Upcoming Seminars & Events

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February 27, 2017
3:00pm - 4:00pm
Stable shock formation for solutions to the multidimensional compressible Euler equations in the presence of non-zero vorticity

It is well-known since the foundational work of Riemann that plane symmetric solutions to the compressible Euler equations may form shocks in finite time. For a class of simple plane symmetric solutions, we prove that the phenomenon of shock-formation is stable under perturbations of the initial data that break the plane symmetry with potentially non-vanishing vorticity. In particular, this is the first constructive shock-formation result for which the vorticity is allowed to be non-vanishing at the shock. We show that the vorticity remains bounded all the way up to the shock, and that the dynamics are well-described by the irrotational compressible Euler equations. This is a joint work with J. Speck (MIT), which is partly an extension of an earlier joint work with J. Speck (MIT), G. Holzegel (Imperial) and W. Wong (Michigan State).

 

Speaker: Jonathan Luk , Stanford University
Location:
Fine Hall 314
February 27, 2017
4:00pm - 5:00pm
Sampling Nodes and Constructing Expanders Locally

In many real world applications we have only limited access to networks. For example when we crawl a social network or we design a peer-to-peer system we are restricted to access nodes only locally. In this talk we will analyze two classic problems in this setting. First we consider the problem of sampling nodes from a large graph according to a prescribed distribution by using only local random walks as the basic primitive. Our goal is to obtain algorithms that make a small number of queries to the graph but output a node that is sampled according to the prescribed distribution. Focusing on the uniform distribution case, we study the query complexity of three algorithms and show a near-tight bound expressed in terms of the parameters of the graph such as average degree and the mixing time. Both theoretically and empirically, we show that some algorithms are preferable in practice than the others. We also extend our study to the problem of sampling nodes according to some polynomial function of their degrees; this has implications for designing efficient algorithms for applications such as triangle counting. Then we focus on the following classic distributed problem with applications to peer-to-peer networks. Given an n-node d-regular network for d = Ω(log n), we want to design a decentralized, local algorithm that transforms the graph into one that has good connectivity properties (low diameter, expansion, etc.) without affecting the sparsity of the graph. To this end, Mahlmann and Schindelhauer introduced the random “flip” transformation, where in each time step, a random pair of vertices that have an edge decide to ‘swap a neighbor’. They conjectured that performing O(nd) such flips at random would convert any connected d-regular graph into a d-regular expander graph, with high probability. However, the best known upper bound for the number of steps is roughly O(n^17d^23), obtained via a delicate Markov chain comparison argument. Our main result is to prove that a natural instantiation of the random flip produces an expander in at most O(n^2d^2\sqrt{log n}) steps, with high probability. We also show that our technique can be used to analyze another well-studied random process known as the ‘random switch’, and show that it produces an expander in O(nd) steps with high probability. (Joint work with Zeyuan Allen-Zhu, Aditya Bhaskara, Flavio Chierichetti, Anirban Dasgupta, Ravi Kumar, Lorenzo Orecchia and Tamas Sarlos)

Speaker: Silvia Lattanzi, Google Research
Location:
Fine Hall 214
February 28, 2017
2:00pm - 3:00pm
The Normalized Volume of a Valuation

Please note special time and location.   Motivated by work in Kahler-Einstein geometry, Chi Li defined the normalized volume function on the space of valuations over a singularity and proposed the problem of both finding and studying the minimizer of this function. While Li's problem is closely connected to the notion of K-semistability, it also relates to an invariant of singularities previously explored in the work of de Fernex, Ein, and Mustata. I will explain the motivation for this problem and discuss a recent result proving the existence of normalized volume minimizers.

Speaker: Harold Blum , University of Michigan
Location:
Fine Hall 401
February 28, 2017
5:00pm - 6:00pm
Episodes from Quantitative Topology: 3. Gromov-Hausdorff space and homeomorphisms

Gromov-Hausdorff space is a metric space of compact metric spaces and is useful in many areas of geometry.  Motivated by Cheeger's thesis, there are a number of results proving that for many pre-compact sets in GH space, there are only finitely many homeomorphism types of manifolds.  I will explain some work with Sasha Dranishnikov and Steve Ferry that shows infinite dimensional phenomena arise in certain effective versions of this result.  This indirectly leads to a certain kind of metric-topological rigidity that holds for all manifolds whose fundamental groups are lattices in real Lie groups (or are word-hyperbolic), but not shared by all those whose fundamental groups are linear.

