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APRIL 26 - 28, 2006 |
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Number Theory Seminar |
Topic: |
Identities in theta correspondence |
Presenter: |
Zhengyu Mao, Rutgers at Newark |
Date: |
Wednesday, April 26, 2006, Time: 2:00 p.m., Location: Fine Hall 314 |
Abstract: |
I will describe some identities of linear forms and
distributions attached to representations in theta correspondence. These
identities can be used to prove formula of special values of L-functions. |
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Algebraic Topology Seminar ***Please note special date and location |
Topic: |
Stable Homology of Aut(F_n) |
Presenter: |
Soren Galatius, Stanford University |
Date: |
Wednesday, April 26, 2006, Time: 3:00 p.m., Location: Fine Hall 801 |
Abstract: |
Let Aut(F_n) denote the automorphism group of a free group
on n generators. It is known that H_k(Aut(F_n)) is independent of n
as long as n >> k. There is a natural homomorphism from the symmetric
group S_n to Aut(F_n), I will sketch a proof that it induces an
isomorphism from H_k(S_n) to H_k(Aut(F_n)) for n >> k. An important
point of view here is that BAut(F_n) can be thought of as a moduli
space of metric graphs, i.e. graphs equipped with metrics, considered
up to isometry. |
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Department Colloquium |
Topic: |
Coarse geometry of mapping class groups |
Presenter: |
Yair Minsky, Yale University |
Date: |
Wednesday, April 26, 2006, Time: 4:30 p.m., Location: Fine Hall 314 |
Abstract: |
The mapping class group of a compact surface can be studied from the
point of view of geometric group theory, in which coarse or
large-scale properties of its Cayley graph can be related to algebraic
properties of the group. I will give a partial picture of this coarse
geometry, including a distance formula, a description of some
quasi-isometrically embedded subgroups, and a solution of the
Brock-Farb rank conjecture. This represents joint work with Masur and
with Behrstock. |
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Ergodic Theory and Statistical Mechanics Seminar |
Topic: |
Geometric Coincidence Conjecture for Pisot Substitutions |
Presenter: |
Jaroslaw Kwapisz, Montana State University |
Date: |
Thursday, April 27, 2006, Time: 2:00 p.m., Location: Fine Hall 322 |
Abstract: |
The talk will focus on the central conjecture in the theory of Pisot substitutions (i.e. substitutions over a finite alphabet with the spectral
radius of the abelianization that is a P.V. number) asserting that the translation
action on the tiling spaces (or substitutive systems) of unimodular Pisot
substitutions has pure discrete spectrum. I will identify the discrete spectrum,
recast the conjecture as a problem about injectivity of a natural geometric
realization map, and further boil it down to an algorithmically verifiable
Geometric Coincidence Condition (GCC). I will indicate how the GCC can be
verified for some families of substitutions (including a broad class of beta shifts). |
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Algebraic Topology Seminar |
Topic: |
Moduli Space of Nodal curves and Homotopy Theory |
Presenter: |
Soren Galatius, Stanford University |
Date: |
Thursday, April 27, 2006, Time: 3:00 p.m., Location: Fine Hall 401 |
Abstract: |
Riemann's moduli space M_g classifies isomorphism classes of
genus g Riemann surfaces (or algebraic curves). M_g is a non-compact
algebraic variety and has a natural compactification due to Deligne,
Mumford and Knudsen. A point in the compactification is an isomorphism
class of a nodal curve, ie. a Riemann surface with a certain mild kind of
singularities. Madsen and Weiss' proof of Mumford's conjecture tells much
about M_g. I will describe an attempt to understand the compactification
from a similar point of view. This is joint work with Y. Eliashberg. |
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Analysis Seminar |
Topic: |
Optimal transportation and Ricci curvature for metric measure spaces |
Presenter: |
Karl-Theodor Sturm, University of Bonn |
Date: |
Thursday, April 27, 2006, Time: 4:00 p.m., Location: Fine Hall 322 |
Abstract: |
We introduce and analyze generalized Ricci curvature bounds for
metric measure spaces (M,d,m), based on convexity properties of
the relative entropy Ent(. | m). For Riemannian manifolds,
Curv(M,d,m) \ge K if and only if Ric_M\ge K on M. For the
Wiener space, Curv(M,d,m)=1. One of the main results is that these lower
curvature bounds are stable under (e.g. measured Gromov-Hausdorff)
convergence. Moreover, we introduce a curvature-dimension condition
CD(K,N) being more restrictive than the curvature bound
Curv(M,d,m)\ge K. For Riemannian manifolds, CD(K,N) is equivalent to
Ric_M(\xi,\xi)\ge K\cdot |\xi|^2 and dim}(M)\le N.
