Seminars Sorted by Series

We consider the problem of finding complete conformal metrics determined by a symmetric function of Ricci tensor in a negative convex cone on compact manifolds. A consequence of our main results is that any smooth bounded domain in Euclidean space...

I will discuss how some questions in conformal geometry can be answered using twistor theory. One such application is to the classification of locally conformally flat Hermitian surfaces. Another application is to determining the conformal...

A well-known example by N. N. Ural'tseva suggests that for fixed p > 2 there is no unique $W^2_p$-solvability of elliptic equations under p > the condition that the leading coefficients are measurable in two spatial variables. We will present a...

It has been known for a long time that varieties of general type are the most complicated algebraic varieties. It is only recently, however, that the correct definition of the "simplest" algebraic varieties was established. These are the rationally...

Hermann Weyl Lecture

Lecture 4. The Ax conjecture and degenerations. Originating with his studies on the logic of finite fields, James Ax conjectured that every PAC field is $C_1$. The recent proof of this in characteristic 0 is connected with degenerations of...

Summary: These lectures are devoted to the interplay between cohomology and Chow groups of a complex algebraic variety, and also to the consequences, on the topology of a family of smooth projective varieties, of statements concerning Chow groups of...

Random graphs and expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We formalize this notion of approximation and ask how well an arbitrary graph can be...

We will explain our recent solution of the Kadison-Singer Problem and the equivalent Bourgain-Tzafriri and Paving Conjectures. We will begin by introducing the method of interlacing families of polynomials and use of barrier function arguments to...

We explain what Ramanujan graphs are, and prove that there exist infinite families of bipartite Ramanujan graphs of every degree. Our proof follows a plan suggested by Bilu and Linial, and exploits a proof of a conjecture of theirs about lifts of...

These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In...

These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In...

These lectures will be about enumerative K-theory of curves (and more general 1-dimensional sheaves) in algebraic threefolds. In the first lecture, we will set up the enumerative problem and survey what we know and what we conjecture about it. In...

On the reality of black holes. I will give a quick introduction to the initial value problem in GR and overview of the problems of Rigidity, Stability and Collapse and how they fit with regard to the Final State Conjecture.The gravitational waves...

I will discuss in some detail the main difficulties of the problem of nonlinear stability of black holes and the recent advances on the related issue of linear stability.The gravitational waves detected recently by LIGO were produced in the final...

I will discuss a recent result in collaboration with J. Szeftel concerning the nonlinear stability of the Schwarzschild spacetime under axially symmetric, polarized perturbations.The gravitational waves detected recently by LIGO were produced in the...

This introductory lecture will describe results about counting rational points on certain non-algebraic sets and sketch how they can be used to attack certain problems in diophantine geometry and functional transcendence.

This lecture will describe the historical context and some key properties of o-minimality. It will then describe certain results in functional transcendence, generalizing the classical results on exponentiation due to Ax, and sketch how they can be...

The Zilber-Pink conjecture is a far reaching finiteness conjecture in diophantine geometry, unifying and extending Mordell-Lang and Andre-Oort. This lecture will state the conjecture, illustrate its varied faces, and indicate how the point-counting...

The PCP theorem says that any mathematical proof can be written in a special "PCP" format such that it can be verified, with arbitrarily high probability, by sampling only a few symbols in the proof. Hence the name, Probabilistically Checkable...

High dimensional expansion generalizes edge and spectral expansion in graphs to higher dimensional hypergraphs or simplicial complexes. Unlike for graphs, it is exceptionally rare for a high dimensional complex to be both sparse and expanding. The...

The unique games conjecture gives a very strong PCP theorem, which, if true, leads to a clean understanding of a broad family of approximation problems. We will describe recent progress on the conjecture and how certain type of expansion and...

The 4th Clay Millenium problem has a very simple formulation: may viscous incompressible fluids form a singularity in finite time? The answer is no in dimension two as proved by Leray in 1932, but the three dimensional problem is out of reach. More...

In the last forty years, the study of singularity formation has mostly concerned model problems and focusing non linearities. In this second lecture, we will try to give a unified overview on the known mechanisms of singularity formation, with in...

The defocusing Non Linear Schrödinger equation $iu_{t} = \Delta {u-u}|u|^{p-1}$ is a classical model of mathematical physics. For energy subcritical non linearities, Ginibre and Velo proved in the ’80s that all solutions are global in time and...

Given several real numbers $\alpha_1,...,\alpha_k$, how well can you simultaneously approximate all of them by rationals which each have the same square number as a denominator? Schmidt gave a clever iterative argument which showed that this can be...

Given any non-negative function $\f:\mathbb{Z}\rightarrow\mathbb{R}$, it follows from basic ergodic ideas that either 100% of real numbers $\alpha$ have infinitely many rational approximations $a/q$ with $a,q$ coprime and $|\alpha-a/q|

I'll describe a recent resolution of this conjecture, which recasts the problem in combinatorial language, and then uses a general 'structure vs randomness' principle combined with an iterative argument to solve this combinatorial problem.

Many important consequences of the Riemann Hypothesis would remain true even if there were some zeros off the critical line, provided these exceptions to the Riemann Hypothesis are suitably rare. We can unconditionally prove some results on the...

Anisha Thomas is hosting the festival of Holi for the IAS community.