Upcoming Seminars & Events

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October 1, 2014
11:15am - 12:15pm
The topology of proper toric maps

Please note special day and time.  I will discuss some of the topology of the fibers of proper toric maps and a combinatorial invariant that comes out of this picture.  Joint with Luca Migliorini and Mircea Mustata.

Speaker: Mark de Cataldo , Stony Brook University
Location:
IAS Room S-101
October 1, 2014
2:00pm - 3:00pm
Increasing subsequences on the plane and the Slow Bond Conjecture

For a Poisson process in R^2 with intensity 1, the distribution of the maximum number of points on an oriented path from (0,0) to (N,N) has been studied in detail, culminating in Baik-Deift-Johansson's celebrated Tracy-Widom fluctuation result. We consider a variant of the model where one adds, on the diagonal, additional points according to an independent one dimensional Poisson process with rate \lambda. The question of interest here is whether for all positive values of \lambda, this results in a change in the law of large numbers for the the number of points in the maximal path. A closely related question comes from a variant of Totally Asymmetric Exclusion Process, introduced by Janowsky and Lebowitz. Consider a TASEP in 1-dimension, where the bond at the origin rings at a slower rate r<1. The question is whether for all values of r<1, the single slow bond produces a macroscopic change in the system. We provide affirmative answers to both questions. Based on joint work with Riddhipratim Basu and Vladas Sidoravicius

Speaker: Allan Sly , UC Berkeley
Location:
Fine Hall 322
October 1, 2014
4:30pm - 5:30pm
The 15-theorem & the 290-theorem

The citation for Manjul Bhargava's recent Fields Medal mentions his improvement of my proof (partly with Will Schneeberger) of the 15- theorem, and his proof (with John Hanke) of the 290-theorem. I shall talk about the history of universal quadratic forms, which was started in 1770 by Lagrange's four squares theorem, and culminated about 20 years ago in these two theorems. I'll also talk about some related problems, some of which are still open.

Speaker: John Conway, Princeton University
Location:
Fine Hall 314
October 2, 2014
12:30pm - 1:30pm
Uniform strong primeness in matrix rings

A ring $R$ is uniformly strongly prime if some finite $S \subseteq R$ is such that for $a,b \in R$, $aSb = \{0\}$ implies $a$ or $b$ is zero, in which case the bound of uniform strong primeness of $R$ is the smallest possible size of such an $S$. The case of matrix rings $R$ is considered. Via vector multiplication and bilinear equations, we obtain alternative definitions of uniform strong primeness, together with new theorems restricting the bound of uniform strong primeness of these rings. 

Speaker: Henry Thackeray , Princeton University
Location:
Fine Hall 314
October 2, 2014
2:00pm - 3:30pm
On well-posedness and small data global existence for a damped free boundary fluid-structure model

We address a fluid-structure system which consists of the incompressible Navier-Stokes equations and a damped linear wave equation defined on two dynamic domains. The equations are coupled through transmission boundary conditions and additional boundary stabilization effects imposed on the free moving interface separating the two domains. We first discuss the local in time existence and uniqueness of solutions. Given sufficiently small initial data, we prove the global in time existence of solutions.   This is a joint work with I. Kukavica, I. Lasiecka, and A. Tuffaha.

Speaker: Mihaela Ignatova , Princeton University
Location:
Fine Hall 214
October 2, 2014
3:00pm - 4:00pm
Finiteness properties for the fundamental groups of complex algebraic varieties

We describe some relations obtained in joint work with S.Papadima and A. Suciu between finiteness properties of fundamental groups and resonance and characteristic varieties. 

Speaker: Alex Dimca, IAS
Location:
Fine Hall 314
October 2, 2014
3:00pm - 4:00pm
Bipartite decomposition of graphs

For a graph G, let bc(G) denote the minimum possible number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to (exactly) one of them. The study of this quantity and its variants received a considerable amount of attention and is related to problems in communication complexity and geometry. After a brief discussion of the background and earlier results on the subject I will focus on the problem of determining the typical value of bc(G) for the binomial random graph G=G(n,p), showing that a conjecture of Erdos about it is false, and suggesting a modified version. Partly based on joint work with Tom Bohman and Hao Huang.

Speaker: Noga Alon, Tel Aviv University and IAS, Princeton
Location:
Fine Hall 224
October 2, 2014
4:30pm - 5:30pm
Dynamics and polynomial invariants of free-by-cyclic groups

The theory developed by Thurston, Fried and McMullen provides a near complete picture of the various ways a hyperbolic 3-manifold M can fiber over the circle. Namely, there are distinguished convex cones in the first cohomology H^1(M;R) whose integral points all correspond to fibrations of M, and the dynamical features of these fibrations are all encoded by McMullen's "Teichmuller polynomial." This talk will describe recent work developing aspects of this picture in the setting of a free-by-cyclic group G (the mapping torus group of an automorphism of a finite rank free group). Specifically, we will describe a polynomial invariant that determines a convex polygonal cone C in the first cohomology of G whose integral points all correspond to algebraically and dynamically interesting splittings of G. The polynomial invariant additionally provides a wealth of dynamical and algebraic information about these splittings. This is joint work with Spencer Dowdall and Christopher Leininger.

