Upcoming Seminars & Events

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May 2, 2016
11:00am - 12:00pm
A message passing algorithm for cryo-EM and synchronization problems

In synchronization tasks, a collection of entities have latent 'alignments' drawn from some symmetry group, and the task is to recover the relative alignments based on very noisy pairwise observations. A central example is the recovery of molecule rotations in cryo-electron microscopy. We present a new algorithm following the framework of approximate message passing, which statistical physics suggests may yield the optimal reconstruction. Our approach leverages the representation theory of compact groups to give a unified approach for all such problems. The algorithm is efficient, resembling power iteration for PCA -- but after each matrix-vector product, we add an Onsager correction term originating in statistical physics, and then perform a nonlinear transformation that combines data from different 'frequency channels' -- that is, from different irreducible representations of the group. By contrast, many previous algorithms use only one Fourier component of the observations (a notable exception being the Non-Unique Games semidefinite programming approach). The algorithm has a natural derivation from belief propagation, and an interpretation that alternates between belief distributions and an additive 'confidence' representation. We will present empirical results of its strength over many groups and noise models, and discuss theoretical questions and results about the quality of its recovery.
Joint work with Afonso Bandeira and Alex Wein.

Speaker: William Perry, Massachusetts Institute of Technology
Location:
Fine Hall 214
May 2, 2016
3:00pm - 4:30pm
Lecture Series on “Arithmetic in Geometry” - #4

In the next two talks, I will discuss another well-known problem which was very nicely formulated by Mark Kac as "Can one hear the shape of a drum?", and its solution, for arithmetic quotients of symmetric spaces, obtained in a joint work (in Publ Math IHES) with Andrei Rapinchuk. For the solution, we introduced a notion of "weak commensurability" of arithmetic, and more general Zariski-dense, subgroups and derive very strong consequences of weak commensurability.

 

Speaker: Professor Gopal Prasad , University of Michigan
Location:
Fine Hall 322
May 2, 2016
3:15pm - 4:30pm
Lipschitz Metrics for Nonlinear Wave Equations

The talk is concerned with some classes of nonlinear wave equations: of first order, such as the Camassa-Holm equation, or of second order, as the variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$. In both cases, it is known that the equations determine a unique flow of conservative solutions within the natural ``energy" space $H^1(\mathbb{R})$. However, this flow is not continuous w.r.t.~the $H^1$ distance. Local well-posedness is usually recovered only on spaces with higher regularity. Our goal is to construct a new metric, which renders this flow uniformly Lipschitz continuous on bounded subsets of $H^1$. For this purpose, $H^1$ is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time.  To complete the construction, one needs an additional argument showing that the family of piecewise smooth solutions is dense. This generic regularity property can be proved using a variable transformation that reduces the equations to a semilinear system, followed by an application of Thom's transversality theorem.

Speaker: Alberto Bressan , Penn State
Location:
Fine Hall 110
May 2, 2016
4:00pm - 5:00pm
How Robust are Reconstruction Thresholds for Community Detection?

The stochastic block model is a model for community detection in graphs: n nodes are partitioned into two hidden communities and we observe a random graph where within-community edges occur with probability a/n and between-community edges occur with probability b/n (for some constants a > b). The goal is to observe this graph and (approximately) recover the hidden community structure. This problem is known to exhibit sharp threshold behavior: it is information-theoretically possible to recover the communities (better than random guessing, in the limit of large n) precisely when (a-b)^2 > 2(a+b). In order to investigate robustness to unknown noise distributions, we consider the 'semirandom' stochastic block model in which we first generate a random graph as above, but then an adversary gets to make seemingly-helpful changes (adding edges within communities and removing edges between them). We show that surprisingly, this 'helpful' adversary can actually shift the threshold, making the problem strictly harder. This challenges the 'robustness' of algorithms that are able to achieve the threshold. On the flip side, we also give an algorithm based on semidefinite programming which succeeds against the semirandom model in a regime of parameters that is not too far from the threshold. Joint work with Ankur Moitra and William Perry.

Speaker: Alex Wein, Massachusetts Institute of Technology
Location:
Fine Hall 214
May 4, 2016
2:30pm - 3:30pm
ShapeFit: Exact location recovery from corrupted pairwise directions

We consider the problem of recovering a set of locations given observations of the direction between pairs of these locations.   This recovery task arises from the Structure from Motion problem, in which a three-dimensional structure is sought from a collection of two-dimensional images.  In this context, the locations of cameras and structure points are to be found from epipolar geometry and point correspondences among images.  These correspondences are often incorrect because of lighting, shadows, and the effects of perspective.  Hence, the resulting observations of relative directions contain significant corruptions.  Over the past few years, researchers have introduced several algorithms for outlier-tolerant location recovery.  For example, Wilson and Snavely introduced the 1dSfM method, and Ozyesil and Singer introduced an second-order cone program (SOCP) based solution known as LUD.  Both of these methods are empirically tolerant to outliers, though they currently lack rigorous guarantees of corruption tolerance.  To solve the location recovery problem in the presence of corrupted relative directions, we introduce a SOCP called ShapeFit.  Empirically, ShapeFit can succeed on synthetic data with over 50% corruption.  Rigorously, we prove that ShapeFit can recover a set of locations exactly when a fraction of the measurements are adversarially corrupted and when the data model is random.  This and subsequent work was done in collaboration with Choongbum Lee, Vladislav Voroninski, Tom Goldstein, and Stefano Soatto.

