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MARCH 2011 |
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| Discrete Mathematics Seminar |
| Topic: |
Tournament heroes |
| Presenter: |
Maria Chudnovsky, Columbia University |
| Date: |
Thursday, March 30, 2011, Time: 2:15 p.m., Location: Fine Hall 314 |
| Abstract: |
The chromatic number of a tournament T is the smallest number of transitive tournaments that partition V(T). Let us say that a tournament S is a hero if for every tournament T not containing S, the chromatic number of T is at most a constant c(S). Recently, in joint work with Eli Berger, Krzysztof Choromanski, Jacob Fox, Martin Loebl, Alex Scott, Paul Seymour and Stephan Thomasse, we proved a theorem that gives a complete description of all heroes. This talk will describe the result, and survey some of the proof ideas. |
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| Department Colloquium |
| Topic: |
Rational points on algebraic varieties |
| Presenter: |
Yuri Tschinkel, NYU |
| Date: |
Wednesday, March 30, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
I will discuss several geometric techniques and constructions that emerged in the study of rational points on higher-dimensional algebraic varieties over global fields |
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| Ergodic Theory and Statistical Mechanics Seminar |
| Topic: |
Bifurcations of solutions of the 2-dimensional Navier-Stokes system |
| Presenter: |
Dong Li, University of Iowa |
| Date: |
Thursday, March 31, 2011, Time: 2:00 p.m., Location: Fine Hall 801 |
| Abstract: |
I will explain recent joint work with Sinai on the bifurcations of solutions to the 2-dimensional Navier-Stokes system |
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| Algebraic Topology Seminar |
| Topic: |
An integral lift of the Gamma-genus |
| Presenter: |
Jack Morava, Johns Hopkins University |
| Date: |
Thursday, March 31, 2011, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
The Hirzebruch genus of a complex-oriented manifold M associated (by Kontsevich) to Euler's Gamma-function has an analytic interpretation as the index of a family of deformations of a Dirac operator, parametrized by the homogeneous space Sp/U; in more homotopy-theoretic terms, it is the homomorphism MU --> MU \smash_{MSp} KO of ring spectra. It also has an interpretation as a kind of equivariant Euler characteristic of the free loopspace of M, suitably polarized. There are further intriguing connections with the theory of asymptotic expansions, involving the values of the zeta function at odd positive integers. |
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| Topology Seminar |
| Topic: |
Pseudo-Anosov maps with small dilatation |
| Presenter: |
Joan Birman, Columbia University |
| Date: |
Thursday, March 31, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Fix an orientable surface $S$. It is known that the set of dilatations of all pseudo-Anosov maps acting on $S$ is a family of real numbers that is bounded below by 1, and has a minimum value $\lambda_{min,S}>1$ which is realized geometrically. We will discuss recent work on the problem of determining $\lambda_{min,S}$ and show how a little-known theorem, the `Coefficient Theorem for Digraphs',can be used to gain insight into this set. The study of small dilatation pA maps appears to be related to the study of small volume fibered hyperbolic 3-manifolds, and an example from 3-manifolds has played a role in understanding the dilatation problem. |
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| APRIL 2011 |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Geometrical variational problems in economics |
| Presenter: |
Robert McCann, University of Toronto |
| Date: |
Friday, April 1, 2011, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory. The solution to this problem when the space of products and of buyers can each be parameterized by a single variable (say quality X, and income Y) garnered Mirrlees (1971) and Spence (1974) their Nobel prizes in 1996 and 2001. The multidimensional version of this question is a largely open problem in the calculus of variations (see Basov's book "Multidimensional Screening".) I plan to describe recent work with A Figalli and Y-H Kim, identifying structural conditions on the value b(X,Y) of product X to buyer Y which reduce this problem to a convex program in a Banach space--- leading to uniqueness and stability results for its solution, confirming robustness of certain economic phenomena observed by Armstrong (1996) such as the desirability for the monopolist to raise prices enough to drive a positive fraction of buyers out of the market, and yielding conjectures about the robustness of other phenomena observed Rochet and Chone (1 998), such as the clumping together of products marketed into subsets of various dimension. The passage to several dimensions relies on ideas from differential geometry / general relativity, optimal transportation, and nonlinear PDE. |
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| Joint Analysis Seminar and PACM Colloquium |
| Topic: |
Product Formulas for Measures and Applications to Analysis |
| Presenter: |
Peter Jones, Yale University |
| Date: |
Monday, April 4, 2011, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: |
