Dedekind Zeta functions at s=-1 and the Fitting ideal of the tame kernel in a relative quadratic extension

Speaker: 

Jonathan Sands

Institution: 

Univ. of Vermont and UCSD

Time: 

Tuesday, February 13, 2007 - 12:00pm

Location: 

MSTB 254

Brumer's conjecture states that Stickelberger elements combining values of L-functions at s=0 for an abelian extension of number fields E/F should annihilate the ideal class group of E when it is considered as module over the appropriate group ring. In some cases, an ideal obtained from these Stickelberger elements has been shown to equal a Fitting ideal connected with the ideal class group. We consider the analog of this at s=-1, in which the class group is replaced by the tame kernel, which we will define. For a field extension of degree 2, we show that there is an exact equality etween the Fitting ideal of the tame kernel and the most natural higher Stickelberger ideal; the 2-part of this equality is conditional on the Birch-Tate conjecture.

Brain Surface Parameterization using Riemann Surface Structure

Speaker: 

Yalin Wang

Institution: 

UCLA

Time: 

Monday, January 29, 2007 - 4:00pm

Location: 

MSTB 254

We develop general approaches that parameterize brain anatomical
surfaces with Riemann surface structure. All metric orientable surfaces
are Riemann surfaces and admit conformal structure. With harmonic
energy minimization, holomorphic 1-form and the Ricci flow methods, we
can parameterize brain surfaces with various canonical surfaces such as
sphere, Euclidean plane and punched hole disks. The resulting surface
subdivision and the parameterizations of the components are intrinsic
and stable. Our parameterization scheme offers a way to explicitly
match landmark curves in anatomical surfaces such as the cortex,
providing a surface-based framework to compare anatomy statistically and
to generate grids on surfaces for PDE-based signal processing. Various
applications of our research will also be discussed.

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.

Wavepacket Spreading on the Fibonacci Chain

Speaker: 

David Damanik

Institution: 

Rice University

Time: 

Thursday, February 8, 2007 - 2:00pm

Location: 

MSTB 254

We discuss the spreading of an initially localized wavepacket
on the Fibonacci chain under Schr\"odinger dynamics. After briefly
recalling the known results that bound the associated dynamical
quantities from above and below, we present a combinatorial approach to
this problem that leads to improved lower bounds (joint work with Mark
Embree and Serguei Tcheremchantsev)

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