Upcoming Seminars & Events

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February 8, 2016
3:15pm - 4:30pm
On the subcritical transition of the 3D Couette flow

We discuss the dynamics of small perturbations of the plane, periodic Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number. For sufficiently regular initial data, we determine the stability threshold for small perturbations and characterize the long time dynamics of solutions near this threshold. For rougher data, we obtain an estimate of the stability threshold which agrees closely with numerical experiments. The primary linear stability mechanism is an anisotropic enhanced dissipation resulting from the mixing caused by the large mean shear; the main linear instability is a non-normal instability known as the lift-up effect. Understanding the variety of nonlinear resonances and devising the correct norms to estimate them form the core of the analysis we undertake. Joint work with Pierre Germain and Nader Masmoudi.

Speaker: Jacob Bedrossian , University of Maryland
Location:
Fine Hall 110
February 8, 2016
4:30pm - 5:30pm
Julia Computing (with some random matrices)

This talk is designed to be of interest for a computer science and mathematics audience, I will combine my two “passions” by  (1)  Explaining why the Julia Computing language is so fast.  Julia in some ways resembles Python, or R, or MATLAB, which makes Julia easy to use, but Julia is deeply architected very differently from these earlier languages, making Julia's modern approach  fast, flexible, and productive.  Julia is being used worldwide by scientists, engineers, in classes,  and in industry for big data, financial applications, and the internet of things. (2) Showing how Julia has enabled research in one of my own favorite topics: random matrices.  We will demonstrate several results that quantify and illuminate finite random matrix theory.

Speaker: Alan Edelman , Massachusetts Institute of Technology
Location:
Fine Hall 214
February 8, 2016
4:30pm - 5:30pm
variance of sums of arithmetic functions over primes in short intervals

This is a special Analysis/Princeton-IAS Number Theory seminar.  Goldston & Montgomery and Montgomery & Soundararajan have established formulae for the variance of sums of the von Magoldt function over short intervals (i.e. for the variance of the number of primes in these intervals) assuming, respectively, the pair-correlation conjecture and the Hardy-Littlewood conjecture. I will discuss the generalisation of these formulae to other arithmetic functions associated with the Selberg class of L-functions, in the context of both zero statistics and arithmetic correlations. I also hope to discuss the function-field analogues of these generalisations.

Speaker: J.Keating , University of Bristol
Location:
IAS Room S-101
February 9, 2016
4:30pm - 5:30pm
Interpolation for normal bundles of general curves

This talk will address the following question: When does there exist a curve of given degree d and genus g, passing through n general points p_1, p_2,..., p_n in P^r?

Speaker: Eric Larson , MIT
Location:
Fine Hall 322
February 10, 2016
3:00pm - 4:00pm
Sudakov-type minoration

The classical Sudakov minoration principle gives a lower bound for suprema of Gaussian processes in terms of the metric entropy. We will discuss bounds of similar type for suprema of canonical processes and norms of logarithmically concave vectors. We will also present some applications based on chaining techniques.

Speaker: Rafal Latala, University of Warsaw
Location:
Fine Hall 214
February 10, 2016
4:30pm - 5:30pm
Effective dynamics of the dispersive equations and the Hardy-Littlewood circle method

The long-time behavior of small amplitude solutions to nonlinear dispersive equations on $\mathbb{R}^n$ is relatively well understood. However, the situation is markedly different when these equations are considered on a bounded domain. In this talk, we will consider nonlinear Schrodinger equations on rational tori and show how to derive limiting equations that govern the long-time dynamics of solutions. The derivation of the limiting equations and the proofs rely heavily on results from analytic number theory. 

Speaker: Jalal Shatah , NYU
Location:
Fine Hall 314
February 11, 2016
10:40am - 11:40am
Restrictions on the fundamental group of some Lagrangian cobordisms.

In this talk we will describe two methods which shows that, under some rigidity assumptions on the involved Legendrian submanifolds, a Lagrangian cobordism is simply connected. The first one uses the functoriality of the fundamental class in Legendrian contact homology with twisted coefficients. The second uses a L^2-completion of the Floer complex associated to the cobordism. This is joint work with G. Dimitroglou Rizell, P. Ghiggini and R. Golovko.

