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DECEMBER 2007 |
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| Statistical Mechanics Seminar |
| Topic: |
Wave Functions in Thermal Equilibrium -- GAP Measures and Canonical Typicality |
| Presenter: |
Roderick Tumulka, Rutgers University |
| Date: |
Wednesday, December 5, 2007, Time: 2:00 p.m., Location: Jadwin 343 |
| Abstract: |
I will talk about the claim that a quantum system in thermal equilibrium at temperature 1/beta has a random wave function whose distribution is a particular probability measure on Hilbert space called GAP(beta). This is roughly analogous to the familiar claim that a classical system in thermal equilibrium at temperature 1/beta has a random phase point (q,p) whose distribution has density proportional to exp(-beta H(q,p)) with H(q,p) the Hamiltonian function. I will explain how the GAP measures are defined in terms of the system's Hamiltonian operator H, what precisely the claim about quantum systems means, and how it is connected to Schrodinger's cat and to the typicality of the canonical density matrix exp(-beta H). |
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| Geometry, Representation Theory, and Moduli Seminar |
| Topic: |
On the geometry of genus 1 Gromov-Witten invariants |
| Presenter: |
Aleksey Zinger, Stony Brook University |
| Date: |
Wednesday, December 5, 2007, Time: 3:00 p.m., Location: Fine Hall 214 |
| Abstract: |
The mirror symmetry principle of string theory has led to astounding predictions for counts of holomorpic curves, especially for a quintic threefold (a degree 5 hypersurface in P^4). There has been much success in verifying these predictions in genus 0, in part due to a good undertanding of the geometry of genus 0 GW-invariants. In this talk, I will give an overview of geometric properties of genus 1 GW-invariants, including a relation between GW-invariants of a hypersurface and of the ambient projective space. These properties mimic well-known genus 0 properties. Taken together, they provide a method for computing genus 1 GW-invariants of all complete intersections and have led to the verification of the 1993 BCOV mirror symmetry prediction for genus 1 GW-invariants of a quintic threefold. |
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| Department Colloquium |
| Topic: |
Contact structures in dimension 3 and the Seiberg-Witten equations |
| Presenter: |
Clifford Taubes, Harvard University |
| Date: |
Wednesday, December 5, 2007, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
I hope to give some indication of how the Seiberg-Witten equations are used to study the dynamics of vector fields on 3-dimensional manifolds. One result of this research is a proof of Alan Weinstein's conjecture about the existence of closed integral curves of the Reeb vector field for a contact 1-form. |
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| Symplectic Geometry Seminar *** Please note special date, time and location |
| Topic: |
Mirror symmetry for Gromov-Witten invariants of a quintic threefold |
| Presenter: |
Aleksey Zinger, Stony-Brook University |
| Date: |
Thursday, December 6, 2007, Time: 1:00 p.m., Location: Fine Hall 224 |
| Abstract: |
The mirror symmetry principle of string theory provides closed formulas for GW-invariants, with special attention devoted to a quintic threefold, Q3. The genus 0 mirror prediction for Q3 was verified 12 years ago by using the Atiyah-Bott localization theorem. In this talk, I will outline how the analoguos genus 1 localization problem is solved by making use of a number of its relations with the genus 0 localization problem. This approach confirms the 1993 BCOV mirror symmetry prediction for genus 1 GW-invariants of Q3. It also produces mirror formulas for genus 1 GW-invariants of a degree n hypersurface in P^{n-1} (Q3 is n=5), confirming a recent prediction of Klemm-Pandharipande for a sextic fourfold (n=6) and producing a puzzling combinatorial identity related to unbranched covers of tori (n=3). |
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| Discrete Mathematics Seminar |
| Topic: |
Critical triangle-free graphs with lots of edges |
| Presenter: |
Wesley Pegden, Rutgers University |
| Date: |
Thursday, December 6, 2007, Time: 2:15 p.m., Location: Fine Hall 224 |
| Abstract: |
How close are the triangle-free graphs to the bipartite graphs? The different ways of asking this question give different answers. For example: on the one hand, triangle-free graphs can have arbitrarily large chromatic number; on the other, natural constructions to demonstrate this fact are typically very sparse. One type of question with this theme is the recently solved question of Erdos and Simonovits, which asks about triangle-free graphs with large chromatic number and large minimum degree.