Speaker: Ilya Kapovich, University of Illinois
Location:
Fine Hall 314
October 2, 2014
4:30pm - 6:00pm
Onsager's Conjecture

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Phil Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

Speaker: Tristan Buckmaster, NYU - Courant Institute
Location:
Fine Hall 322
October 2, 2014
4:30pm - 5:30pm
The standard L-function for G_2: a "new way" integral

We consider a Rankin-Selberg integral representation of a cuspidal (not necessarily generic) representation of the exceptional group G2. Although the integral unfolds with a non-unique model, it turns out to be Eulerian and represents the standard L-function of degree 7. We discuss a general approach to the integrals with non-unique models. The integral can be used to describe the representations of G2 for which the (twisted) L-function has a pole as functorial lifts. This is a joint work with Avner Segal.

Speaker: Nadia Gurevich , Ben Gurion University
Location:
IAS Room S-101
October 3, 2014
2:00pm - 3:00pm
Regularity of manifolds with bounded Ricci curvature and the codimension $4$ conjecture

This talk will concern joint work with Aaron Naber on the regularity of noncollapsed Riemannian manifolds $M^n$ with bounded Ricci curvature $|{\rm Ric}_{M^n}|\leq n-1$, as well as their Gromov-Hausdorff limit spaces, $(M^n_j,d_j)\stackrel{d_{GH}}{\longrightarrow}(X,d)$, where $d_j$ denotes the Riemannian distance. We will explain a proof of the conjecture that $X$ is smooth away from a closed subset of  codimension $4$. By combining this with the ideas of quantitative stratification, we obtain a priori $L^q$ estimates on the full curvature tensor, for all $q<2$.  We also prove a conjecture of Anderson stating that for all $v>0$, $ <\infty$, the collection of $4$-manifolds $(M^4,g)$ with $|{\rm Ric}_{M^n}|\leq 3$, ${\rm Vol}(M^3)\geq v$, ${\rm diam}(M^4)\leq d$, contains a most a finite number of diffeomorphism types. A local version of this is used to how that noncollapsed $4$-manifolds with bounded Ricci curvature have a priori $L^2$ Riemannian curvature estimates. 

Speaker: Jeff Cheeger, NYU
Location:
Fine Hall 314
October 6, 2014
3:15pm - 4:30pm
On two extremal problems for the Fourier transform

One of the most fundamental facts about the Fourier transform is the Hausdorff-Young inequality, which states that for any locally compact Abelian group, the Fourier transform maps $L^p$ boundedly to $L^q$, where the two exponents are conjugate and $p \in [1,2]$. For Euclidean space, the optimal constant in this inequality was found by Babenko for $q$ an even integer, and by Beckner for general exponents. Lieb showed that all extremizers are Gaussian functions. This is a uniqueness theorem; these Gaussians form the orbit of a single function under the group of symmetries of the inequality. We establish a stabler form of uniqueness for $1<p<2$: If a function $f$ nearly achieves the optimal constant in the inequality, then $f$ must be close in norm to a Gaussian, with a quantitative bound involving the square of the distance to the nearest Gaussian.
Related problems concern the size of Fourier coefficients of indicator functions of sets. Here less seems to be known. Some partial results will be announced. Common to the analyses of both problems are recompactness theorems, which guarantee that extremizers exist, and that functions/sets that nearly extremize the inequalities must be close to exact extremizers. After stating theorems concerning two extremal problems, I will outline the proofs of the relevant precompactness theorems, at the heart of which lie principles of additive combinatorics. 

Speaker: Michael Christ , UC Berkeley
Location:
Fine Hall 314
October 6, 2014
4:30pm - 5:30pm
Community detection with the non-backtracking operator

Community detection consists in identification of groups of similar items within a population. In the context of online social networks, it is a useful primitive for recommending either contacts or news items to users. We will consider a particular generative probabilistic model for the observations, namely the so-called stochastic block model and prove that the non-backtracking operator provides a significant improvement when used for spectral clustering.  joint work with C. Bordenave and L. Massoulie.