Speaker: Baptiste Chantraine , Nantes
Location:
IAS Room S-101
February 11, 2016
1:00pm - 2:00pm
Toward a contact Fukaya category

Please note special time.  I will describe some work in progress (maybe more accurately, wild speculation) regarding a version of the derived Fukaya category for contact 1-jet spaces J^1(X). This category is built from Legendrian submanifolds equipped with augmentations, and the full subcategory corresponding to a fixed Legendrian submanifold \Lambda is the augmentation category Aug(\Lambda), which I will attempt to review. The derived Fukaya category is generated by unknots, with the corollary that all augmentations ``come from unknot fillings''. I will also describe a potential application to proving that ``augmentations = sheaves''. This is work in progress with Tobias Ekholm and Vivek Shende, building on joint work with Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow.

Speaker: Lenny Ng , Duke University
Location:
IAS Room S-101
February 11, 2016
2:00pm - 3:30pm
New interactions between Analysis and Number Theory

I will discuss two new (unrelated) phenomena. (1) Taking maximal averages of functions has connections to transcendental number theory and (2) the Ulam sequence (1,2,3,4,6,8,11,...) defined via additive combinatorics has very strange distribution behavior when multiplied with 2.571.... 

Speaker: Stefan Steinerberger , Yale University
Location:
Fine Hall 601
February 11, 2016
2:30pm - 3:30pm
A frontal view on Lefschetz fibrations II

Please note special time.    In this series of two talks we will discuss Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. The main focus is on Weinstein manifolds which admit a Weinstein Lefschetz fibration with an $A_k$--fibre; this provides a large class of Weinstein structures ranging from flexible Weinstein manifolds to rich rigid examples. In particular, we will describe the computation of their symplectic homologies and discuss its implications to Legendrian submanifolds and their Lagrangian fillings.

Speaker: Roger Casals , MIT
Location:
IAS Room S-101
February 11, 2016
3:00pm - 4:00pm
Some interesting algebraic aspects of graph chordality

I will describe some interesting phenomena within toric geometry related to chordality of graphs, and outline how classical techniques to analyze chordal graphs can be used to prove interesting algebraic results.

Speaker: Karim Adiprasito, Hebrew University of Jerusalem and IAS
Location:
Fine Hall 224
February 11, 2016
3:00pm - 4:00pm
Schubert Calculus and Positivity

Schubert calculus is the study of certain intersections of varieties in the flag manifold.  These intersections are /positive/ in the differential-geometric sense, but they also have “positivity properties” in several associated rings, notably equivariant cohomology and equivariant K-theory. We show how one can get new and old formulas for the structure constants in these rings using Bott-Samelson manifolds, a sequence of projective bundles lying over the flag manifold. Time permitting, we discuss the notion of positivity in the symplectic category, and explore how to extend these ideas in the more general context.

Speaker: Rebecca Goldin, George Mason University
Location:
Fine Hall 214
February 11, 2016
4:30pm - 5:30pm
TBA - Zineb Hassainia
Speaker: Zineb Hassainia , New York University
Location:
Fine Hall 322
February 11, 2016
4:30pm - 5:30pm
Bridge trisections of knotted surfaces in the four-sphere

A trisection is a decomposition of a four-manifold into three trivial pieces and serves as a four-dimensional analogue to a Heegaard decomposition of a three-manifold. In this talk, I will discuss an adaptation of the theory of trisections to the relative setting of knotted surfaces in the four-sphere that serves as a four-dimensional analogue to bridge splittings of classical knots and links - every such surface admits a decomposition into three standard pieces called a bridge trisection. I'll describe how every such decomposition can be represented diagrammatically as a triple of trivial tangles and give a calculus of moves for passing between diagrams of a fixed surface. This is joint work with Alexander Zupan.

Speaker: Jeffrey Meier , University of Indiana
Location:
Fine Hall 314
February 11, 2016
4:30pm - 5:30pm
Statistics of abelian varieties over finite fields

Joint work with Jacob Tsimerman. Let B(g,p) denote the number of isomorphism classes of g-dimensional abelian varieties over the finite field of size p. Let A(g,p) denote the number of isomorphism classes of
principally polarized g dimensional abelian varieties over the finite field of size p. We derive upper bounds for B(g,p) and lower bounds for A(g,p) for p fixed and g increasing. The extremely large gap between the lower
bound for A(g,p) and the upper bound B(g,p) implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field. 

