SEMINARS Updated: 4-13-2011     APRIL 2011     Department Colloquium Topic: Eigenfunctions and nodal sets Presenter: Steve Zelditch, Northwestern University Date: Wednesday, April 13, 2011, Time: 4:30 p.m., Location: Fine Hall 314 Abstract: Nodal sets are the zero sets of eigenfunctions of the Laplacian on a Riemannian manifold (M, g). If Delta phi = lambda^2 \phi, then phi is somewhat analogous to a polynomial of degree lambda and its nodal set is somewhat analogous to a real algebraic variety of this degree. The analogy is closest if (M, g) is real analytic. Donnelly-Fefferman then showed that the hypersurface volume is of order lambda. The physicists make the bold conjecture that, if the geodesic flow is chaotic, then the nodal sets become equidistributed with respect to the volume form. This is far beyond current technology, but I will show that if one complexifies everything--the manifold, the eigenfunctions and the nodal sets--then one can obtain equidistribution results in the ergodic case. The main new result is that intersections of the complex nodal set with a typical complex geodesic in the ergodic case becomes concentrated on the real geodesic in a uniform way with respect to arc-length. This is almost the physics conjecture.     Ergodic Theory and Statistical Mechanics Seminar Topic: Local semicircle law at the edge of spectrum for Gaussian $\beta $-ensembles Presenter: Percy Wong, Princeton University Date: Thursday, April 14, 2011, Time: 2:00 p.m., Location: Fine Hall 801 Abstract: We prove, via the moments method, a local semicircle law for the Gaussian $\beta$-ensembles at the almost optimal scaling of $n^{-2/3 +\epsilon}$ at the edge of the spectrum for all $\beta \geq 1$.     Discrete Mathematics Seminar Topic: Upper tails for cliques Presenter: Robert DeMarco, Rutgers University Date: Thursday, April 14, 2011, Time: 2:15 p.m., Location: Fine Hall 224 Abstract: For a fixed graph H, what is the probability a random graph G(n,p) contains more than twice the expected number of copies of H. In this talk I will sketch the answer to this question in the case when H is a clique. A conjecture for general H will also be presented. (Joint work with Jeff Kahn)     Algebraic Topology Seminar Topic: On the rational homotopy type of Moment-angle complexes Presenter: Sam Gitler, IAS Date: Thursday, April 14, 2011, Time: 3:00 p.m., Location: Fine Hall 314 Abstract: In this talk I will discuss the following theorem, A moment-angle complex is rationally elliptic if it is homeomorphic to a product of odd dimensional spheres and discs.     Topology Seminar Topic: The Rank versus Genus Conjecture Presenter: Tao Li, Boston College Date: Thursday, April 14, 2011, Time: 4:30 p.m., Location: Fine Hall 314 Abstract: We construct a counterexample to the Rank versus Genus Conjecture (also known as the Rank Conjecture), i.e., a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus.     Differential Geometry and Geometric Analysis Seminar Topic: Stability of hyperbolic manifolds with cusps under Ricci flow Presenter: Richard Bamler, Princeton University Date: Friday, April 15, 2011, Time: 3:00 p.m., Location: Fine Hall 314 Abstract: We show that every hyperbolic manifold of finite volume and dimension greater or equal to 3 is stable under normalized Ricci flow, i.e. that every small perturbation of the hyperbolic metric flows back to the hyperbolic metric again. Here we do not need to make any decay assumptions on this perturbation. As we will see, the main difficulty in the proof comes from a weak stability of the cusps which has to do with the existence of certain cusp deformations. We will overcome this weak stability by using a new analytical method developed by Koch and Lamm.     Differential Geometry and Geometric Analysis Seminar Topic: Symmetries of extremal metrics on projective bundles and stability Presenter: Hongnian Huang, CIRGET University of Quebec at Montreal Date: Friday, April 15, 2011, Time: 4:00 p.m., Location: Fine Hall 314 Abstract: In this talk, we will discuss some partial results in a project initiated by Vestislav Apostolov aiming to solve a conjecture about the specific form of an extremal Kahler metric on a projective bundle over a curve. The conjecture, on one hand, connects to the Donaldson-Uhlenbeck-Yau Theorem and, on the other hand relates the existence of extremal metrics to the stability of the vector bundle, which is a natural extension of Yau's conjecture in the case of Kaehler Einstein metric when the first Chern class of the Kaehler manifold is positive.     