Upcoming Seminars & Events

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October 3, 2016
3:00pm - 4:30pm
Global Existence, Blowup and Scattering for large data Supercritical and other wave equations

PLEASE NOTE NEW START TIME OF 3:00.   I present a new approach to classify the asymptotic behavior of certain classes of wave equations, supercritical and others, with large initial data. In some cases, as for Nirenberg type equations, a fairly complete classification of the solutions (finite time blowup or global existence and scattering) is proved. New results are obtained for the well known monomials wave equations in the sub, critical and super critical cases. This approach, developed jointly with M. Beceanu, is based on a new decomposition into incoming and outgoing waves for the wave equation, and the positivity of  the fundamental solution of the wave equation in three dimensions. 

Speaker: Avy Soffer , Rutgers University
Location:
Fine Hall 314
October 3, 2016
4:00pm - 5:00pm
Network clustering with higher order structures

Spectral clustering is a well-known way to partition a graph or network into clusters or communities with provable guarantees on the quality of the clusters. This guarantee is known as the Cheeger inequality and it holds for undirected graphs. We'll discuss a new generalization of the Cheeger inequality to higher-order structures in networks including network motifs. This is easy to implement and seamlessly generalizes spectral clustering to directed, signed, and many other types of complex networks. In particular, our generalization allow us to re-use the large history of existing ideas in spectral clustering including local methods, overlapping methods, and relationships with kernel k-means. We will illustrate the types of clusters or communities found by our new method in biological, neuroscience, ecological, transportation, and social networks.
This is joint work with Austin Benson and Jure Leskovec at Stanford.

Speaker: David Gleich, Purdue University
Location:
Fine Hall 214
October 4, 2016
1:30pm - 2:30pm
Packaging the construction of Kuranishi structure on the moduli space of pseudo-holomorphic curve

This is a part of my joint work with Oh-Ohta-Ono and is a part of project to rewrite the whole story of virtual fundamental chain in a way easier to use. In general we can construct virtual fundamental chain on (basically all) the moduli space of pseudo-holomorphic curve. It depends on the choices. In this talk I want to provide a statement to clarify which is the data we need to start with and in which sense the resulting structure is well defined. A purpose of writing such statement is then it can be a black box and can be used without looking the proof. Also it is useful to see some properties of it such as its relation to the (target space) group action or compatibility with forgetful map.

Speaker: Kenji Fukaya , Stony Brook University
Location:
IAS Room S-101
October 4, 2016
3:00pm - 4:00pm
Projective Dehn twist via Lagrangian cobordism

In this talk, I would like to explain my joint work with Weiwei Wu about understanding projective Dehn twist using Lagrangian cobordism.

 

Speaker: Cheuk Yu Mak , IAS
Location:
IAS Room S-101
October 4, 2016
4:30pm - 5:30pm
Elliptic Calabi-Yau 3-folds, Jacobi forms, and derived categories

By physical considerations, Huang, Katz and Klemm conjectured in 2015 that topological string partition functions for elliptic Calabi-Yau 3-folds are governed by certain Jacobi forms. This gives strong structure results for curve counting invariants of elliptic CY 3-folds. I will explain a mathematical approach to prove (part of) the HKK Conjecture. Our method is to construct an involution on the derived category and use wall-crossing techniques.  The talk is based on joint work with Georg Oberdieck.

Speaker: Junliang Shen, ETH Zurich
Location:
Fine Hall 322
October 5, 2016
3:00pm - 4:00pm
TBD - Christos Mantoulidis
Speaker: Christos Mantoulidis, Stanford University
Location:
Fine Hall 314
October 5, 2016
4:30pm - 5:30pm
Mean field evolution of fermonic systems