We consider a different question with the same motivation: are there k-chromatic-critical triangle-free graphs with lots of edges for arbitrarily large k? We'll talk about a construction that shows that the answer is yes. We'll also show how one can nonconstructively extend the result to graphs without pentagons as well. Results about the density of general critical graphs typically leave wide gaps between what we know as upper and lower bounds; the case of triangle-free graphs seems unusual in this context, since our construction actually shows the exact maximum asymptotic density. |
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| Joint Princeton University/IAS Number Theory Seminar |
| Topic: |
Heegner Divisors, L-Functions and Harmonic Weak Maass Forms |
| Presenter: |
Jan Bruinier, University of Cologne |
| Date: |
Thursday, December 6, 2007, Time: 4:30 p.m., Location: Fine Hall 214 |
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| Topology Seminar |
| Topic: |
Comultiplication and the Ozsvath-Szabo contact invariant |
| Presenter: |
John Baldwin, Columbia University |
| Date: |
Thursday, December 6, 2007, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Let S be a surface with boundary and suppose that g and h are diffeomorphisms of S which restrict to the identity on the boundary. I'll describe how the Ozsvath-Szabo contact invariants associated to the open books (S,g), (S,h), and (S,hg) are natural with respect to a comultiplication on the corresponding Heegaard-Floer homology groups. In particular, it follows that if the contact invariants associated to the open books (S,g) and (S,h) are non-zero, then so is the contact invariant associated to the open book (S,hg). I plan to discuss an extension of this comultiplication to HF^+ and an obstruction to the compatibility of a contact structure with a planar open book. |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Gluing Monopoles |
| Presenter: |
Pedram Safari, Harvard University |
| Date: |
Friday, December 7, 2007, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
A canonical method is established to glue Seiberg-Witten monopoles over a closed 4-manifold split along a 3-dimensional submanifold. The method is quite generic and only requires mild conditions. |
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| PACM Colloquium |
| Topic: |
Collective motion and decision-making in animal groups |
| Presenter: |
Iain Couzin, Ecology & Evolutionary Biology, Princeton University |
| Date: |
Monday, December 10, 2007, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: |
Animal groups such as bird flocks, insect swarms and fish schools are spectacular, ecologically important and sometimes devastating features of the biology of various species. Outbreaks of the desert locust, for example, can invade approximately one fifth of the Earth's land surface and are estimated to affect the livelihood of one in ten people on the planet. Using a combined theoretical and experimental approach involving insect and vertebrate groups I will address how, and why, individuals move in unison and investigate the principals of information transfer in these groups, particularly focusing on leadership and collective consensus decision-making.
For very large animal groups, despite huge differences in the size and cognitive abilities of group members, recent models from theoretical physics ('self-propelled particle', SPP, models) have suggested that general principles underlie collective motion. Such models demonstrate that some group-level properties may be largely independent of the types of animals involved. I shall present recent experimental work on locusts that validates some of the predictions of simple mechanistic models including a density-dependent "phase transition" from disordered to ordered motion.
Details of the mechanism by which individuals interact, however, also provide important biological insights into swarm behaviour. Using laboratory studies involving nerve manipulation and field experiments we demonstrate that some swarming insects are in effect on a "forced march" driven by cannibalism.