Condition CD(K,N) implies sharp version of the
Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison
theorem and of the Bonnet-Myers theorem. Moreover, it allows to
construct canonical Dirichlet forms with {Gaussian upper and lower
bounds} for the corresponding heat kernels. |
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Topology Seminar |
Topic: |
Contact structures, Giroux torsion and contact invariants |
Presenter: |
Andras Stipsicz, Renyi Institute of Mathematics |
Date: |
Thursday, April 27, 2006, Time: 4:30 p.m., Location: Fine Hall 314 |
Abstract: |
Contact structures with positive Giroux torsion are expected to behave quite differently than the ones with vanishing torsion. For example
it is conjectured that the positivity of the Giroux torsion
of a contact structure is an obstruction for fillability. We give
some evidence for this conjecture through verifying a vanishing result for
the contact Ozsvath-Szabo invariants of contact structures with large enough torsion on certain 3-manifolds. |
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PACM Seminar - Distinguished Lecture Series |
Topic: |
Quantum Computers: How physics experiments might solve mathematical problems |
Presenter: |
Peter Shor, Mathematics, Massachusetts Institute of Technology |
Date: |
Thursday, April 27, 2006, Time: 8:00 p.m., Location: A02 McDonnell Hall |
Abstract: |
Quantum computers are hypothetical devices which use the principles of quantum mechanics to perform computations. For some difficult computational problems, including the cryptographically important problems of prime factorization and finding discrete logarithms, the best algorithms known for classical computers are exponentially slower than the algorithms known for quantum computers. Although they have not yet been built, quantum computers do not appear to violate any fundamental principles of physics. I will explain how quantum mechanics provides this extra computational power. One of the main difficulties in building quantum computers is in manipulating coherent quantum states without introducing errors or losing coherence. This problem can be alleviated by the use of quantum error correcting codes; if a quantum computer can be built with only moderately reliable hardware, then software can be used to make it extremely reliable. I will discuss these results as well. |
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Geometric Analysis Seminar |
Topic: |
Classification of extremal metrics on geometrically ruled surfaces
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Presenter: |
Christina W. Tonnessen-Friedman, Union College |
Date: |
Friday, April 28, 2006, Time: 3:00 p.m., Location: Fine Hall 314 |
Abstract: |
This talk will cover some recent results from joint work with V. Apostolov, D. Calderbank, and P. Gauduchon on extremal K\"ahler metrics on so-called admissible projective bundles. Since it is an excellent motivator and simplifies the presentation significantly, I will focus on the case of a geometrically ruled complex surface, where we now have complete answers to some long-standing questions. |
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MAY 1 - 5, 2006 |
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PACM Seminar |
Topic: |
Rare events in complex systems: How to determine their transition pathways and rate? |
Presenter: |
Eric Vanden-Eijnden, Courant Institute of Mathematical Sciences, New York University |
Date: |
Monday, May 1, 2006, Time: 4:00 p.m., Location: Fine Hall 214 |
Abstract: |
The dynamical behavior of many systems arising in physics, chemistry, biology, etc. is dominated by rare but important transition events between long lived states. Important examples include nucleation events during phase transition, conformational changes of macromolecules, or chemical reactions. Understanding the mechanism and computing the rate of these transitions is a topic that has attracted a lot of attention for many years. In this talk, I will discuss the theoretical background and algorithmic details of the finite-temperature string method, which gives a firm theoretical background to the concept of reaction coordinate to describe these transitions, and allows to determine their pathways and rate. The string method will be illustrated via several examples. |
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Algebraic Geometry Seminar |
Topic: |
Filtrations on cohomology arising from geometry |
Presenter: |
Mark Andrea de Cataldo, Stony Brook, State University of New York |
Date: |
Tuesday, May 2, 2006, Time: 4:30 p.m., Location: Fine Hall 322 |
Abstract: |
I will report on work in progress with Luca Migliorini at Bologna. Given a map of complex algebraic varieties, the Leray spectral sequence and the perverse Leray spectral sequence induce filtrations on the (intersection) cohomology (with compact supports) of the domain. We describe these filtrations geometrically in terms of hyperplane sections and deduce various new Hodge-theoretic consequences for the homology of complex varieties. A very interesting precursor of this new point of view, concerning the Leray spectral sequence for cohomology, is due to Donu Arapura. |
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Operation Research and Financial Engineering Seminar |
Topic: |
Climate Risk, Securitization, and Equilibrium Bond Pricing |
Presenter: |
Ulrich Horst, University of BC Vancouver |
Date: |
Tuesday, May 2, 2006, Time: 4:30 p.m., Location: Friend Center Bowl 008 |
Abstract: |
We propose a method of pricing financial securities written on non-tradable underlyings such as temperature or precipitation
levels. To this end, we analyze a financial market where agents
are exposed to financial and non-financial risk factors. The
agents hedge their financial risk in the stock market and trade a
risk bond issued by an insurance company. From the issuer's point
of view the bond's primary purpose is to shift insurance risks
related to non-catastrophic weather events to financial markets.
As such its terminal payoff and yield curve depend on an
underlying climate or temperature process whose dynamics is
independent of the randomness driving stock prices. We prove that
if the bond's payoff function is monotone in the external risk
process, it can be priced by an equilibrium approach. The
equilibrium market price of climate risk and the equilibrium price process
are characterized as solution of non-linear backward stochastic differential
equations. Transferring the BSDEs into PDEs, we represent the bond prices as
smooth functions of the underlying risk factors.