Analysis Seminar Topic: Resonances and null structures in weakly dispersive equations Presenter: Fabio Pusateri, New York University Date: Monday, April 18, 2011, Time: 4:00 p.m., Location: Fine Hall 314 Abstract: We show a result about global existence of small solutions for a class of scattering critical nonlinear dispersive equations. The problem we consider originates from a classical result of S. Klainerman on the wave equation. Our proof is based on the method of space-time resonances, recently introduced by Germain, Masmoudi and Shatah. In particular, we focus on the relation between resonances and null structures. We will also discuss some related application to scattering critical equations of Schrodinger type.     PACM Colloquium Topic: Likelihood and algebraic maps for stochastic biochemical network models Presenter: Gregory Rempala, Medical College of Georgia Date: Monday, April 18, 2011, Time: 4:00 p.m., Location: Fine Hall 214 Abstract: With the development of new sequencing technologies of modern molecular biology, it is increasingly common to collect time-series data on the abundance of molecular species encoded within the genomes. This presentation shall illustrate how such data may be used to infer the parameters as well as the structure of the biochemical network under mass-action kinetics. Given certain constraints on the geometry of the stoichiometric space, we use algebraic methods as an alternative to conventional hierarchical graphical models, to carry out network structure inference by identifying reaction rate constants which are significantly different from zero.     Algebraic Geometry Seminar Topic: TBA Presenter: Charles Doran, University of Alberta Date: Tuesday, April 19, 2011, Time: 4:30 p.m., Location: Fine Hall 322     Department Colloquium Topic: TBA Presenter: Alexander Bufetov, Rice University Date: Wednesday, April 20, 2011, Time: 4:30 p.m., Location: Fine Hall 314     Ergodic Theory and Statistical Mechanics Seminar Topic: TBA Presenter: Alexander Bufetov, Rice University Date: Thursday, April 21, 2011, Time: 2:00 p.m., Location: Fine Hall 801     Discrete Mathematics Seminar Topic: TBA Presenter: Paul Wollan, Sapienza University of Rome and Georgia Tech Date: Thursday, April 21, 2011, Time: 2:15 p.m., Location: Fine Hall 224     Algebraic Topology Seminar Topic: Unstable operations in etale and motivic cohomology Presenter: Chuck Weibel, Rutgers University Date: Thursday, April 21, 2011, Time: 3:00 p.m., Location: Fine Hall 314 Abstract: Epstein constructed cohomology operations P^i in etale cohomology, and Voevodsky constructed cohomology operations P^i in motivic cohomology. These differ from the topological operations because of twisting. We modify Henri Cartan's classification of unstable operations to classify all etale operations (they are the expected operations), and classify all weight one motivic operations. Here new operations arise, and we paint a conjectural picture in all weights. This is joint work with Bert Guillou.     Joint Princeton University and IAS Number Theory Seminar Topic: Algebraic cycles and Euler systems for real quadratic fields Presenter: Henri Darmon, McGill University Date: Thursday, April 21, 2011, Time: 4:30 p.m., Location: Fine Hall 214 Abstract: I will discuss some possible applications of algebraic cycles and p-adic families of modular forms to the arithmetic of elliptic curves over abelian extensions of real quadratic fields. This is a report on work in progress with Victor Rotger and Ignacio Sols, and on earlier work with Massimo Bertolini and Kartik Prasanna.     Differential Geometry and Geometric Analysis Seminar Topic: The Yamabe problem for the conformal fractional Laplacian Presenter: Maria del Mar Gonzalez, Universitat Politècnica de Catalunya Date: Friday, April 22, 2011, Time: 4:00 p.m., Location: Fine Hall 314 Abstract: Based on the relations between scattering operators of asymptotically hyperbolic metrics and Dirichlet-to-Neumann operators of uniformly degenerate elliptic boundary value problems, we formulate fractional Yamabe problems that include the boundary Yamabe problem studied by Escobar. We observe an interesting Hopf type maximum principle together with interplays between analysis of weighted trace Sobolev inequalities and conformal structure of the underlying manifolds, which extend the phenomena displayed in the classic Yamabe problem and boundary Yamabe problem. This is joint work with J. Qing.     