These results will be discussed in the context of the evolution of functional complexity and pattern formation in biological systems. |
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| Operations Research and Financial Engineering Seminar |
| Topic: |
Optimality in Large-Scale Multiple Testing
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| Presenter: |
Tony Cai, The Wharton School, University of Pennsylvania |
| Date: |
Monday, December 10, 2007, Time: 4:30 p.m., Location: Friends Center Room 008 |
| Abstract: |
See http://orfe.princeton.edu/papers/cai-abstract.pdf |
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| Algebraic Geometry Seminar |
| Topic: |
TBA |
| Presenter: |
Paul Hacking, University of Washington |
| Date: |
Tuesday, December 11, 2007, Time: 4:30 p.m., Location: Fine Hall 322 |
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| Statistical Mechanics Seminar |
| Topic: |
Random matrices, statistical mechanics and hyperbolic supersymmetry |
| Presenter: |
Thomas Spencer, IAS |
| Date: |
Wednesday, December 12, 2007, Time: 2:00 p.m., Location: Jadwin 343 |
| Abstract: |
We present a statistical mechanics model with a hyperbolic supersymmetry. This model is expected to qualitatively describe properties of random band matrices in N dimensions eg localization and delocalization. The "spins" in this model may be thought of taking values in a Poincare super-disc. In three dimensions we show that this model has a diffusive phase. In one dimension there is only the localized phase. The analysis of this model relies a family of identities coming from SUSY together with estimates of a random walk on a percolation cluster. The surprising relation of this model to linearly reinforced random walk will also be highlighted. No knowledge of SUSY is needed. This is joint work with M. Disertori and M. Zirnbauer. |
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| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
Attractors with Large Invisible Parts |
| Presenter: |
Andrei Negut, Princeton University |
| Date: |
Thursday, December 13, 2007, Time: 2:00 p.m., Location: Fine Hall 401 |
| Abstract: |
Philosophically, an attractor of a dynamical system is a closed subset of the phase space which orbits "approach" as time goes to infinity. Different meanings of the word "approach" produce different versions of attractors: maximal, Milnor, statistical, minimal etc. The question of generic non-coincidence between these various types of attractors has not yet been answered. We will be concerned with a different point of view. Physically, if one looks at the attractor, then one knows where most orbits will go to as time goes to infinity. But it is possible that a large part of the attractor is redundant, in the sense that orbits spend very very little time near it. Thus, it would be more significant to look only at the non-redundant part of the attractor. Concretely, we will present an example of a random dynamical system, given by parameters of "reasonable magnitude" (e.g. 1000). For this dynamical system, roughly half of the attractor is "invisible" in the sense that orbits spend near it only a fraction of 2^{-500} of all time. The number 2^{-500} is equal to zero for all physical or computer experiments, and therefore an observer should not bother with the "invisible" half of the attractor. Moreover, small perturbations of this dynamical system exhibit the same property, and therefore the phenomenon is generic. |
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| Topology Seminar |
| Topic: |
Components of Springer fibers and Khovanov's arc algebra |
| Presenter: |
Ben Webster, IAS |
| Date: |
Thursday, December 13, 2007, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Using the structure of certain Springer fibers and their components, I'll describe a geometric construction of an algebra which is painfully close to being isomorphic to the arc algebra defined by Khovanov, but in fact, isn't. I'll then hopefully explain why this is actually a good thing. |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
On the $\sigma_2$-scalar curvature and its application |
| Presenter: |
Yuxin Ge, University Paris 12 |
| Date: |
Friday, December 14, 2007, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
In this talk, we establish an analytic foundation for a fully non-linear equation $\frac{\sigma_2}{\sigma_1}=f$ on manifolds with positive scalar curvature. This equation arises from conformal geometry. As application, we prove that, if a compact 3-dimensional manifold $M$ admits a riemannian metric with positive scalar curvature and $\int \sigma_2\ge 0$, then topologically $M$ is a quotient of sphere. |
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| Topology Seminar |
| Topic: |
Subdirect products of surfaces, homological finiteness, and residually-free groups |
| Presenter: |
Martin Bridson, Imperial College |
| Date: |
Thursday, December 20, 2007, Time: 4:30 p.m., Location: Fine Hall 314 |
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