Speaker: Marc LeLarge , ENS - France
Location:
Fine Hall 214
October 7, 2014
2:00pm - 3:30pm
TBA - O'Grady
Speaker: Kieran O'Grady, Sapienza University, Rome
Location:
IAS Room S-101
October 7, 2014
3:30pm - 4:30pm
TBA - Lesieutre
Speaker: John Lesieutre , IAS
Location:
IAS Room S-101
October 8, 2014
4:30pm - 5:30pm
Riemann Sums and Mobius

$S =$ square-free natural numbers. An Hilbert-Schmidt operator, $\mathcal{A}$, associated to the M\"obius function has the property that $\mathcal{A}: \bigcup_{0<r<\infty} l^r(S) \to \bigcap_{0<r<\infty} l^r(S),$ injectively. If $0<r<2$ and $\xi \in l^r(S)$, the series $f_\xi = \sum_{n\in S} \mathcal{A} \xi(n) \cos 2\pi n x$ converges uniformly to $f_\xi \in \mathcal{R}_0$, i.e., a periodic, even, continuous function with equally spaced Riemann sums, $\sum_{j=0}^{N-1} f_\xi\left(\frac{j}{n}\right) = 0$, $N = 1,2,\dots$. If $\mathcal{A} \xi_\lambda = \lambda \xi_\lambda$, $\xi_\lambda(1) = 1$, then $\xi_\lambda$ is multiplicative. If $f_\xi \in \mathrm{Lip}(\alpha)$ for some $\alpha>0$, and if $\chi$ is any Dirichlet character, then $L(s,\chi)\neq 0$, $\mathrm{Re} (s)>1-\alpha$. Conjecturally, GRH is equivalent to $f_\xi \in \Lambda_\alpha$, $\alpha<\frac{1}{2}$, $\xi\in l^r(S)$, $0<r<2$. Using a 1991 estimate by R.\ C.\ Baker and G.\ Harman, one finds that GRH implies $f_\xi \in \Lambda_\alpha$, $\alpha<\frac{1}{4}$, $\xi\in l^r(S)$, $0<r<2$. The question whether $\mathcal{R}_0 \cap \Lambda_\alpha \neq \{0\}$ for some positive $\alpha$ is open.

Speaker: William A. Veech , Rice University
Location:
Fine Hall 314
October 8, 2014
4:45pm - 5:45pm
Ancient solutions to geometric flows

We will discuss  ancient   or eternal solutions to  geometric parabolic partial differential equations.  These are  special  solutions that  appear as blow up limits near a singularity. They often represent  models of ingularities.  We will address  the  classification of  ancient  solutions to geometric flows such as the Mean Curvature flow,  the Ricci  flow and  the Yamabe flow, as well as methods of  constructing new ancient solutions from  the gluing of two or more solitons.  We will also include  future  research  directions. 

Speaker: Panagiota Daskalopoulos , Columbia University
Location:
Rutgers - Hill Center, Room 705
October 8, 2014
5:45pm - 6:45pm
Geometry of asymptotically flat graphical hypersurfaces in Euclidean space

We consider a special class of asymptotically flat manifolds of nonnegative scalar curvature that can be isometrically embedded in Euclidean space as graphical hypersurfaces. In this setting, the scalar curvature equation becomes a fully nonlinear equation with a divergence structure, and we prove that the graph must be weakly mean convex. The arguments use some intriguing relation between the scalar curvature and mean curvature of the graph and the mean curvature of its level sets. Those observations enable one to give a direct proof of the positive mass theorem in this setting in all dimensions, as well as the stability statement that if the ADM masses of a sequence of such graphs approach zero, then the sequence converges to a flat plane in both Federer-Flemings flat topology and Sormani-Wenger's intrinsic flat topology. 

Speaker: Lan-Hsuan Huang, University of Connecticut
Location:
Rutgers - Hill Center, Room 705
October 9, 2014
12:30pm - 1:30pm
Counting n particles in the plane

How do algebraic geometers "count" the number of ways of putting n points in the plane? I'll explain what Euler characteristic is, what a Hilbert scheme is, and how to compute the Euler characteristic of the Hilbert scheme of n points in the plane. The answer may surprise you, especially if you don't know what those words mean yet. Everything will be defined from scratch.

Speaker: Daniel Litt , Stanford University
Location:
Fine Hall 314
October 9, 2014
2:00pm - 3:30pm
Generalized Morse-Kaktuani Flows

The Prouhet-Thue-Morse sequence and its generalizations have occured in many settings. ``Morse-Kakutani flow'' refers to Kakutani's 1967 generalization of the Morse minimal flow (1922). These flows are
 $\mathbb{Z}_2$ skew products of almost one-to-one extensions of the adding machine ($x \to x+1$ on the $2$-adic completion of $\mathbb{Z}$). ``Generalized Morse-Kakutani flow'' is a $K$ skew product of similar construct, with base flow factor $x\to x+1$ on the profinite completion $\mathbb{Z}$ and $K$ any compact group of countable density. A review of definitions and some old theorems will be followed by a sketch of a proof that Sarnak's Mobius Orthogonality Conjecture holds for a restricted class generalized Morse-Kakutani flows. 

Speaker: William A. Veech , Rice University
Location:
Fine Hall 601

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