Upcoming Seminars & Events

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October 26, 2016
3:00pm - 4:00pm
The free-boundary Brakke flow
Differential Geometry & Geometric Analysis Seminar

A surface has geometric free-boundary if it meets some barrier hypersurface orthogonally, like a bubble on a bathtub.  We extend Brakke's weak notion of mean curvature flow to have a free-boundary condition, which allows the surface to pop'' upon tangential contact with the barrier.

Speaker: Nick Edelen , MIT
Location:
Fine Hall 314
October 26, 2016
3:00pm - 4:00pm
Stationary Aggregation Processes
Probability Seminar

In this talk I'll introduce stationary versions of known aggregation models e.g., DLA, Hastings Levitov, IDLA and Eden. Using the additional symmetry and ergodic theory, one obtains new geometric insight on the aggregation processes.

Speaker: Eviatar Procaccia , Texas A&M University
Location:
Fine Hall 214
October 27, 2016
12:30pm - 1:30pm
Tunnell's Work on the Congruent Number Problem

The congruent number problem is a classical diophantine problem which asks to determine which integers are the areas of right triangles with rational sides. We explain Tunnell's theorem on congruent numbers using elliptic curves and modular forms.

Speaker: Zhaorong Jin , Princeton University
Location:
Fine Hall 110
October 27, 2016
3:00pm - 4:00pm
Topology and Combinatorics of 'unavoidable' complexes and the family tree of the Van Kampen-Flores theorem
Algebraic Topology Seminar

Special classes of simplicial complexes (chessboard, unavoidable’, threshold, simple games’, etc.) play a central role in applications of algebraic topology in discrete geometry and combinatorics. As an illustration we outline the proof of a (so far) the most general theorem of Van Kampren-Flores type  (arXiv:1502.05290 [math.CO], Theorem 1.2), confirming a conjecture of Blagojevic, Frick, and Ziegler (Tverberg plus constraints, Bull. London Math. Soc., 46 (2014) 953-967).  The results presented in the lecture are a joint work with Dusko Jojic (University of Banja Luka) and Sinisa Vrecica (University of Belgrade).

Location:
Fine Hall 322
October 27, 2016
3:00pm - 4:30pm
Three graph classes: mock threshold, cute, and nice graphs
Discrete Mathematics Seminar

A graph class, defined in one way, can be characterized in several other ways. Forbidden induced subgraphs, intersection of subobjects, relationship among invariants, and vertex ordering are some of the most common ways. A graph is mock threshold if every induced subgraph of it has a vertex with degree or codegree at most 1. I will present the forbidden induced subgraph characterization for this class due to Richard Behr, Thomas Zaslavsky, and myself. A graph is cute if every induced cycle in it is a triangle or a quadrilateral. We discuss chi-boundedness, recognition and optimization for this class. A graph is nice if the difference between the chromatic number and clique number is either 0 or 1 for every induced subgraph of it. Perfect graphs, planar graphs, tripartite graphs, and line graphs are nice. We will address the following question: Is there anything nice about nice graphs?

Speaker: Vaidy Sivaraman , Binghamton University
Location:
Fine Hall 224
October 27, 2016
4:30pm - 5:30pm
Exponential self-similar mixing by incompressible flows
Analysis of Fluids and Related Topics

I will address the problem of the optimal mixing of a passive scalar under the action of an incompressible flow in two space dimensions. The scalar solves the continuity equation with a divergence-free velocity field which satisfies a bound in the Sobolev space $W^{s,p}$, where $s \geq 0$ and $1\leq p\leq \infty$. The mixing properties are given in terms of a characteristic length scale, called the mixing scale. We consider two notions of mixing scale, one functional, expressed in terms of the homogeneous Sobolev norm $\dot H^{-1}$, the other geometric, related to rearrangements of sets. We study rates of decay in time of both scales under self-similar mixing. For the case $s=1$ and $1 \leq p \leq \infty$ (including the Lipschitz case, and the case of physical interest of enstrophy-constrained flows), we present examples of velocity fields and initial configurations for the scalar that saturate the exponential lower bound established in previous works for the decay in time of both scales. We also obtain several consequences for the geometry of regular Lagrangian flows associated to Sobolev velocity fields and for the loss of regularity for continuity equations with non-Lipschitz velocity field. The talk will be based on joint works with G. Alberti (University of Pisa, Italy) and A. L. Mazzucato (Penn State).