Speaker: Shmuel Weinberger , University of Chicago
Location:
McDonnell Hall A02
March 1, 2017
2:00pm - 3:00pm
On a fully nonlinear version of the Min-Oo Conjecture

Please note:  This is an additional DGGA seminar for this date.  Please note different room and time.   In this talk, we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations in subdomains of the standard $n$-sphere $\s^n$ under suitable conditions on the boundary.  This proves rigidity for compact connected locally conformally flat manifolds $(M,g)$ with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere $\partial D(r)$, $D(r)$ a geodesic ball of radius $r\in (0,\pi/2]$ in $\mathbb{S}^n$, and totally umbilic with mean curvature bounded bellow by the mean curvature of this geodesic sphere. Under the above conditions, $(M,g)$ must be isometric to the closed geodesic ball $\overline{D(r)}$. In particular, we recover the solution by F.M. Spiegel  to the Min-Oo conjecture for locally conformally flat manifolds.  As a side product, our methods in dimension $2$ provide a new proof to a classical theorem of Toponogov. Roughly speaking, Toponogov's Theorem is equivalent to a rigidity theorem for spherical caps in the Hyperbolic three-space $\mathbb H^3$.  This is a joint work with E. Barbosa and M.P. Cavalcante.

Speaker: Jose Espinar, IMPA
Location:
Fine Hall 110
March 1, 2017
2:30pm - 3:30pm
TBA - Stefan Karpinski
Speaker: Stefan Karpinski, Julia Computing
Location:
Fine Hall 224
March 1, 2017
3:00pm - 4:00pm
The isoperimetric problem for Lens spaces

Given a Riemannian manifold, the isoperimetric problem consists in classifying the regions that minimize perimeter among regions of same volume.  In this talk, we show that the solutions of the isoperimetric problem in the Lens space with large fundamental group are either geodesic spheres or tori of revolution about geodesics.

Speaker: Celso Viana , University College London
Location:
Fine Hall 314
March 2, 2017
10:00am - 11:00am
Liouville sectors and local open-closed map

I will describe joint work-in-progress with Sheel Ganatra and Vivek Shende. We introduce a class of Liouville manifolds with boundary, called Liouville sectors, in which Floer theory is well behaved (a condition on the characteristic foliation of the boundary controls holomorphic curves from escaping). A corollary of this setup is a local-to-global argument for verifying Abouzaid's generation criterion for the wrapped Fukaya category. We verify this criterion for Liouville sectors associated to Nadler's arboreal singularities, and so any Liouville manifold covered by such sectors has a finite generating collection of Lagrangians (conjecturally, all Weinstein manifolds admit such a cover). We expect the language of Liouville sectors also suffices to formulate and prove statements such as "the wrapped Fukaya category is a homotopy cosheaf" with respect to some reasonable class of open covers.

 

Speaker: John Pardon , Princeton University
Location:
Fine Hall 214
March 2, 2017
3:00pm - 4:00pm
An Algorithm for Hecke Operators

Hecke operators act on the cohomology of locally symmetric spaces for SL(n,R), and the Hecke eigenvalues are important in number theory and automorphic forms.  When n = 2, the case of classical modular forms, modular symbols (due to Manin) give an algorithm for computing the Hecke operators.  Ash and Rudolph extended the algorithm to all n, but only in the top non-vanishing degree of the cohomology--the virtual cohomological dimension, or vcd.  Gunnells found an algorithm for all n, but only in degree one less than the vcd.  This talk is on work in progress on an algorithm that computes the Hecke operators in all degrees.  This is joint work with Robert MacPherson.

Speaker: Mark McConnell, Princeton University
Location:
Fine Hall 322
March 2, 2017
4:30pm - 5:30pm
Real structures on ordinary Abelian varieties

The "moduli space" for principally polarized complex n dimensional Abelian varieties with real structure (that is, anti-holomorphic involution) may be identified with a certain locally symmetric space for the group GL(n) over the real numbers.  Is it possible to make sense of the points of this space over a finite field?  This talk describes joint work with Yung-sheng Tai (Haverford College) in which we propose an approach to this question for ordinary Abelian varieties. 

Speaker: Mark Goreski, IAS
Location:
Fine Hall 214
March 2, 2017
4:30pm - 5:30pm
On a Slightly Compressible Water Wave

In this talk, I would like to go over some recent results on a compressible water wave. We generalize the apriori energy estimates for the compressible Euler equations established in Lindblad-Luo to when the fluid domain is unbounded. In addition, we establish weighted elliptic estimates that allow us to find initial data in some weighted Sobolev spaces with weight $w(x)=(1+|x|^2)^{\mu}, \mu \geq 2$, and we show this propagates within short time; in other words, we are able to prove weighted energy estimates for compressible water waves. These results serve as good preparation for proving long time existence also for compressible water waves.

Speaker: Chenyun Luo , Johns Hopkins University
Location:
Fine Hall 322
March 2, 2017
4:30pm - 5:30pm
Pin(2)-equivariant Floer homology and homology cobordism

We review Manolescu's construction of the Pin(2)-equivariant     Seiberg-Witten Floer stable homotopy type, and apply it to the study of the 3-dimensional homology cobordism group. We introduce the `local     equivalence' group, and construct a homomorphism from the homology cobordism group to the local equivalence group. We then apply Manolescu's Floer homotopy type to obstruct cobordisms between Seifert 0
spaces. In particular, we show the existence of integral homology spheres not homology cobordant to any Seifert space. We also introduce connected Floer homology, an invariant of homology cobordism taking
values in isomorphism classes of modules. 