We will discuss elementary product formalisms for positive measures. These appeared in analysis for purposes of examining "harmonic measures" related to elliptic equations (work of R. Fefferman, J. Pipher, C. Kenig). We will discuss three topics where product formulas appear: applied projects related to signal processing; SLE; and Geometric measure theory. For the first topic we will explain some work arising in analysis of network failures. For the second topic (SLE) we will show the relations between some models of random measures, and relations to SLE. The third topic (geometric measure theory) will be a discussion of joint work with Marianna Csörnyei. The main point here is how product formulas can detect directionality in sets. The new result concerns Lebesgue measurable sets E of finite measure in the unit cube (in any dimension). The set E can be decomposed into a bounded number of sets with the property that each (sub)set has a nice "tangent cone". This yields strong results on points of non- differentiability for Lipschitz functions. The main technical result needed is a d dimensional, measure theoretic version of (a geometric form of) the Erdös-Szekeres theorem, which holds when d = 2. In what is perhaps a small surprise, certain ideas from random measures can be used effectively in the deterministic setting. |
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| Algebraic Topology Seminar **please note special day and time** |
| Topic: |
Compact aspherical manifolds whose fundamental groups have center |
| Presenter: |
Sylvain Cappell, NYU |
| Date: |
Tuesday, April 5, 2011, Time: 5:00 p.m., Location: Fine Hall 314 |
| Abstract: |
Classical work of Borel had shown that an action of the circle on a manifold with contractible universal cover yields non-trivial center in the manifold's fundamental group. In the early 70's, Conner and Raymond made further deep investigations which led them to conjecture a converse to Borel's result. We construct counter-examples to this conjecture, i.e., we exhibit aspherical manifolds (in all dimensions greater than or equal to 6) which have non-trivial center in their fundamental groups but no circle actions (and hence no compact Lie group actions). The constructions involve synthesizing rather disparate methods of geometric topology, geometric group theory and hyperbolic geometry. (This is joint work with Shmuel Weinberger and Min Yan.) |
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| PACM Colloquium - Distinguished Lecture Seminar **Please note special date** |
| Topic: |
Stirring Tails of Evolution |
| Presenter: |
Ray Goldstein, Cambridge University |
| Date: |
Wednesday, April 6, 2011, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: |
One of the most fundamental issues in biology is the nature of evolutionary transitions from single cell organisms to multicellular ones. Not surprisingly for microscopic life in a fluid environment, many of the processes involved are related to transport and locomotion, for efficient exchange of chemical species with the environment is one of the most basic features of life. This is particularly so in the case of flagellated eukaryotes such as green algae, whose members serve as model organisms for the study of transitions to multicellularity. In this talk I will focus on recent experimental and theoretical studies of the stochastic nonlinear dynamics of these flagella, whose coordinated beating leads to graceful locomotion but also to fluid flows that can out-compete diffusion. A synthesis of high-speed imaging, micromanipulation, and three-dimensional tracking has quantified the underlying stochastic dynamics of flagellar beating, allowed for tests of the hydrodynamic origins of flagellar synchronization, and revealed a eukaryotic equivalent of the run-and-tumble locomotion of peritrichously flagellated bacteria. Challenging problems in applied mathematics, fluid dynamics, and biological physics that arise from these findings will be highlighted. |
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| Department Colloquium |
| Topic: |
Endoscopic transfer of the Bernstein center |
| Presenter: |
Thomas Haines, University of Maryland |
| Date: |
Wednesday, April 6, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
The Langlands-Shelstad theory of endoscopy plays a central role in the study of Shimura varieties and the Arthur-Selberg trace formula. The fundamental lemma and a deep consequence, endoscopic transfer, have now been established in works of Ngo, Waldspurger, and Hales. Both of these statements are identities involving orbital integrals of a function $f$ on a $p$-adic group $G$ and those of certain "transfer" functions $f^H$ on related groups $H$, called endoscopic. This talk will give background on the fundamental lemma, some of its variants, and the roles they played. Then I will describe a conjectural construction of a large class of matching functions in the Bernstein centers of $G$ and $H$. This conjecture has been verified in some encouraging cases. The functions being transferred arise in connection with Shimura varieties with bad reduction, and some applications to that subject will also be briefly discussed. |
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| Algebraic Topology Seminar |
| Topic: |
Moment-angle complexes from simplicial posets |
| Presenter: |
Taras Panov, Moscow State University |
| Date: |
