Upcoming Seminars & Events

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December 5, 2016
3:00pm - 4:30pm
The Kakeya needle problem for rectifiable sets

We show that the classical results about rotating a line segment in arbitrarily small area, and the existence of a Besicovitch and a Nikodym set hold if we replace the line segment by an arbitrary rectifiable set.
This is a joint work with Alan Chang. 

Speaker: Marianna Csörnyei , University of Chicago
Location:
Fine Hall 314
December 5, 2016
4:00pm - 5:00pm
Turbulent weak solutions of the Euler equations

This joint Math/PACM colloquium will be held at 4:00, Monday, December 5, in Fine 214.

Motivated by Kolmogorov's theory of hydrodynamic turbulence, we consider dissipative weak solutions to the 3D incompressible Euler equations. We show that there exist infinitely many weak solutions of the 3D Euler equations, which are continuous in time, lie in a Sobolev space $H^s$ with respect to space, and they do not conserve the kinetic energy. Here the smoothness parameter $s$ is at the Onsager critical value $1/3$, consistent with Kolmogorov's $-4/5$ law for the third order structure functions. We shall also discuss bounds for the second order structure functions, which deviate from the classical Kolmogorov 1941 theory. This talk is based on joint work with T. Buckmaster and N. Masmoudi. 

Speaker: Vlad Vicol, Princeton University
Location:
Fine Hall 214
December 6, 2016
3:00pm - 4:00pm
Contact manifolds with flexible fillings

In this talk, I will prove that all flexible Weinstein fillings of a given contact manifold have isomorphic integral cohomology. As an application, I will show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. Using similar methods, I will construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. These results are proven by studying Reeb chords of loose Legendrians and using positive symplectic homology.

 

Speaker: Oleg Lazarev , Stanford University
Location:
Fine Hall 224
December 6, 2016
4:30pm - 5:30pm
On Noether's inequality for stable log surfaces

In this talk I report on some recent progress on the geography problem of stable log surfaces. This is about restrictions on their holomorphic invariants, such as the volume K^2 and the geometric genus p_g. Compared to the case of surfaces of general type, a new feature here is that the volume of a stable log surface is not necessarily an integer. Extending the work of Tsunoda and Zhang in the nineties, I will give an optimal lower bound of the volume when the geometric genus is one. Then I will use an example to illustrate that a speculated Noether type inequality for stable log surfaces does not hold in general.

Speaker: Wenfei Liu , Xiamen University
Location:
Fine Hall 322
December 7, 2016
3:00pm - 4:00pm
Uniqueness of weak solutions to the Ricci flow

In his resolution of the Poincaré and Geometrization Conjectures, Perelman constructed Ricci flows in which singularities are removed by a surgery process. His construction depended on various auxiliary parameters, such as the scale at which surgeries are performed. At the same time, Perelman conjectured that there must be a canonical flow that automatically "flows through its surgeries”, at an infinitesimal scale.  Recently, Kleiner and Lott constructed so-called Ricci flow space-times, which exhibit this desired behavior. In this talk, I will first review their construction. I will then present recent work of Bruce Kleiner and myself, in which we show that these Ricci flow space-times are in fact unique and fully determined by their initial data. Therefore, these flows can be viewed as “canonical”, hence confirming Perelman’s Conjecture. I will also discuss further applications of this uniqueness statement.

Speaker: Richard Bamler, UC Berkeley
Location:
Fine Hall 314
December 7, 2016
3:00pm - 4:00pm
The spectral gap of dense random regular graphs

Let G be uniformly distributed on the set of all simple d-regular graphs on n vertices, and assume d is bigger than some (small) power of n. We show that the second largest eigenvalue of G is of order √d with probability close to one. Combined with earlier results covering the case of sparse random graphs, this settles the problem of estimating the magnitude of the second eigenvalue, up to a multiplicative constant, for all values of n and d, confirming a conjecture of Van Vu. Joint work with Pierre Youssef.