Speaker: Michael Lipnowski, Duke University
Location:
Fine Hall 214
February 12, 2016
10:40am - 11:40am
Satellite operations and Legendrian knot theory

Please note special day and time.    Satellite operations are a common way to create interesting knot types in the smooth category. It starts with a knot K, called the companion knot, in some manifold M and another knot P, called the pattern, in S^1\times D^2 and then creates a third knot P(K), called the satellite knot, as the image of P when S^1\times D^2 is identified with a neighborhood of K. In this talk we will discuss the relation between Legendrian knots representing K, P, and P(K). Sometimes the classification of Legendrian representatives for K and P yields a classification for P(K), but other times it does not. We will discuss why this happens and a general framework for studying Legendrian Satellites.

Speaker: John Etnyre , Georgia Tech
Location:
IAS Room S-101
February 12, 2016
1:00pm - 2:00pm
A Quantitative Look at Lagrangian Cobordisms

Please note special day and time.   Lagrangian cobordisms between Legendrian submanifolds arise in Relative Symplectic Field Theory. In recent years, there has been much progress on answering qualitative questions such as: For a fixed pair of Legendrians, does there exist a Lagrangian cobordism?  I will address two quantitative questions about Lagrangian cobordisms:  For a fixed pair of Legendrians, what is the minimal “length” of a Lagrangian cobordism?  What is the relative Gromov width of a Lagrangian cobordism? Regarding length, I will give examples of pairs of Legendrians where Lagrangian cobordisms are flexible in that the non-cylindrical region can be arbitrarily short; I will also give examples of other pairs of Legendrians where Lagrangian cobordisms are rigid in that there is a positive lower bound to their length. For the second quantitative measure, I will give some calculations and estimates of the relative Gromov width of particular Lagrangian cobordisms.  This is joint work with Joshua M. Sabloff.

Speaker: Lisa Traynor , Bryn Mawr
Location:
IAS Room S-101
February 12, 2016
2:30pm - 3:30pm
A frontal view on Lefschetz fibrations I

Please note special day and time.   In this series of two talks we will discuss Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. The main focus is on Weinstein manifolds which admit a Weinstein Lefschetz fibration with an $A_k$--fibre; this provides a large class of Weinstein structures ranging from flexible Weinstein manifolds to rich rigid examples. In particular, we will describe the computation of their symplectic homologies and discuss its implications to Legendrian submanifolds and their Lagrangian fillings.

Speaker: Emmy Murphy , MIT
Location:
IAS Room S-101
February 12, 2016
3:00pm - 4:00pm
Mean convex level set flow in general ambient manifolds

Mean curvature flow is a geometric heat equation for hyper-surfaces, which is the gradient flow of the surface area functional. The flow typically becomes singular at finite time, after which it can be extended by an object called the "level set flow". In general, the level set flow is not that well behaved, but in the important mean convex case, where the initial hypersurface is a boundary of a domain which starts moving inward, a beautiful regularity and structure theory was developed in the last 20 years by Brian White. While parts of this theory work in full generality, parts were only known to hold in either the Euclidean setting or in low dimensions. We prove two new estimates for the level set flow of mean convex domains in general Riemannian manifolds. Our estimates give control - exponential in time - for the infimum of the mean curvature, and the ratio between the norm of the second fundamental form and the mean curvature. In particular, the estimates remove the above mentioned stumbling block that has been left after the work of White and thus allow us to extend the structure theory for weak mean convex level set flow to general ambient manifolds of arbitrary dimension.

Speaker: Or Hershkovits , Courant Institute/NYU
Location:
Fine Hall 314
February 15, 2016
4:30pm - 5:30pm
Laplacian growth, sandpiles and scaling limits

How can repeating simple local operations lead to an intricate large scale structure? This phenomenon arises in several growth models originating in Physics: Internal diffusion limited aggregation (IDLA) and the Abelian sandpile. The first of these is closely related to free boundary problems for the Laplacian and an algebraic operation introduced by Diaconis and Fulton known as ``smash sum’’. These connections allow a precise description of large scale geometry, using a least action principle. The abelian sandpile, discovered independently by Statistical Physicists and Combinatorialists is harder to analyze, yet has recently yielded many of its secrets in works of Pegden, Smart and Levine. I will also discuss the rotor-router model, where (with random initial conditions) the range is conjectured to grow like t^{2/3} at time t; recently, with L. Florescu, we showed this holds as a lower bound.  This talk is based on joint works with Lionel Levine.
 

Speaker: Yuval Peres , Microsoft Research & UC Berkeley
Location:
Fine Hall 214

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