In a recent breakthrough Campos, Griffiths, Morris and Sahasrabudhe obtained the first exponential improvement of the upper bound on the classical Ramsey numbers since 1935. I will outline a reinterpretation of their proof, replacing the underlying book algorithm with a simple inductive statement. In particular, I will present a complete proof of an improved upper bound on the off-diagonal Ramsey numbers and describe the main steps involved in improving their upper bound for the diagonal Ramsey numbers to $R(k,k)\le(3.8)^k$ for sufficiently large $k$.

Based on joint work with Parth Gupta, Ndiame Ndiaye, and Louis Wei.

Let $(V(u))$ be a branching random walk and $(\eta(u))$ be i.i.d marks on the vertices. To each path $\xi$ of the tree, we associate the discounted sum $D(\xi)$ which is the sum of the $\exp(V(u))\eta_u$ along the path. We study conditions under which $\sup_\xi D(\xi)$ is finite, answering an open question of Aldous and Bandyopadhyay. We will see that this problem is related to the study of the local time process of the branching random walk along a path. It is a joint work with Yueyun Hu and Zhan Shi.

The celebrated Szemeredi-Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $\mathcal{O}(n^{4/3})$, which is asymptotically tight.

Solymosi conjectured that this bound drops to $o(n^{4/3})$ if we exclude subconfigurations isomorphic to any fixed point-line configuration. We describe a construction disproving this conjecture. On the other hand, we prove new upper bounds on the number of incidences for configurations that avoid certain subconfigurations. Joint work with Martin Balko.

A successful strategy to handle problems involving primes is to approximate them by a more 'simple' function. Two aspects need to be balanced. On the one hand, the approximant should be simple enough so that the considered problem can be solved for it. On the other hand, it needs to be close enough to the primes in order to make it an admissible to replacement. In this talk I will present how one can construct general approximants in the context of the Circle Method and will use this to give a different perspective on Goldbach type applications.

In "Higher Operations in Perturbation Theory", Gaiotto, Kulp, and Wu discussed Feynman integrals that control certain deformations in quantum field theory. The corresponding integrands are differential forms in Schwinger parameters. Specifically, the integrand $\alpha$ is associated to a single topological direction of the theory.
I will show how the combinatorial properties of graph polynomials lead to a relatively simple, explicit formula for $\alpha$, that can be evaluated quickly with a computer. This is interesting for two reasons. Firstly, knowing the explicit formula leads to an elementary proof of the fact that $\alpha$ squares to zero, which asserts the absence of quantum corrections in topological field theories of two (or more) dimensions, known as Kontsevich's formality theorem. Secondly, the underlying constructions and proofs are not intrinsically limited to topological theories. In this sense, they serve as a particularly instructive example for simplifications that can occur in Feynman integrals with numerators.

 I will first give a short review of the concepts of Asymptotically Flat Spacetimes, IR triangle and Noether's theorems. I will then present what Asymptotic Higher Spin Symmetries are and how they were introduced as a candidate for an approximate symmetry of General Relativity and the S-matrix. Next, I'll move on to the recent developments of establishing these symmetries as Noether symmetries and describing how they are canonically and non-linearly realized on the asymptotic gravitational phase space. I will discuss how the introduction of dual equations of motion encapsulates the non-perturbativity of the analysis. Finally I'll emphasize the relation to twistor, especially with 2407.04028. Based on 2409.12178 and 2410.15219

Physics beyond relativistic invariance and without Lorentz (or Poincaré) symmetry and the geometry underlying these non-Lorentzian structures have become very fashionable of late. This is primarily due to the discovery of uses of non-Lorentzian structures in various branches of physics, including condensed matter physics, classical and quantum gravity, fluid dynamics, cosmology, etc. In this talk, I will be talking about one such theory - Carrollian theory, where the Carroll group replaces the Poincare group as the symmetry group of interest. Interestingly, any null hypersurface is a Carroll manifold and the Killing vectors on the null manifold generate Carroll algebra. Historically, Carroll group was first obtained from the Poincaré group via a contraction by taking the speed of light going to zero limit as a “degenerate cousin of the Poincaré group”.  I will shed some light on Carrollian fermions, i.e. fermions defined on generic null surfaces. Due to the degenerate nature of the Carroll manifold, there exist two distinct Carroll Clifford algebras and, correspondingly, two different Carroll fermionic theories. I will discuss them in detail. Then, I will show some examples; when the dispersion relation becomes trivial, i.e. energy bands flatten out, there can be a possibility of the emergence of Carroll symmetry.