Speaker: Konstantin Tikhomirov , Princeton University
Location:
Fine Hall 214
December 7, 2016
4:30pm - 5:30pm
Energy Identity for Stationary Yang Mills

The first part of this talk will introduce and discuss the basics of Yang Mills connections, which are connections over a principle bundle which are critical points of the energy functional \int |F|^2, the L^2 norm of the curvature of A, and thus A may be viewed as a solution to a nonlinear pde. In many problems, e.g. compactifications of moduli spaces, one considers sequences A_i of such connections which converge to a potentially singular limit connection A_i-> A . The convergence may not be smooth, and we can understand the blow up region by converging the energy measures |F_i|^2 dv_g -> |F|^2dv_g +\nu, where \nu=e(x)d\lambda^{n-4} is the n-4 rectifiable defect measure (e.g. think of \nu as being supported on an n-4 submanifold). It is this defect measure which explains the behavior of the blow up, and thus it is a classical problem to understand it. The conjectural picture is that e(x) may be computed explicitly as the sum of the bubble energies which arise from blow ups at x, a formula known as the energy identity. This talk will primarily be spent explaining in detail the concepts above, with the last part focused on the recent proof of the energy identity, which is joint with Daniele Valtorta. The techniques may also be used to prove a conjectured L^1 estimate on the hessian of the curvature, and we will discuss these results as well.

Speaker: Aaron Charles Naber, Northwestern University
Location:
Fine Hall 314
December 8, 2016
12:30pm - 1:30pm
TBD - Eric Naslund
Speaker: Eric Naslund, Princeton University
Location:
Fine Hall 110
December 8, 2016
2:00pm - 3:30pm
Quasi-periodic solutions to nonlinear PDE

We discuss time quasi-periodic solutions to nonlinear Schroedinger (NLS) and nonlinear wave equations (NLW) on the torus in arbitrary dimensions. The latter is hyperbolic and uses additionally, a Diophantine property of algebraic numbers.  We mention also a work in progress, on space-time quasi-periodic solutions to non-integrable NLS, whose analysis rests on semi-algebraic geometry.

Speaker: Wei-Min Wang , Université Paris-Sud
Location:
Jadwin Hall 111
December 8, 2016
2:30pm - 3:30pm
Non-Convex Phase Retrieval from STFT Measurements

The problem of recovering a one-dimensional signal from its Fourier transform magnitude, called phase retrieval, is ill-posed in most cases. We consider the closely-related problem of recovering a signal from its phaseless short-time Fourier transform (STFT) measurements. This problem arises naturally in several applications, such as ultra-short pulse characterization and ptychography. We suggest to recover the signal by a gradient algorithm, minimizing a non-convex loss function. The algorithm is initialized by the leading eigenvector of a designed matrix. We show that under appropriate conditions, this initialization is close to the underlying signal. We analyze the geometry of the loss function and show empirically that the gradient algorithm converges to the underlying signal even with small redundancy in the measurements. The last part of the talk will be devoted to a new class of problems, called high-order phase retrieval.

Speaker: Tamir Ben Dory, PACM
Location:
Fine Hall 214
December 8, 2016
3:00pm - 4:30pm
Improperly coloring K_t minor-free graphs

We show that for every t > 0 there exists a constant c=c(t) such that, if a graph G does not contain K_t as a minor, then its vertex set can be partitioned into at most t-1 parts such that every part induces a subgraph with maximum component of size at most c. This relaxation of Hadwiger's conjecture improves previous results of Kawarabayashi and Mohar, Wood, and Liu and Oum, who proved that the same conclusion holds for partitions into 31t/2, 7t/2 and 3t parts respectively.  We also discuss applications of our results to extremal questions on bootstrap percolation for minor-closed graph families. Based on joint work with Zdenek Dvorak.

Speaker: Sergey Norin , McGill University
Location:
Fine Hall 224
December 8, 2016
4:30pm - 5:30pm
TBD - Dinakar Ramakrishnan
Speaker: Dinakar Ramakrishnan, California Institute of Technology
Location:
IAS Room S-101
December 8, 2016
4:30pm - 5:30pm
TBA - Boyu Zhang
Speaker: Boyu Zhang , Harvard University
Location:
Fine Hall 314
December 9, 2016
3:30pm - 5:00pm
New Junior Faculty Lectures II

The Department of Mathematics is holding the second of two events where instructors and assistant professors wh joined the department this fall will speak briefly about their research.

