Array Imaging

Speaker: 

Robert Grimmett Professor George Papanicolaou

Institution: 

Stanford University

Time: 

Tuesday, November 13, 2007 - 11:00am

Location: 

NatSci II, Room 1201

I will introduce the emerging interdisciplinary field of array imaging with several examples such as sonar, seismic imaging, radar, ultrasonic non-destructive testing, etc. I will explain how resolution issues can be addressed in a unified
mathematical way, along with some new ideas about optimizing the image formation process. I will also show the results of several numerical experiments.

Non-commutative Iwasawa theory

Speaker: 

John Coates

Institution: 

University of Cambridge

Time: 

Thursday, May 10, 2007 - 4:00pm

Location: 

MSTB 254

Iwasawa theory, and especially its main conjectures, is the main tool for
studying the mysterious exact formulae in number theory linking the very
different mathematical worlds of purely arithmetic questions on the one
hand, with special values of complex L-functions on the other (typified by
the conjecture of Birch and Swinnerton-Dyer). My lecture will attempt to
explain how, in the special case of elliptic curves, non-commutative
phenomena which arise in each of these worlds lead to very unexpected
consequences in the other world.

Evolutionary Perspectives on Ecosystem Patterning

Speaker: 

Professor Simon Levin

Institution: 

Princeton University

Time: 

Wednesday, February 14, 2007 - 2:00pm

Location: 

MSTB 254

There is a rich mathematical literature, going back at least to Vito Volterra, describing the dynamics of populations in ecological time. There are many reasons, however, to place such dynamics in an evolutionary framework, allowing examination of how behaviors and interactions change over longer time scales. Such approaches can inform the description of interactions on the faster time scales and explain why we observe the systems we do; furthermore, they can provide insights about how systems will change in response to changing environmental conditions. General approaches to such questions will be discussed, with specific application to problems of resource use and stoichiometry.

Individual Choices, Cooperation and the Global Commons: Mathematical Challenges in Uniting Ecology and Socioeconomics for a Sustainable Environment.

Speaker: 

Professor Simon Levin

Institution: 

Princeton University

Time: 

Tuesday, February 13, 2007 - 11:00am

Location: 

NS2 1201

We live in a Global Commons, in which the actions of individuals bear costs for society as a whole. The resources we extract for our own uses are no longer available to others, and the toxicants we discharge affect others. The result of this mismatch between individual actions and individual costs is evidenced in the depletion of common resources, the toxification of the environment, and even the frightening loss of effectiveness of the antibiotics that are so fundamental to public health. In the terminology of economists, conventional markets have failed to restrain our harmful activities, like over-consumption, because those markets do not adequately incorporate the social costs, the externalities.

How can we resolve this situation, and develop patterns of social behavior that hold out greater hope for a sustainable future? What can we learn from evolutionary theory, and how can mathematical approaches improve our ability to devise strategies? Not only individuals and corporations, but also societies and nations, act in their own selfish interest, leading to problems for the biosphere as a whole. This lecture will explore how, and under what conditions, cooperation and altruism have arisen in the process of evolution; why social norms, including punishment, have arisen to reinforce socially beneficial behavior; and how those social norms can lead to inter-group conflicts. Attention will be addressed to the socioeconomic systems in which environmental management is based, and ask what lessons can be learned from our examination of natural systems, and how we can modify social norms to achieve global cooperation in managing our common future.

Recent progress on Serre's conjectures.

Speaker: 

Professor Kenneth Ribet

Institution: 

UC Berkeley

Time: 

Thursday, May 24, 2007 - 4:00pm

Location: 

MSTB 254

Over the last 15 years, there have been tremendous advances in our understanding of the connections among modular forms, Galois representations and algebraic varieties. Undoubtedly, the most spectacular development in this subject was the proof of Fermat's Last Theorem, which was completed in 1994. Beginning in the late 1960s, J-P. Serre proposed links of various kinds between modular forms and representations of Galois groups. In 1987, Serre wrote a seminal article that included precise conjectures relating mod p Galois representations and mod p modular forms. These conjectures were so powerful and general that they were inaccessible by then-current methods. Amazingly, these conjectures have been proved over the last two years, with the final step being contributed only several months ago. The main ideas are due to Khare and Wintenberger, with major contributions from Kisin and others. My talk will explain the history of the conjectures and some elements of the ingenious proof.

Knoted Solitons in the Faddeev and Skyrme Models

Speaker: 

Fanghua Lin

Institution: 

Courant Institute of New York University

Time: 

Wednesday, January 11, 2006 - 2:00pm

Location: 

MSTB 254

Here we give a brief survey on recent mathematical works concerning the Faddeev and Skyrme models.One of the most facinating phenomena descibed by these models are the knoted topological soliton solutions which are fundamentally different from many other well-known feild theory models such as instantons and monopoles in the Yang-Mills or the general gauge field theory,bubbles in the nonlinear sigma models or ferromagnetisms and vortices in superconductors and superfluids.In this lecture we shall illustrate some key features of these models that lead to the exisitence of stable knoted solitons and to discuss some possible implications in other problems.

Conserved Quantities and Analysis on Multiscale Problems

Speaker: 

Fanghua Lin

Institution: 

Courant Institute of New York University

Time: 

Monday, January 9, 2006 - 2:00pm

Location: 

MSTB 254

The importance of the conserved quantities were well recoganized in the physical sciences. In this lecture, through several examples, we shall illustrate the fundamental roles played by such conserved quatities in the multiscale analysis. One therefore has to put such quatities into serious considerations also in both modelings and computations.

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