PACM Colloquium Topic: Learning from Labeled and Unlabeled Data: Global vs. Multiple Approaches Presenter: Boaz Nadler, Weizmann Institute - Israel Date: Monday, April 25, 2011, Time: 4:00 p.m., Location: Fine Hall 214 Abstract: In recent years there is increasing interest in learning from both labeled and unlabeled data (a.k.a. semi-supervised learning, or SSL). The key assumption in SSL, under which an abundance of unlabeled data may help, is that there is some relation between the unknown response function to be learned and the marginal density of the predictor variables. In the first part of this talk I'll present a statistical analysis of two popular graph based SSL algorithms: Laplacian regularization method and Laplacian eigenmaps. In the second part I'll present a novel multiscale approach for SSL as well as supporting theory. Some intimate connections to harmonic analysis on abstract data sets will be discussed. Joint work with Nati Srebro (TTI), Xueyuan Zhou (Chicago), Matan Gavish (WIS/Stanford) and Ronald Coifman (Yale).     Algebraic Topology Seminar Topic: Integral and Rigid Elliptic Genera Presenter: Victor Buchstaber, Steklov Mathematical Institute, Russian Academy of Sciences Date: Thursday, April 28, 2011, Time: 3:00 p.m., Location: Fine Hall 314 Abstract: Each Hirzebruch genus is dened by the exponential of the corresponding formal group law. We will describe the formal group law associated with the general Weierstrass model of the elliptic curve with parameters ¹ = (¹1; ¹2; ¹3; ¹4; ¹6) and arithmetic Tate uniformization. We obtain the corresponding general elliptic genus which is Z[¹]-integral. Krichever genus is dened by the Baker-Akhiezer function of an elliptic curve and has 4 parameters. This genus has the fundamental rigidity property on S1-equivariant SU-manifolds. We will introduce the universal Krichever formal group law and describe the conditions necessary and sucient for the elliptic formal group law to become the Krichever formal group law. We will obtain the list of corresponding integral genera with the rigidity property on S1-equivariant SU-manifolds. Main denitions will be introduced during the talk. New results presented in the talk were obtained in recent joint works with E. Yu. Bunkova.   Joint Princeton University and IAS Number Theory Seminar **please note special location** Topic: Serre´s conjectures on the number of rational points of bounded height Presenter: P. Salberger, Chalmers University Date: Thursday, April 28, 2011, Time: 4:30 p.m., Location: S-101 at IAS Abstract: We give a survey of recent results on conjectures of Heath-Brown and Serre on the asymptotic density of rational points of bounded height. The main tool in the proofs is a new global determinant method inspired by the local real and p-adic determinant methods of Bombieri-Pila and Heath-Brown.     Topology Seminar Topic: TBA Presenter: Peter Ozsvath, MIT Date: Thursday, April 28, 2011, Time: 4:30 p.m., Location: Fine Hall 314     Differential Geometry and Geometric Analysis Seminar Topic: TBA Presenter: Artem Pulemotov, University of Chicago Date: Friday, April 29, 2011, Time: 3:00 p.m., Location: Fine Hall 314     MAY 2011     Department Colloquium Topic: TBA Presenter: Gautam Chinta, The City University of New York Date: Wednesday, May 4, 2011, Time: 4:30 p.m., Location: Fine Hall 314     Topology Seminar Topic: TBA Presenter: Liam Watson, UCLA Date: Thursday, May 5, 2011, Time: 4:30 p.m., Location: Fine Hall 314     Differential Geometry and Geometric Analysis Seminar Topic: TBA Presenter: Yannick Slre, Universite de Aix-Marseille III Date: Friday, May 6, 2011, Time: 3:00 p.m., Location: Fine Hall 314     Department Colloquium Topic: TBA Presenter: Balazs Szegedy, University of Toronto Date: Wednesday, May 11, 2011, Time: 4:30 p.m., Location: Fine Hall 314     Topology Seminar Topic: A Heegaard Floer characterization of Borromean knots Presenter: Yi Ni, Caltech Date: Thursday, May 12, 2011, Time: 4:30 p.m., Location: Fine Hall 314 Abstract: If the total rank of the knot Floer homology of a knot is equal to the total rank of the Heegaard Floer (hat) homology of the ambient 3-manifold, we say that the knot has simple knot Floer homology (or the knot is Floer simple). It is known that the unknot is the only Floer simple knot in S3. However, the question of determining all the Floer simple knots in general 3-manifolds is far from being solved. In this talk we will answer this question for the connected sums of S1\times S2: Such knots are essentially the Borromean knots.