Speaker: Gianluca Crippa , Universitat Basel
Location:
Fine Hall 322
October 27, 2016
4:30pm - 5:30pm
Co-oriented Taut Foliations
Topology Seminar

I will describe a new construction of  (codimension one) co-oriented taut foliations (CTFs) of 3-manifolds. It follows from this construction that if K is  either an alternating knot or a Montesinos knot, then K is not an L-space knot if and only if every nontrivial Dehn filling on K yields a 3-manifold that contains a CTF. This work is joint with Charles Delman.

Speaker: Rachel Roberts , Washington University
Location:
Fine Hall 314
October 27, 2016
4:30pm - 5:30pm
The Arithmetic of Noncongruence Subgroups of SL(2,Z)
Princeton University/IAS Number Theory Seminar

After beginning by giving a brief overview of how one can think of noncongruence modular curves as moduli spaces of elliptic curves with G-structures, we will discuss how these moduli interpretations fits into the greater body of knowledge concerning noncongruence subgroups, in particular focusing on the Unbounded Denominators Conjecture for their modular forms.

Speaker: William Chen , IAS
Location:
Fine Hall 214
October 31, 2016
11:00am - 12:00pm
Moduli of Riemann surface and Bers conjecture
Special Lecture

This is a continuation of the October 24 talk.  It was Koebe who first proved that closed Riemann surface can be uniformized  by Schottky groups. However Marden (1974) showed that not every Schottky group is generated by geometric circle reflections in the complex plane, which is called "classical"(original definition by Schottky himself) Schottky group.  Bers (1975) and Hejhal (1975) and Ahlfors made detailed studies on Schottky space of moduli space of Riemann surface. And Bers made the following conjecture: "Every closed Riemann surface can be uniformized by classical Schottky group."  In this talk I will describe and present resolution of this conjecture based on two recent works. In fact, I will present the solution which actually answer a lot more to the original problem. First I will talk about smooth moduli space of Riemann surface, which we show that
every closed Riemann surface is uniformizable by a Schottky group of Hausdorff dimension less than one. Second, I will give complete and sharp classification of Kleinian groups of Hausdorff dimension at most one. These two part works are independent and is based on completely different ideas proofs. We prove the result on moduli space by developing ideas of Cayley graph measure decompositions and norm of homological markings. The prove of the classification is based on application of deformation theory on local existence result and rectifiability of invariant curves.

Speaker: Yong Hou , Princeton University
Location:
Fine Hall 110
November 1, 2016
1:30pm - 2:30pm
Symplectic Geometry Seminar
Speaker: Ivan Smith , Cambridge University
Location:
IAS Room S-101
November 3, 2016
2:30pm - 3:30pm
Clique-based semidefinite relaxation of the quadratic assignment problem
PACM IDeAS

The matching problem between two adjacency matrices, A and B, can be formulated as the NP-hard quadratic assignment problem (QAP). While the QAP admits a semidefinite (SDP) relaxation that is often tight in practice, this SDP scales badly as it involves a matrix variable of size n^2 by n^2. To achieve a speed up, a further relaxation of the SDP will be described, where the number of variables scale as O(bn^2), where b is the number of non-zero entries in B. The dual problem of this relaxation has a natural three-block structure that can be solved via Alternating Direction Method of Multipliers (ADMM) in a distributed manner. I will show results that suggest this relaxation offers a good compromise between speed and tightness in practice, and will discuss how the assignment problem in Nuclear Magnetic Resonance Spectroscopy can be formulated as a QAP with sparse B. This is joint work with Yuehaw Khoo and Amit Singer.

Speaker: Jose Simoes Bravo Ferreira, Princeton University
Location:
Fine Hall 224
November 7, 2016
3:00pm - 4:30pm
Non-linear stability of Kerr-de Sitter black holes
Analysis Seminar

In joint work with András Vasy, we recently proved the stability of the Kerr-de Sitter family of black holes as solutions of the initial value problem for the Einstein vacuum equations with positive cosmological constant, for small angular momenta but without any symmetry assumptions on the initial data. I will explain the general framework which enables us to deal systematically with the diffeomorphism invariance of Einstein's equations, and thus how our solution scheme finds a suitable (wave map type) gauge within a carefully chosen finite-dimensional family of gauges; I will also address the issue of finding the mass and the angular momentum of the final black hole.

