Upcoming Seminars & Events

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February 27, 2015
1:30pm - 2:30pm
Disc filling and connected sum

In my talk I will report on recent work with Hansjörg Geiges about a strong connection between the topology of a contact manifold and the existence of contractible periodic Reeb orbits. Namely, if the contact manifold appears as non-trivial contact connected sum and has non-trivial fundamental group or torsion-free homology, then the existence is ensured. This generalizes a result of Helmut Hofer in dimension three.

Speaker: Kai Zehmisch, Universität Münster
Location:
IAS Room S-101
March 2, 2015
4:30pm - 5:30pm
Manifolds on the verge of a regularity breakdown

Invariant manifolds are landmarks that organize the long term behavior of dynamical systems with complicated  trajectories.  From the point of view of applications, it is interesting to know that they persist under perturbations or that one can validate the the numerical calculations finding them. There are two main theorems in this direction: Normal hyperbolicity and KAM theory. We will present numerical explorations and rigorous results on what happens on the boundary of validity of these theorems. There are some surprising regularities and scaling relations, some of which can be proved rigorously. This is joint work with many people including R. Calleja, A. Haro, T. Blass.

Speaker: Rafael de la Llave, Georgia Tech
Location:
Fine Hall 214
March 2, 2015
4:30pm - 5:30pm
From Knots to Clusters: the Path via Sheaves

Please note special day, time and location.  Given a Legendrian knot, I will construct a category of constructible sheaves, invariant under Legendrian isotopy up to equivalence.  The constructble sheaves live on the front plane and are stratified by the front diagram of the Legendrian knot.  I will then prove that the "rank-one" subcategory is equivalent to a category of augmentations of the Chekanov-Eliashberg differential graded algebra. The proof boils down to local calculations, which are easy to describe. Next I will apply a similar construction to alternating strand diagrams (such as those generated by bipartite graphs) on surfaces and show how the mooduli spaces of rank-one objects are cluster varieties. The relation to the previous work is that an exact Lagrangian filling defines and augmentation, and the cluster chart near an augmentation is given by the space of line bundles on the filling, recovering the integrable system that Goncharov-Kenyon associate to a bipartite graph. The first part is based on joint works with Vivek Shende and David Treumann (arXiv:1402.0490) and with Lenhard Ng, Dan Rutherford, Vivek Shende, and Steven Sivek (arXiv:1502.04939). The second part is based on work in progress with Vivek Shende, David Treumann and Harold Williams.

Speaker: Eric Zaslow, Northwestern University
Location:
Fine Hall 110
March 3, 2015
4:30pm - 5:30pm
Airy diffusion and N^{1/3} fluctuations in the 2D Ising model

For the two-dimensional Ising model at low temperatures consider a floating droplet of the (+) phase floating in the sea of (-) phase, pressed against a horizontal wall within a box of linear size N. I will explain that the fluctuations of the boundary of the droplet near the contact with the wall are of the order of N^{1/3}. When scaled by N^{1/3} vertically and by N^{2/3} horizontally, the limiting behavior of the boundary as N goes to infinity is given by the Airy diffusion process. This diffusion process has appeared earlier in a paper by Ferrari and Spohn, where the brownian motion above the parabolic barrier is considered. Work in progress with D. Ioffe and Y. Velenik.

Speaker: Senya Shlosman, CPT, Univ. Marseille, Luminy
Location:
Jadwin Hall 343
March 4, 2015
1:30pm - 2:30pm
Counting genus one partitions and permutations

The study of counting rooted maps was initiated by Tutte, who was motivated by the four color conjecture, and the study of hypermaps grew out of this initiative. Hypermaps are pairs of permutations that can be topologically represented by labeled maps, the genus of the underlying surface may be expressed purely algebraically in terms of these permutations. Looking at the special case of rooted hypermonopoles leads to the definition of the genus of a permutation. In this talk we prove a conjecture of Martha Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Our method also provides another way to count rooted hypermonopoles of genus one, and puts the spotlight on a class of genus one permutations that is invariant under an obvious extension of the Kreweras duality map to genus one permutations. This is joint work with Robert Cori. 

Speaker: Gabor Hetyei, University of North Carolina at Charlotte
Location:
Fine Hall 214
March 4, 2015
3:00pm - 4:00pm
Diffusions with Rough Drifts and Stochastic Symplectic Maps

This is a joint seminar with Analysis of Fluids and Related Topics.  According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a $d$-dimensional diffusion with a drift in $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $d/r+2/q<1$. As an application one show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function $H$ satisfies $\nabla H\in L^{r,q}$ with $d/r+2/q<1$. As our second application we derive a Constantin-Iyer type circulation formula for certain weak solutions of Navier-Stokes equation.