Please note different day (Wednesday) and location (Jadwin 303).   In this talk I will discuss the dynamics of interacting fermionic systems in the mean field regime. Compared to the bosonic case, fermionic mean field scaling is naturally coupled with a semiclassical scaling, making the analysis more involved. As the number of particles grows, the quantum evolution of the system is expected to be effectively described by Hartree-Fock theory. The next degree of approximation is provided by a classical effective dynamics, corresponding to the Vlasov equation.  I will consider initial data which are close to quasi-free states, at zero (pure states) or at positive temperature (mixed states), with an appropriate semiclassical structure. Under mild regularity assumptions on the interaction potential, I will show that the time evolution of such initial data stays close to a quasi-free state, with reduced one-particle density  matrix given by the solution of the time-dependent Hartree-Fock equation. The result can be extended to Coulomb interactions, under the assumption that the solution of the time-dependent Hartree-Fock equation preserves the semiclassical structure of the initial data. If time permits, the convergence from the time-dependent Hartree-Fock equation to the Vlasov equation will also be discussed. The results hold for all semiclassical times, and give effective bounds on the rate of convergence towards the effective dynamics as the number of particles goes to infinity.

Speaker: Marcello Porta, Zurich University
Location:
Jadwin 303
October 6, 2016
12:30pm - 1:30pm
TBD - Jack Sempliner
Speaker: Jack Sempliner , Princeton University
Location:
Fine Hall 110
October 6, 2016
2:00pm - 3:30pm
TBA - Tetiana Shcherbyna
Speaker: Tetiana Shcherbyna , Princeton University
Location:
Jadwin Hall 111
October 6, 2016
2:30pm - 3:30pm
From Integrability to Medical Imaging and to the Asymtotics of the Riemann Zeta Function

It is often realized that this technique can actually be used for the solution of a plethora of other problems,and thus it becomes a mathematical method.In this lecture, a review will be presented of how a concrete problem in the area of integrability led to the development of a new method in mathematical physics for analyzing boundary value problems for linear and for integrable nonlinear PDEs,called  the "Unified Transform". This method has been acclaimed by the late Israel Gelfand as "the most important development on the exact analysis of PDE since the work of the classics in the 18th century." Remarkable connections with the development of several effective algorithms for Medical Imaging,and with the Riemann hypothesis will also be reviewed.

Speaker: Thanasis Fokas, University of Cambridge
Location:
Fine Hall 224
October 6, 2016
3:00pm - 4:30pm
Complexity-separating graph classes for vertex, edge and total colouring

Given a class A of graphs and a decision problem X belonging to NP, we say that a full complexity dichotomy of A was obtained if one describes a partition of A into subclasses such that X is classified as polynomial or NP-complete when restricted to each subclass. The concept of full complexity dichotomy is particularly interesting for the investigation of NP-complete problems: as we partition a class A into NP-complete subclasses and polynomial subclasses, it becomes clearer why the problem is NP-complete in A. The class C of graphs that do not contain a cycle with a unique chord was studied by Trotignon and Vušković who proved a structure theorem which led to solving the vertex-colouring problem in polynomial time. We apply the structure theorem to study the complexity of edge-colouring and total-colouring, and show that even for graph classes with strong structure and powerful decompositions, the edge-colouring problem may be difficult. We discuss several surprising complexity dichotomies found in subclasses of C, and the concepts of separating class and of separating problems.

Speaker: Celina de Figueiredo, Rio de Janeiro
Location:
Fine Hall 224
October 6, 2016
4:30pm - 6:00pm
Generated Jacobian equations and regularity: optimal transport, geometric optics, and beyond

PLEASE NOTE SPECIAL DAY, TIME AND LOCATION.  Equations of Monge-Amp{\`e}re type arise in numerous contexts, and solutions often exhibit very subtle properties; due to the highly nonlinear nature of the equation, and its degenerate ellipticity. Motivated by an example from geometric optics I will talk about the class of Generated Jacobian Equations, recently introduced by Trudinger. This class includes optimal transport, the Minkowski problem, and the classical Monge-Amp{\`e}re equation. I will present a new regularity result for weak solutions of these equations, which is new even in the case of equations arising from near-field problems in geometric optics. This talk is based on joint works with Nestor Guillen.