Speaker: Peter Hintz , UC Berkeley
Location:
Fine Hall 314
November 7, 2016
4:00pm - 5:00pm
Statistics, Machine Learning, and Understanding the 2016 Election
PACM/Applied Mathematics Colloquium

Although 2016 is a highly unusual political year, elections and public opinion follow predictable statistical properties. I will review how the Presidential, Senate, and House races can be tracked and forecast from freely available polling data. Missing data can be filled in using a Google-Wide Association Study (GoogleWAS). Finally, simple statistics can be used to identify inequities such as partisan gerrymandering, and provide a tool for possible judicial relief. These examples show how statistics and machine learning can deepen an understanding of the U.S. political scene, even under extreme circumstances.

Samuel S.-H. Wang, Ph.D., Professor, Neuroscience Institute and Department of Molecular Biology, Princeton University

Speaker: Samuel Wang, Princeton University
Location:
Fine Hall 214
November 8, 2016
4:30pm - 5:30pm
Tropical curve counting in superabundant geometries
Algebraic Geometry Seminar

I will discuss a general framework using Artin fans -- certain logarithmic algebraic stacks -- in which to understand the relationship between logarithmic stable maps and tropical curve counting. These objects provide a flexible tool to study correspondences between algebraic and tropical curves. In particular, we obtain new realization theorems for tropical curves in superabundant settings. After explaining some remarkably cheap consequences of this setup, I will discuss an application, joint with Yoav Len, to the enumerative geometry of elliptic curves on toric surfaces.

Speaker: Dhruv Ranganathan , MIT
Location:
Fine Hall 322
November 9, 2016
2:30pm - 3:30pm
Hermite interpolation and approximation in manifolds
PACM IDeAS

In this talk we study the Hermite interpolation and approximation problem. It aims at producing a function together with its derivatives, which interpolate or approximate given discrete point-vector data. The classical Hermite method interpolates data in linear spaces using polynomial functions.  We are interested in interpolating or approximating manifold-valued point-vector data using curves defined solely by the intrinsic geometry of the underlying manifold. For this purpose we use iterative refinement algorithms, called Hermite subdivision schemes. These algorithms successively refine given point-vector data and, via a limit process, produce a function and its derivatives solving the Hermite problem. Subdivision algorithms are well-suited for our purpose, as they can be adapted easily from the linear situation to the manifold setting. We give an introduction to linear Hermite subdivision schemes and present an adaptation to manifolds using geodesics and parallel transport. Furthermore, we show that the resulting nonlinear algorithms solve the manifold-valued Hermite problem by providing convergence and smoothness analysis.

Speaker: Caroline Moosmüller, TU Graz, Austria
Location:
Fine Hall 224
November 9, 2016
3:00pm - 4:00pm
Location:
Fine Hall 214
November 9, 2016
3:00pm - 4:00pm
TBA - Jim Isenberg
Differential Geometry & Geometric Analysis Seminar
Speaker: Jim Isenberg, University of Oregon
Location:
Fine Hall 314
November 9, 2016
4:30pm - 5:30pm
The stability of Kerr-de Sitter black holes
Department Colloquium

In this lecture I will discuss Kerr-de Sitter black holes, which are rotating black holes in a universe with a positive cosmological constant, i.e. they are explicit solutions (in 3+1 dimensions) of Einstein's equations of general relativity. They are parameterized by their mass and angular momentum.  I will discuss the geometry of these black holes as well as that of the underlying de Sitter space, and then talk about the stability question for these black holes in the initial value formulation. Namely, appropriately interpreted, Einstein's equations can be thought of as quasilinear wave equations, and then the question is if perturbations of the initial data produce solutions which are close to, and indeed asymptotic to, a Kerr-de Sitter black hole, typically with a different mass and angular momentum. In this talk, based on joint work with Peter Hintz, I will discuss geometric aspects of the stability problem, in particular showing that Kerr-de Sitter black holes with small angular momentum are stable in this sense.

Speaker: Andras Vasy , Stanford University
Location:
Fine Hall 314
November 10, 2016
12:30pm - 1:30pm