3:30 p.m. — Otis Chodosh, Veblen Research Instructor
                    “Global geometric questions involving scalar curvature”

3:50 p.m. — Ziyang Gao, Instructor
                     “Heights in families of abelian varieties and the Geometric Bogomolov Conjecture”

4:10 p.m. — Evita Nestoridi, Instructor
                     "A random walk on the upper triangular matrices”

4:30 p.m. — Konstantin Tikhomirov, Instructor
                    “Geometric aspects of the random matrix theory”

Speaker: ,
Location:
McDonnell Hall A01
December 12, 2016
3:00pm - 4:30pm
Maximizers for the Stein–Tomas Inequality

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein–Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the  existence of an optimizer in the Stein–Tomas inequality. Our result is valid in any dimension.

The talk is based on joint work with E. Lieb and J. Sabin.

Speaker: Rupert Frank , Caltech
Location:
Fine Hall 314
December 12, 2016
4:00pm - 5:00pm
Quantum Oracle Classification: The Case of Group Structure

The Quantum Oracle Classification (QOC) problem is to classify a function, given only quantum black box access, into one of several classes without necessarily determining the entire function. Generally, QOC captures a very wide range of problems in quantum query complexity. However, relatively little is known about many of these problems. In this work, we analyze the a subclass of the QOC problems where there is a group structure. That is, suppose the range of the unknown function A is a commutative group G, which induces a commutative group law over the entire function space. Then we consider the case where A is drawn uniformly at random from some subgroup A of the function space. Moreover, there is a homomorpism f on A, and the goal is to determine f(A). This class of problems is very general, and covers several interesting cases, such as oracle evaluation; polynomial interpolation, evaluation, and extrapolation; and parity. These problems are important in the study of message authentication codes in the quantum setting, and may have other applications. We exactly characterize the quantum query complexity of every instance of QOC with group structure in terms of a particular counting problem. That is, we provide an algorithm for this general class of problems whose success probability is determined by the solution to the counting problem, and prove its exact optimality. Unfortunately, solving this counting problem in general is a non-trivial task, and we resort to analyzing special cases. Our bounds unify some existing results, such as the existing oracle evaluation and parity bounds. In the case of polynomial interpolation and evaluation, our bounds give new results for secret sharing and information theoretic message authentication codes in the quantum setting.

Speaker: Mark Zhandry, Princeton University
Location:
Fine Hall 214
December 13, 2016
3:00pm - 4:00pm
TBD - Guogang Liu
Speaker: Guogang Liu , Nantes
Location:
Fine Hall 224
December 13, 2016
4:30pm - 5:30pm
Log geometric techniques for open invariants in mirror symmetry

This is a joint Algebraic Geometry and Symplectic Geometry seminar.  Please note different room (322) and start time (4:30). We would like to discuss an algebraic-geometric approach to some open invariants arising naturally on the A-model side of mirror symmetry. The talk will start with a smooth overview of the use of logarithmic geometry in the Gross-Siebert program. We then will discuss various illustrations of the use in open invariants, including a description of the symplectic Fukaya category via certain stable logarithmic curves. For this, our main object of study will be the degeneration of elliptic curves, namely the Tate curve. However, the results are expected to generalise to higher dimensional Calabi-Yau manifolds. This is joint work with Bernd Siebert, with general ideas based on discussions of Bernd Siebert and Mohammed Abouzaid.

Speaker: Hülya Argüz , IAS
Location:
Fine Hall 322
December 14, 2016
2:00pm - 3:00pm
Properly immersed CMC surfaces in hyperbolic 3-manifolds of finite volume

Please note special time:  2:00.    If $N$ is a noncompact hyperbolic 3-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of $\Sigma$ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in $N$.

Speaker: Álvaro Krüger Ramos, UFRGS-Brazil
December 14, 2016
2:30pm - 3:30pm
Alternating projections for phase retrieval with random sensing vectors

Phase retrieval problems are a subclass of low-rank matrix recovery problems, that have been studied for a long time because of their important applications in physics. They have traditionally been solved with non-convex algorithms, that came with no theoretical convergence guarantees, and were known to sometimes get stuck in local optima. In the past few years, new algorithms have been proposed, that provably reconstruct the true solution with high probability for random instances of phase retrieval problems. We will show that, in this "random" setting, the most well-known traditional algorithm actually enjoys essentially the same convergence guarantees as the newer methods, provided that it is preceded by a suitable initialization procedure. We will discuss the importance of the initialization.

Speaker: Irene Waldspurger, MIT
Location:
Fine Hall 224

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