A probabilistic Takens theorem

Yonatan Gutman
Institute of Mathematics of the Polish Academy of Sciences
May 16, 2019
Let $X \subset \R^N$ be a Borel set, $\mu$ a Borel probability measure on $X$ and $T:X \to X$ a Lipschitz and injective map. Fix $k \in \N$ greater than the (Hausdorff) dimension of $X$ and assume that the set of $p$-periodic points has dimension smaller than $p$ for $p=1, \ldots, k-1$. We prove that for a typical polynomial perturbation $\tilde{h}$ of a given Lipschitz map $h : X \to \R$, the $k$-delay coordinate map $x \mapsto (\tilde{h}(x), \tilde{h}(Tx), \ldots, \tilde{h}(T^{k-1}x))$ is injective on a set of full measure $\mu$.

Two Important Milestones in the History of the Universe: The Last Scattering Surface, the Black Body Photosphere of the Universe and Distortions of the CMB Spectrum

Rashid Sunyaev
Distinguished Visiting Professor, School of Natural Sciences, IAS; Max-Planck-Institute für Astrophysik; Space Research Institute, Moscow
May 15, 2019

Our Universe is filled with Cosmic Microwave Background (CMB) radiation having an almost perfect black body spectrum with a temperature of To=2.7K. The number density of photons in our Universe exceeds the number density of electrons by a factor of more than a billion. In the expanding Universe the temperature at early times was higher than today: Tr = To (1+z), where z is the redshift. 

Singularity formation for some incompressible Euler flows

Tarek Elgindi
University of California, San Diego
May 6, 2019

We describe a recent construction of self-similar blow-up solutions of the incompressible Euler equation. A consequence of the construction is that there exist finite-energy $C^{1,a}$ solutions to the Euler equation which develop a singularity in finite time for some range of $a>0$. The approach we follow is to isolate a simple non-linear equation which encodes the leading order dynamics of solutions to the Euler equation in some regimes and then prove that the simple equation has stable self-similar blow-up solutions.

Singularities in reductions of Shimura varieties

Thomas Haines
University of Maryland
May 2, 2019

The singularities in the reduction modulo $p$ of the modular
curve $Y_0(p)$ are visualized by the famous picture of two curves
meeting transversally at the supersingular points. It is a fundamental
question to understand the singularities which arise in the reductions
modulo $p$ of integral models of Shimura varieties. For PEL type
Shimura varieties with parahoric level structure at $p$, this question
has been studied since the 1990's. Due to the recent construction of

Globally Consistent Three-family Standard Models in F-theory

Mirjam Cvetic
University of Pennsylvania
April 29, 2019

We present recent advances in constructions of globally consistent 
F-theory compactifications with the exact chiral spectrum of the minimal 
supersymmetric Standard Model. We highlight the first such example and 
then turn to a subsequent systematic exploration of the landscape of 
F-theory three-family Standard Models with a gauge coupling unification. 
Employing algebraic geometry techniques, all global consistency 
conditions of these models can be reduced to a single criterion on the 

Anomalies in the Space of Couplings and Dynamical Applications

Clay Cordova
Member, School of Natural Sciences, IAS
April 26, 2019

Anomalies are invariants under renormalization group flow which lead to powerful constraints on the phases of quantum field theories.  I will explain how these ideas can be generalized to families of theories labelled by coupling constants like the theta angle in gauge theory.  Using these ideas we will be able to prove that certain systems, such as Yang-Mills theory in 4d, necessarily have a phase transition as these parameters are varied.  We will also show how to use the same ideas to constrain the dynamics of defects where coupling constants vary in spacetime.