 

Speaker: Fraydoun Rezakhanlou, UC Berkeley
Location:
Fine Hall 322
March 4, 2015
3:00pm - 4:00pm
Diffusions with Rough Drifts and Stochastic Symplectic Maps

This is a joint seminar with the Probability Seminar. Please note special day and time.  According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a $d$-dimensional diffusion with a drift in $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $d/r+2/q<1$. As an application one show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function $H$ satisfies $\nabla H\in L^{r,q}$ with $d/r+2/q<1$. As our second application we derive a Constantin-Iyer type circulation formula for certain weak solutions of Navier-Stokes equation.

Speaker: Fraydoun Rezakhanlou, UC Berkeley
Location:
Fine Hall 322
March 4, 2015
4:30pm - 5:30pm
Generalization of Selberg’s 3/16 theorem for finitely generated subgroups of SL(2,Z)

A celebrated theorem of Selberg in 1965 states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We will discuss how Selberg’s theorem can be generalized to finitely generated subgroups of SL(2,Z) which are of infinite index.  This talk is based on joint work with Dale Winter

Speaker: Hee Oh, Yale University
Location:
Fine Hall 314
March 5, 2015
12:30pm - 1:30pm
Quantum knot invariants and the volume conjecture

I'll sketch some of the basics of quantum topology, including how to get knot invariants from a ribbon category and how to construct non-cocommutative Hopf algebras.  Next we'll turn to hyperbolic geometry and discuss how hyperbolic structures on knot complements arise.  Finally, I'll discuss how these two topics are linked via the volume conjecture, and give some directions of current work.

Speaker: Daniel Vitek , Princeton University
Location:
Fine Hall 314
March 5, 2015
3:00pm - 4:00pm
Spectral theory of simplicial complexes

While spectral graph theory is a well established and fortuitous field, its higher dimensional analogue is still in its infancy. I will explain some approaches to the definition of a spectral theory for simplicial complexes, their applications to combinatorial questions, and examples such as random and Ramanujan complexes.

Speaker: Ori Parzanchevski, IAS
Location:
Fine Hall 224
March 5, 2015
3:00pm - 4:00pm
TBA - Denham
Speaker: Graham Denham, University of Western Ontario
Location:
Fine Hall 314
March 5, 2015
4:30pm - 5:30pm
Faltings heights of CM abelian varieties

I'll describe ongoing joint work with F. Andreatta, E. Goren, and K. Madapusi Pera towards Colmez's conjecture expressing Faltings heights of CM abelian varieties in terms of values of Artin L-functions.

Speaker: Benjamin Howard, Boston College
Location:
IAS Room S-101
March 5, 2015
4:30pm - 5:30pm
Functoriality in Khovanov-Floer theories

There has been a lot of interest in recent years in connections between Khovanov homology and Floer theory. These connections usually come in the form of spectral sequences, with E_2 page the Khovanov homology of a link and converging to the relevant Floer theory. Important examples include Ozsvath-Szabo’s spectral sequence in Heegaard Floer homology and Kronheimer-Mrowka’s spectral sequence in singular instanton Floer homology. In particular, the latter was used to prove that Khovanov homology detects the unknot. A natural question is whether these constructions are functorial? That is, are the intermediate pages of these spectral sequences link invariants, and do link cobordisms induce well-defined maps on these pages? We answer these questions in the affirmative, as part of a much more general framework. At the end, we will describe how this framework might be used to define a host of new knot and cobordism invariants. This is joint work with Matt Hedden and Andrew Lobb.

Speaker: John Baldwin, Boston College
Location:
Fine Hall 314
March 5, 2015
4:30pm - 5:30pm
New regularity estimates for compressible Navier-Stokes

We present a new approach offering explicit regularity estimates for solutions to transport equations, and in particular for the density of the solutions to the compressible Navier-Stokes equations. This new method removes several constraints of the classical Lions-Feireisl theory. It thus allows us to treat a larger class of models with *non* *monotone* *pressure* terms, or *anisotropic* *viscosity* for instance, leading to many applications from geophysical flows (eddy viscosity) to solar events (virial pressure law) and some biological situations.