Thursday, April 7, 2011, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
The construction of moment-angle complexes may be extended from simplicial complexes to simplicial posets. As a result, a certain T^m-space Z_S is associated to an arbitrary simplicial poset S on m vertices. Face rings Z[S]$ of simplicial posets generalise those of simplicial complexes, but have much more complicated algebraic structure. These rings Z[S] may be studied by topological methods. The space Z_S has many important topological properties of the original moment-angle complex Z_K associated to a simplicial complex K. In particular, the integral cohomology algebra of Z_S is isomorphic to the Tor-algebra of the face ring Z[S]. This leads directly to a generalisation of Hochster's theorem, expressing the algebraic Betti numbers of the ring Z[S] in terms of the homology of full subposets in S. Finally, the total amount of homology of Z_S may be estimated from below, which settles Halperin's toral rank conjecture for the moment-angle complexes Z_S |
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| Topology Seminar |
| Topic: |
Geometric structures on moment-angle manifolds |
| Presenter: |
Taras Panov, Moscow State University |
| Date: |
Thursday, April 7, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
| Abstract: |
Moment-angle complexes are spaces acted on by a torus and parametrised by finite simplicial complexes. They are central objects in toric topology, and currently are gaining much interest in the homotopy theory. Due the their combinatorial origins, moment-angle complexes also find applications in combinatorial geometry and commutative algebra. Moment-angle complexes corresponding to simplicial subdivision of spheres are topological manifolds, and those corresponding to simplicial polytopes admit smooth realisations as intersection of real quadrics in C^m. After an introductory part describing the general properties of moment-angle complexes we shall concentrate on the complex-analytic and Lagrangian aspects of the theory. We show that the moment-angle manifolds corresponding to complete simplicial fans admit nonKaehler complex-analytic structures. This generalises the known construction of complex-analytic structures on polytopal moment-angle manifolds, coming from identifying them as LVM-manifolds. We proceed by describing the Dolbeault cohomology and certain Hodge numbers of moment-angle manifolds by applying the Borel spectral sequence to holomorphic principal bundles over toric varieties. A new wide family of minimal Lagrangian submanifolds N in C^m or CP^m can be constructed from intersections of real quadrics. These submanifolds have the following topological properties: every N embeds in the corresponding moment-angle manifold Z, and every N is the total space of two different fibrations, one over the torus T^{m-n} with fibre a real moment-angle manifold R, and another over a small cover with fibre a torus. These properties are used to produce new examples of Lagrangian submanifolds with quite complicated topology. Different parts of this talk are based on joint works with Victor Buchstaber, Andrei Mironov and Yuri Ustinovsky. |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
Minimal fillings and boundary rigidity - a survey |
| Presenter: |
Dmitri Burago, Penn State |
| Date: |
Friday, April 8, 2011, Time: 3:00 p.m., Location: Fine Hall 314 |
| Abstract: |
A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only. The goal is to show that certain classes of metrics are boundary rigid (and, ideally, to suggest a procedure for recovering the metric). To visualize that, imagine that one wants to find out what the Earth is made of. More generally, one wants to find out what is inside a solid body made of different materials (in other words, properties of the medium change from point to point). The speed of sound depends on the material. One can "tap" at some points of the surface of the body and "listen when the sound gets to other points". The question is if this information is enough to determine what is inside. This problem has been extensively studied from PDE viewpoint: the distance between boundary points can be interpreted as a "travel time" for a solution of the wave equation. Hence this becomes a classic Inverse Problem when we have some information about solutions of a certain PDE and want to recover its coefficients. For instance such problems naturally arise in geophysics (when we want to find out what is inside the Earth by sending sound waves), medical imaging etc. In a joint project with S. Ivanov we suggest an alternative geometric approach to this problem. In our earlier work, using this approach we were able to show boundary rigidity for metrics close to flat ones (in all dimensions), thus giving the first class of boundary rigid metrics of non–constant curvature beyond two dimensions. We were now able to extend this result to include metrics close to a hyperbolic one. The approach is grew up from another long-term project of studying surface area functionals in normed spaces, which we have been working on it for more than ten years. There are a number of related issues regarding area-minimizing surfaces in Riemannian manifold. The talk gives a non-technical survey of ideas involved. It assumes no background in inverse problems and is supposed to be accessible to a general math audience (in other words, we will sweep technical details under the carpet). |