Speaker: Paul Hand, Rice University
Location:
Fine Hall 224
May 4, 2016
4:30pm - 5:30pm
Integrability versus Wave Turbulence in Hamiltonian partial differential equations

In the world of Hamiltonian partial differential equations, complete integrability and wave turbulence are often considered as opposite paradigms. The purpose of this talk is to give a rough idea of these different notions, and to discuss the example of a nonlinear wave toy model which surprisingly displays both properties. The key is a Lax pair structure involving Hankel operators from classical analysis, and is connected to a surprisingly explicit inverse spectral method. 

 

Speaker: Patrick Gerard , University of Paris-Sud
Location:
Fine Hall 314
May 5, 2016
2:00pm - 3:30pm
TBA - Tuomas Sahlsten
Speaker: Tuomas Sahlsten , University of Bristol
Location:
Fine Hall 601
May 5, 2016
4:30pm - 5:30pm
Solitary water waves in finite and infinite depth

We will discuss the qualitative properties of spatially localized traveling water waves. First we will review some results for waves in a 2D finite-depth fluid with vorticity and possibly density stratification but no surface tension. Next we will consider waves in a 2D or 3D infinite-depth fluid with or without surface tension but with an irrotational velocity field. In this second case we prove asymptotic formulas for the velocity potential and free surface, and relate the constants in these formulas to the kinetic energy. As a consequence, we find that nontrivial waves must have infinite angular momentum. The first part includes joint work with Robin Ming Chen and Samuel Walsh, and also Walter Strauss. 

Speaker: Miles Wheeler , New York University
Location:
Fine Hall 322
May 5, 2016
4:30pm - 5:30pm
Yang-Mills gradient flow

 We shall discuss long time existence and convergence properties of Yang-Mills gradient flow over closed, four-dimensional manifolds and applications to Morse theory for the quotient space of connections modulo gauge transformations.

Speaker: Paul Feehan , Rutgers/IAS
Location:
Fine Hall 314
May 5, 2016
4:30pm - 5:30pm
Rational curves on elliptic surfaces

Given a non-isotrivial elliptic curve E over K=Fq(t), there is always a finite extension L of K which is itself a rational function field such that E(L) has large rank. The situation is completely different over complex function fields: For "most" E over K=C(t), the rank E(L) is zero for any rational function field L=C(u).  The yoga that suggests this theorem leads to other remarkable statements about rational curves on surfaces generalizing a conjecture of Lang. 

Speaker: Douglas Ulmer, Georgia Institute of Technology
Location:
IAS Room S-101
May 6, 2016
3:00pm - 4:00pm
TBA - Yevgeny Liokumovich
Speaker: Yevgeny Liokumovich , Imperial College London
Location:
Fine Hall 314
May 9, 2016
2:30pm - 3:30pm
Optimal detection of weak principal components in high-dimensional data

Please note special day (Monday).   Principal component analysis is a widely used method for dimension reduction. In high dimensional data, the ``signal'' eigenvalues corresponding to weak principal components (PCs) do not necessarily separate from the bulk of the ``noise'' eigenvalues. In this setting, it is not possible to decide based on the largest eigenvalue alone whether or not there are "signal" PCs in the data. In this talk we explore this phenomenon in a general model that captures the shape of eigenvalue distributions often seen in applications. We show how to construct statistical tests to detect principal components, based on all eigenvalues. We also explain how recent computational advances in random matrix theory enable the efficient implementation of our methods.
The talk is based on two papers: arxiv.org/abs/1602.06896 and arxiv.org/abs/1507.01649.

Speaker: Edgar Dobriban, Stanford University
Location:
TBD
May 10, 2016
2:00pm - 3:30pm
TBA - Tuomas Sahlsten

Please note special day (Tuesday).

Speaker: Tuomas Sahlsten, Univeristy of Bristol
Location:
TBD
May 10, 2016
4:30pm - 5:30pm
On the Hilbert Property and the fundamental group of algebraic varieties

Please note special day (Tuesday).    This concerns recent work with P. Corvaja in which we relate the Hilbert Property for an algebraic variety (a kind of axiom linked with Hilbert Irreducibility, relevant e.g. for the Inverse Galois Problem) with the fundamental group of the variety. In particular, this leads to new examples (of surfaces) of failure of the Hilbert Property. We also prove the Hilbert Property for a non-rational surface (whereas all previous examples involved rational varieties). 

Speaker: Umberto Zannier , SNS Pisa
Location:
Fine Hall 214
May 11, 2016
2:30pm - 3:30pm
TBA - Kirill Serkh
Speaker: Kirill Serkh, Yale University
Location:
Fine Hall 224
May 12, 2016
2:00pm - 3:30pm
TBA - Wen-Liang Tseng
Speaker: Wen-Liang Tseng , National Taiwan University
Location:
TBD
May 18, 2016
2:30pm - 3:30pm
Randomized Algorithms for Matrix Decomposition

Matrix decompositions, and especially SVD, are very important tools in data analysis. When big data is processed, the computation of matrix decompositions becomes expensive and impractical. In recent years, several algorithms, which approximate matrix decomposition, have been developed. These algorithms are based on metric conservation features for linear spaces of random projections. We present a randomized method based on sparse matrix distribution that achieves a fast approximation with bounded error for low rank matrix decomposition.

Speaker: Yariv Aizenbud, Tel Aviv University
Location:
Fine Hall 224
May 19, 2016
3:00pm - 4:00pm
TBA - Patricia Klaser
Speaker: Patricia Klaser , UFRGS-Brazil
Location:
Fine Hall 1001
May 19, 2016
4:30pm - 5:30pm
TBA - Lillian Pierce
Speaker: Lillian Pierce, Duke University
Location:
Fine Hall 214
May 20, 2016
3:00pm - 4:00pm
TBA - Pablo Mira
Speaker: Pablo Mira , Universidad Politécnica de Cartagena
Location:
Fine Hall 314

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