We survey some of the applications of generalized indiscernible sequences, both in model theory and in structural Ramsey theory.  Given structures $A$ and $B$, a semi-retraction is a pair of  quantifier-free type respecting maps $f: A \rightarrow B$ and $g: B \rightarrow A$ such that $g \circ f: A \rightarrow A$ is quantifier-free type preserving, i.e. an embedding.  In the case that $A$ and $B$ are locally finite ordered structures, if $A$ is a semi-retraction of $B$ and the age of $B$ has the Ramsey property, then the age of $A$ has the Ramsey property.

The index of a saddle point of a smooth function is the number of descending directions of the saddle. While the index can usually be retrieved by counting the number of negative eigenvalues of the Hessian at the critical point, we may not have the luxury of having second derivatives in data deriving from practical applications. To address this problem, we develop a computational pipeline for estimating the index of a non-degenerate saddle point without explicitly computing the Hessian. In our framework, we only require a sufficiently dense sample of level sets of the function near the saddle point. Using techniques in Morse theory and Topological Data Analysis, we show how the shape of saddle points can help us infer the index of the saddle. Furthermore, we derive an explicit upper bound on the density of point samples necessary for inferring the index depending on the curvature of level sets. 

I will give a pedagogical introduction to the Cardy-like limit of the superconformal index of N=4 SYM and generic N=1 SCFTs, highlighting its role in the holographic dual black hole microstate counting problem.

The Alfven waves are fundamental wave phenomena in magnetized plasmas and the dynamics of Alfven waves are governed by the MHD system. In the talk,  we construct and study the long time behavior of (viscous and non-viscous) Alfven waves.

As applications, (1) We provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution at the beginning behave like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number); thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects;

(2) We prove the rigidity aspects of the scattering problem for the MHD equations: We prove that the Alfven waves must vanish if their scattering fields vanish at infinities.

There is a striking relationship between Willmore surfaces of revolution and elastic curves in hyperbolic half-space. Here the term elastic curve refer to a critical point of the energy given by the integral of the curvature squared. In the talk we will discuss this relationship and use it to study long-time existence and asymptotic behavior for the L2-gradient flow of the Willmore energy, under the condition that the initial datum is a torus of revolution. As in the case of Willmore flow of spheres, we show that if an initial datum has Willmore energy below 8 \pi then the solution of the Willmore flow converges to the Clifford Torus, possibly rescaled and translated. The energy threshold of 8 \pi turns out to be optimal for such a convergence result. 

The lecture is based on joint work with M. Müller (Univ. Freiburg), R. Schätzle (Univ. Tübingen) and A. Spener (Univ. Ulm).

There are many open conjectures about the algebraic behaviour of transcendental functions in arithmetic geometry, one of which is the Existential Closedness problem. In this talk I will review recent developments made on this question: the cases where we have unconditional existence of solutions, the conditional existence of generic solutions (depending on the conjecture of periods and Zilber-Pink), and even a few cases of unconditional existence of generic solutions. Many of the results I will mention are joint work with (different subsets of) Vahagn Aslanyan, Jonathan Kibry, Sebastián Herrero, and Roy Zhao. 

In this talk, I will discuss AdS super gluon scattering amplitudes in various spacetime dimensions. These amplitudes are dual to correlation functions in a variety of non-maximally supersymmetric CFTs, such as the 6d E-string theory, 5d Seiberg exceptional theories, etc. I will introduce a powerful method based on symmetries and consistency conditions, and show that it fixes all the infinitely many four-point amplitudes at tree level. I will also point out many interesting properties and structures of these amplitudes, which include the flat space limit, Parisi-Sourlas-like dimensional reduction, hidden conformal symmetry, and a color-kinematic duality in AdS. Along the way, I will also review some earlier progress and the relation with this work. I will conclude with a brief discussion of various open problems.