Speaker: PE Jabin, University of Maryland
Location:
Fine Hall 322
March 5, 2015
5:30pm - 6:30pm
TBA - Angel Castro

Please note special time.

Speaker: Angel Castro, ICMAT, Madrid
Location:
Fine Hall 322
March 6, 2015
9:30am - 9:30am
Columbia-Princeton Probability Day 2015

Speakers

  • Davar Khoshnevisan (Utah)
  • Fraydoun Rezakhanlou (Berkeley)
  • Prasad Tetali (Georgia Tech)
  • Balint Virag (Toronto)

Junior Speakers

  • Alex Drewitz (Columbia)
  • Leonid Petrov (Virginia)

PLEASE CLICK HERE FOR CONFERENCE LINK.  

Contact Mykhaylo Shkolnikov for further details.

Speaker: ,
Location:
TBD
March 9, 2015
3:15pm - 4:30pm
The entropy production problem in kinetic theory

The Boltzmann equation is the central equation of kinetic theory, and it describes the evolution of the phase space density for a dilute gas. Boltzmann's famous H-theorem says that the entropy is strictly monotone increasing for solutions of this equation unless the solution is in equilibrium.  For many years discussion of this result centered on understanding how  such irreversibility could arise from reversible Newtonian dynamics, and important questions  in this direction remain open. However, more recent work has sought a quantitative version of the H-theorem than can be used to quantify the rate of approach to equilibrium. The first results in this direction were obtained by myself and C. Carvalho, and were further developed and sharpened by Toscani and Villani, and they figure in the work for which Villani won the Fields medal. The problem of relating entropy to entropy production has been found to be applicable to many problems beside the Boltzmann equation, and has suggested a number of interesting functional inequalities that are the subject of recent works,
and it has also suggested several open problems. These will be explained here, assuming, of course, no prior knowledge of kinetic theory. 

Speaker: Eric Carlen, Rutgers University
Location:
Fine Hall 314
March 9, 2015
4:30pm - 5:30pm
Computational imaging for phase, 3D and super-resolution imaging

Computational imaging involves the joint design of optical systems and post-processing algorithms, such that computation replaces optical elements, enabling simple experimental setups. This talk will describe new optical microscopes that employ simple experimental architectures and efficient nonlinear inverse algorithms to achieve high-resolution 3D and phase images. By leveraging recent advances in computational illumination, we achieve brightfield, darkfield and phase contrast images simultaneously, with extension to 3D and gigapixel phase imaging. We discuss unique challenges for large-scale real-time imaging of biological samples in vitro and in vivo.  Bio: Laura Waller is an Assistant Professor at UC Berkeley in the Department of Electrical Engineering and Computer Sciences (EECS), with affiliations in Bioengineering and Applied Sciences & Technology. She was a Postdoctoral Research Associate in Electrical Engineering and Lecturer of Physics at Princeton University from 2010-2012 and received B.S., M.Eng., and Ph.D. degrees in EECS from the Massachusetts Institute of Technology in 2004, 2005, and 2010, respectively. She is a Moore Foundation Data-Driven Investigator, Bakar fellow , NSF CAREER awardee and Packard Fellow.

Speaker: Laura Waller, UC-Berkeley - Elec. Eng. & Computer Science
Location:
Fine Hall 214
March 10, 2015
4:30pm - 5:30pm
The Monge ampere and related Hessian equations: Some non local versions.

***Please note this week's colloquium will be on Tuesday in room 214.

An important family of equations in geometry and other applications are those involving symmetric functions of the eigenvalues of the Hessian.
We will describe some possible non local versions, and their properties:
existence regularity, etc

Speaker: Luis Caffarelli, University of Texas, Austin
Location:
Fine Hall 214
March 12, 2015
2:00pm - 3:30pm
Distribution Free Malliavin Calculus

The theory and applications of Malliavin calculus are well developed for Gaussian and Poisson processes. In this talk I will discuss an extension Malliavin calculus to random fields generated by a sequence $\Xi=( \xi_{1},\xi_{2},...) $ of arbitrary square integrable and uncorrelated random variables.  The distribution functions $Pr( \xi_{i}<x) =F^{i}(x)$ will be assumed to be given but the type of each distribution will not be specified.  The above setting constitute the so called "distribution free" paradigm. As the title suggests, our task is to develop a version of Malliavin calculus in the distribution free setting.  Applications of the distribution free calculus to stochastic ODEs and PDEs will be presented. 

Speaker: Boris Razovsky, Brown University
Location:
Fine Hall 601

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