments (possibly combining various experimental tools) and new technological devices, using combinations of various theoretical and experimental methods.

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