Speaker: Matthew Stoffregen , UCLA
Location:
Fine Hall 314
March 3, 2017
2:30pm - 3:30pm
Structure-Aware Dictionary Learning for Graph Signals

Please note different day (Friday).    Dictionary Learning techniques aim to find sparse signal representations that capture prominent characteristics in the given data. For signals residing on non-Euclidean topologies, represented by weighted graphs, an additional challenge is incorporating the underlying geometric structure of the data domain into the learning process. We introduce an approach that aims to infer and preserve the local intrinsic geometry in both dimensions of the data. Combining ideas from spectral graph theory, manifold learning and sparse representations, our proposed algorithm simultaneously takes into account the underlying graph topology as well as the data manifold structure. The efficiency of this approach is demonstrated on a variety of applications, including sensor network data completion and enhancement, image structure inference, and challenging multi-label classification problems.

Speaker: Yael Yankelevsky, Technion
Location:
Fine Hall 224
March 6, 2017
3:00pm - 4:00pm
Schrodinger equations and irrational tori

We prove long-time Strichartz estimates for solutions to the linear Schrodinger equation on generic irrational tori. This improves the recent work of Bourgain and Demeter. As an application, we also establish
polynomial bounds for Sobolev norm of solutions to the energy critical nonlinear Schrodinger equation in 3D. The first part is joint work with P. Germain and L. Guth.

Speaker: Yu Deng , NYU
Location:
Fine Hall 314
March 6, 2017
4:00pm - 5:00pm
How Far Are We From Having a Satisfactory Theory of Clustering?

Unsupervised learning is widely recognized as one of the most important challenges facing machine learning nowadays. However, unlike supervised learning, our current theoretical understanding of those tasks, and in particular of clustering, is very rudimentary. Although hundreds of clustering papers are being published every year, there is hardly any work reasoning about clustering independently of any particular algorithm, objective function, or generative data model.  My talk will focus on such clustering research.  I will discuss two aspects in which theory could play a significant role in guiding the use of clustering tools. The first is model selection - how should a user pick an appropriate clustering tool for a given clustering problem, and how should the parameters of such an algorithmic tool be tuned? In contrast with other common computational tasks, in clustering, different algorithms often yield drastically different outcomes. Therefore, the choice of a clustering algorithm may play a crucial role in the usefulness of an output clustering solution. However, there currently exist no methodical guidance for clustering tool selection for a given clustering task. I will describe some recent proposals aiming to address this crucial lacuna.  The second aspect of clustering that I will address is the complexity of computing a cost minimizing clustering (given some clustering objective function). Once a clustering model (or objective) has been picked, the task becomes an optimization problem.  While most of the clustering objective optimization problems are computationally infeasible, they are being carried out routinely in practice. This theory-practice gap has attracted significant research attention recently. I willdescribe some of the theoretical attempts to address this gap and discuss how close do they bring us to a satisfactory understanding of the computational resources needed for achieving good clustering solutions. The talk is based on joint work with my students Hassan Ashtiani and Shrinu Kushagra

Speaker: Shai Ben-David, University of Waterloo
Location:
Fine Hall 214
March 7, 2017
4:30pm - 5:30pm
TBA - Julie Rana
Speaker: Julie Rana , University of Minnesota
Location:
Fine Hall 322
March 8, 2017
2:30pm - 3:30pm
TBA - Yuxin Chen
Speaker: Yuxin Chen, Princeton University
Location:
Fine Hall 224
March 8, 2017
3:00pm - 4:00pm
TBA - Paul Bourgade
Speaker: Paul Bourgade , Courant Institute/NYU
Location:
Fine Hall 214
March 8, 2017
4:30pm - 5:30pm
Interpolation and Approximation

Suppose f is a real-valued function on an awful set E in R^n. How can we decide whether f extends to a smooth function F on the whole R^n? "Smooth" means that F belongs to our favorite space X of continuous functions, e.g. C^m, C^{m, alpha}, or W^{m,p}. If such an F exists, how small can we take its norm in X? What can we say about its derivatives at a given point in or near E? Can we take F to depend linearly on f? Suppose E is finite. Can we compute an F with close-to-minimal norm in X? How many computer operations does it take? What if we require merely that F agree approximately with f on E? What if we are allowed to discard a few points of E as "outliers"? Which points should we discard to make the norm of F as small as possible? The subject started with Whitney in 1934. I've been working on it for many years. The talk will be completely accessible.

Speaker: Charlie Fefferman, Princeton University
Location:
Fine Hall 314
March 9, 2017
10:45am - 11:45am
TBA - Chris Woodward
Speaker: Chris Woodward , Rutgers University
Location:
IAS Room S-101

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