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. 

The classification of finite simple groups shows that many (those of Lie type) are obtained as (projectivisations of) subgroups of some $GL_{n}(\mathbb{F}_{q})$.

Green first determined the character table of any $GL_{n}(\mathbb{F}_{q})$ in his influential 1955 paper, while others have since given more explicit constructions of certain `cuspidal' representations.

In this talk, I will introduce parabolic induction as a means of obtaining representations of $GL_{n}(\mathbb{F}_{q})$ from those of $GL_{m}(\mathbb{F}_{q})$ where $m<n$.

Finding the irreducible representations of any $GL_{n}(\mathbb{F}_{q})$ then becomes inductive on $n$ for fixed $q$, with the cuspidal representations serving as atoms for this process.

Harish-Chandra's philosophy of cusp forms reduces the problem to the following two steps:

•  Find the cuspidal representations of any $GL_{n}(\mathbb{F}_{q})$
•  Determine the irreducible components of any representation $\sigma_{1}\circ\dots\circ\sigma_{k}$ parabolically induced from cuspidals $\sigma_{i}$

The majority of my talk will then aim to address each of these points.

The Harish-Chandra Lefschetz principle asserts representation theory for real groups, p-adic groups and automorphic forms should be placed on an equal footing. A particular example in this aspect is that Ciubotaru and Trapa constructed Arakawa-Suzuki type functors between category of Harish-Chandra modules and category of graded Hecke algebra modules, giving an explicit connection on the representation categories between p-adic and real sides. 

This talk plans to begin with comparing the representation theory between real and p-adic general linear groups, such as unitary and unipotent representations. Then I shall explain results in more details on the p-adic branching law from GL(n+1) to GL(n), including branching laws for Arthur type representations (one of the non-tempered Gan-Gross-Prasad conjectures). The analogous results and predictions on the real group side will also be discussed. Time permitting, I will explain a notion of left-right Bernstein-Zelevinsky derivatives and its applications on branching laws.
 

A proposal for the worldsheet string theory that is dual to free N=4 SYM in 4d will be explained. It is described by a free field sigma model on the twistor space of AdS5 x S5, and is a direct generalisation of the corresponding model for tensionless string theory on AdS3 x S3. I will explain how our proposal fits into the general framework of AdS/CFT, and review the various checks that have been performed.
 

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. I will also discuss how this correspondence is modified in more general contexts, including e.g. Chern-Simons terms.