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| PACM Colloquium |
| Topic: |
From (basic) image denoising to surface evolution |
| Presenter: |
Antonin Chambolle, CMAP -- Ecole Polytechnique |
| Date: |
Monday, April 11, 2011, Time: 4:00 p.m., Location: Fine Hall 214 |
| Abstract: |
It is relatively easy to make a connection between the implicit time-discrete approaches for the mean curvature flow and the "Rudin-Osher-Fatemi" total variation based approach for image denoising. This connection has interesting consequences, allowing to build explicit solutions for the flow of the total variation or study regularity issues, up to showing the existence of the crystalline curvature flow of convex sets or building up efficient algorithms. The talk will explain the relationship between all these problems. (Joint works with V. Caselles, M. Novaga, J. Darbon, T. Pock, D. Cremers). |
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| Algebraic Geometry Seminar |
| Topic: |
TBA |
| Presenter: |
Mircea Mustaţă, U. of Michigan |
| Date: |
Tuesday, April 12, 2011, Time: 4:30 p.m., Location: Fine Hall 322 |
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| Algebraic Topology Seminar **Please note special day and time** |
| Topic: |
The Geometry of Music |
| Presenter: |
Dmitri Tymoczko, Princeton University, Department of Music |
| Date: |
Tuesday, April 12, 2011, Time: 5:00 p.m., Location: Fine Hall 314 |
| Abstract: |
In my talk, I explain how to translate basic concepts of music theory into the language of contemporary topology and geometry. Musicians commonly abstract away from five kinds of musical information -- including the order, octave, and specific pitch level of groups of notes. This process produces a family of quotient spaces or orbifolds: for example, two-note chords live on a Mobius strip, while three-note chord-types live on a cone. These spaces provide a general geometrical framework for understanding and interpreting music. Related constructions also appear naturally in other applied-math contexts, for instance in economics. |
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| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
Steve Zelditch, Northwestern University |
| Date: |
Wednesday, April 13, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
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| 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 |
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| 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. |
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| Differential Geometry and Geometric Analysis Seminar |
| Topic: |
TBA |
| Presenter: |
Richard Bamler, Princeton University |
| Date: |
Friday, April 15, 2011, Time: 3:00 p.m., Location: Fine Hall 314 |
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| 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. |
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| 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 |
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| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
Alexander Bufetov, Rice University |
| Date: |
Wednesday, April 20, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
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| 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 |
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| 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 |
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| 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 |
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| 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. |
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| 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). |
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| Algebraic Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Victor Buchstaber, Steklov Mathematical Institute, Russian Academy of Sciences |
| Date: |
Thursday, April 28, 2011, Time: 3:00 p.m., Location: Fine Hall 314 |
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| Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Peter Ozsvath, MIT |
| Date: |
Thursday, April 28, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
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| 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 |
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| MAY 2011 |
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| 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 |
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| Topology Seminar |
| Topic: |
TBA |
| Presenter: |
Liam Watson, UCLA |
| Date: |
Thursday, May 5, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
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| 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 |
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| Department Colloquium |
| Topic: |
TBA |
| Presenter: |
Balazs Szegedy, University of Toronto |
| Date: |
Wednesday, May 11, 2011, Time: 4:30 p.m., Location: Fine Hall 314 |
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| 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. |
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