Speaker: Jun Kitagawa, Michigan State University
Location:
Fine Hall 1001
October 6, 2016
4:30pm - 5:30pm
TBD - Peter Lambert-Cole
Speaker: Peter Lambert-Cole , Indiana University
Location:
Fine Hall 314
October 6, 2016
4:30pm - 5:30pm
A Lagrangian Fluctuation-Dissipation Relation for Scalar Turbulence

A common approach to calculate the solution of a scalar advection-diffusion equation is by a Feynman-Kac representation which averages over stochastic Lagrangian trajectories going backward in time to the initial conditions and boundary data. The trajectories are obtained by solving SDE's with the advecting velocity as drift and a backward Itō term representing the scalar diffusivity.  In this framework, we present an exact formula for scalar dissipation in terms of the variance of the scalar values acquired along each random trajectory. As an important application, we study the connection between anomalous scalar dissipation in turbulent flows for large Reynolds and Péclet numbers and the spontaneous stochasticity of the Lagrangian particle trajectories. The latter property corresponds to the Lagrangian trajectories remaining random in the limit Re,Pe→∞, when the backward Itō term formally vanishes but the advecting velocity field becomes non-Lipschitz. For flows on domains without boundaries (e.g. tori, spheres) and for wall-bounded flows with no-flux Neumann conditions for the scalar, we prove that spontaneous stochasticity is necessary and sufficient for anomalous scalar dissipation. The fluctuation-dissipation relation provides a Lagrangian representation of scalar dissipation also in turbulent flows where present experiments suggest that dissipation is tending to zero as Re,Pe→∞. We discuss an illustrative example of Rayleigh-Bénard convection with imposed heat-flux at the top and bottom plates. Our formula here shows that the scalar dissipation is given by the variance of the local time densities of the stochastic particles at the heated boundaries. The ``ultimate regime'' of turbulent convection predicted by Kraichnan-Spiegel occurs when the near-wall particle densities are mixed to their asymptotic uniform values in a large-scale turnover time. The current observations of vanishing scalar dissipation require that fluid particles be trapped at the wall and remain unmixed for many, many large-scale turnover times.  This talk presents joint work with Gregory Eyink.  

Speaker: Theodore Drivas, Johns Hopkins University
Location:
Fine Hall 322
October 6, 2016
4:30pm - 5:30pm
The Unpolarized Shafarevich Conjecture for K3 Surfaces

Shafarevich made the following conjecture for higher genus curves: the set of isomorphism classes of genus g curves over K (a number field) with good reduction outside S (a fixed finite set of primes) is finite.  Faltings proved this and the analogous conjecture for abelian varieties of given degree. Zarhin proved this finiteness across all degrees. Using Faltings’ theorem, Andre proved the finiteness of K3 surfaces (over K, S) of a given degree. We prove the analog of Zarhin’s theorem, i.e. there are still finitely many K3 surfaces across all degrees.

Speaker: Yiwei She , Columbia University
Location:
Fine Hall 214
October 10, 2016
3:00pm - 4:30pm
A proof of Onsager's Conjecture
Speaker: Phillip Isett , MIT
Location:
Fine Hall 314
October 10, 2016
4:00pm - 5:00pm
TBD - Eitan Bachmat
Speaker: Eitan Bachmat, Ben-Gurion University (Israel)
Location:
Fine Hall 214
October 11, 2016
1:30pm - 2:30pm
TBD - Luis Diogo
Speaker: Luis Diogo , Columbia University
Location:
IAS Room S-101
October 11, 2016
3:00pm - 4:00pm
TBA - Joshua Sabloff
Speaker: Joshua Sabloff , Haverford College
Location:
IAS Room S-101
October 11, 2016
4:30pm - 5:30pm
Compactification of strata of abelian differentials

Many questions about Riemann surfaces are related to study their flat structures induced from abelian differentials. Loci of abelian differentials with prescribed type of zeros form a natural stratification. The geometry of these strata has interesting properties and applications to moduli of complex curves. In this talk we focus on the question of compactifying the strata of abelian differentials from the viewpoints of algebraic geometry, complex analytic geometry, and flat geometry. In particular, we provide a complete description of the strata compactification over the Deligne-Mumford moduli space of pointed stable curves. The upshot is a global residue condition compatible with a full order on the dual graph of a stable curve. This is joint work with Bainbridge, Gendron, Grushevsky and Moeller, based on arXiv:1604.08834.

Speaker: Dawei Chen , Boston College
Location:
Fine Hall 322

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