The talk is based on joint work with Matthias Muller. |
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Geometry, Representation Theory, and Moduli Seminar |
Topic: |
TBA |
Presenter: |
Alexander Braverman, Brown University |
Date: |
Wednesday, May 3, 2006, Time: 3:00 p.m., Location: Fine Hall 214 |
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Department Colloquium |
Topic: |
Asymptotics for prime specialization over finite fields |
Presenter: |
Brian Conrad, University of Michigan |
Date: |
Wednesday, May 3, 2006, Time: 4:30 p.m., Location: Fine Hall 314 |
Abstract: |
It is a classical and extremely difficult problem to prove theorems about prime values of irreducible polynomials over the integers. For example, it is still not known if there are infinitely many primes of the form n2 + 1. There is a long history of analogies between the integers and polynomials (in one variable) over a finite field, and so one can formulate an analogous problem in this other setting. It was discovered several years ago (joint work with K. Conrad and R. Gross) that there are some surprises. We illustrate the unexpected behavior by means of some explicit examples, and discuss theorems that predict these phenomena. The main goal of the talk is to motivate (by examples) and (briefly!) discuss the proofs of recent asymptotic results as the finite field and polynomial being specialized are allowed to vary; these asymptotics accord well with a general philosophy of Katz concerning limiting behavior over large finite fields and behavior over number fields. The case of characteristic 2 is not suitable for a general audience, but anyone interested can ask me about it at the colloquium dinner. |
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Analysis Seminar |
Topic: |
Hyperbolicity and the stability of the Hamilton system |
Presenter: |
Antonio Bove, University of Bologna |
Date: |
Thursday, May 4, 2006, Time: 4:00 p.m., Location: Fine Hall 322 |
Abstract: |
We prove that the Cauchy Problem for a class of hyperbolic operators
with double characteristics and whose simple null bicharacteristics
have limit points on the set of double points is not well-posed in the
$ C^{\infty} $ category, even though the usual Ivrii-Petkov conditions
on the lower order terms are satisfied.
According to the standard linear algebra classification these
operators, at a double point, have fundamental matrices exhibiting a
Jordan block of size 4. |
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Geometric Analysis Seminar |
Topic: |
Bergman kernels, Bergman metrics and Monge-Ampere geodesics |
Presenter: |
Steve Zelditch, Johns Hopkins University |
Date: |
Friday, May 5, 2006, Time: 3:00 p.m., Location: Fine Hall 314 |
Abstract: |
The space of Hermitian metrics of positive curvature on a positive holomorphic line bundle $L \to M$ is an infinite dimensional symmetric space whose geodesics are defined by a Monge-Ampere equation (Mabuchi, Semmes, Donaldson). Each hermitian metric may be approximated by Bergman metrics of high level k, i.e. metrics induced by holomorphic embeddings in projective space by sections of L^k (Tian-Yau-Z expansion). Recently, Phong-Sturm have raised the question of how well the geodesics in the space B_k of Bergman metrics of level k approximate the Monge-Ampere geodesics. My talk is about joint work in progress with Jian Song, in which we show that on a toric variety, the Bergman geodesics approach the Monge Ampere geodesics in certain norms. |
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Geometric Analysis Seminar ***Please note special time |
Topic: |
Blowup analysis for the Yamabe equation in high dimensions |
Presenter: |
Fernando Marques, Stanford University |
Date: |
Friday, May 5, 2006, Time: 4:00 p.m., Location: Fine Hall 314 |
Abstract: |
In this talk we will address the problem of analyzing the blow-up behavior of a sequence of solutions to the Yamabe equation in high dimensions. We will show how sharp pointwise estimates can be used to prove that the Weyl tensor vanishes up to order $[\frac{n-6}{2}]$ at a blowup point. This gives a proof of the compactness of the set of solutions to the Yamabe problem, a result conjectured by R. Schoen around 15 years ago. This is joint work with Marcus Khuri and Richard Schoen. |
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MAY 8 - 12, 2006 |
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Analysis Seminar *** Please note special day and location |
Topic: |
The local equivalence problem and symmetries of Levi degenerate
hypersurfaces |
Presenter: |
Martin Kolar, Masaryk University Brno |
Date: |
Monday, May 8, 2006, Time: 4:00 p.m., Location: Fine Hall 314 |
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MAY 15- 19, 2006 |
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Operation Research and Financial Engineering Seminar |
Topic: |
Dynamic risk measures |
Presenter: |
Pauline Barrieu, London School of Economics and Political Science |
Date: |
Tuesday, May 16, 2006, Time: 4:30 p.m., Location: Room E-219, Engineering Quad |
Abstract: |
We develop a methodology to optimally design financial instruments, written on a non-tradeable underlying risk. The idea is to minimize the risk of the issuer under the participation constraint imposed by the buyer. The problem is reduced to a unique inf-convolution problem involving a transformation of the initial convex risk measures of the agents. Introducing dynamic risk measures defined through their local specifications using Backward Stochastic Differential Equations (BSDEs) enables to achieve some tractability in the risk assessment criterion, but also to solve this inf-convolution problem and obtain an explicit characterization of the optimal transfer in general situations. |
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