The BSD conjecture predicts that a rational elliptic curve $E$ has infinitely many points if and only if its $L$-function vanishes at $s=1$.

There are $p$-adic versions of similar phenomena. If $E$ is $p$-ordinary, there is, for example, a $p$-adic analytic analogue $L_p(E,s)$ of the $L$-function, and if $E$ has good reduction, then it has infinitely many rational points iff $L_p(E,1) = 0$. However if $E$ has split multiplicative reduction at $p$ - that is, if $E/\mathbf{Q}_p$ admits a Tate uniformisation $\mathbf{C}_p^{\times}/q^{\mathbf{Z}}$ - then $L_p(E,1) = 0$ for trivial reasons, regardless of $L(E,1)$; it has an 'exceptional zero'. Mazur--Tate--Teitelbaum's exceptional zero conjecture, proved by Greenberg--Stevens in '93, states that in this case the first derivative $L_p'(E,1)$ is much more interesting: it satisfies $L_p'(E,1) = \mathrm{log}(q)/\mathrm{ord}(q) \times L(E,1)/(\mathrm{period})$. In particular, it should vanish iff $L(E,1) = 0$ iff $E(\mathbf{Q})$ is infinite; and even better, it has a beautiful and surprising connection to the Tate period $q$, via the 'L-invariant' $\mathrm{log}(q)/\mathrm{ord}(q)$.

In this talk I will discuss exceptional zero phenomena and L-invariants, and a generalisation of the exceptional zero conjecture to automorphic representations of GL(3). This is joint work in progress with Daniel Barrera and Andrew Graham.

Recently, work has been done to understand higher-form symmetries in linear gravity. Just like Maxwell theory, which has both electric and magnetic U(1) higher form symmetries, linearised gravity exhibits analogous structure. The authors of
[https://arxiv.org/pdf/2409.00178] investigate electric and magnetic higher form symmetries in linearised gravity, which correspond to shift symmetries of the graviton and the dual graviton respectively. By attempting to gauge the two symmetries, the authors investigate the mixed ’t Hooft anomalies anomaly structure of linearised gravity. Furthermore, if a specific shift symmetry is considered, the corresponding charges are related to Roger Penrose's quasi-local charge construction.

Based on: [https://arxiv.org/pdf/2410.08720][https://arxiv.org/pdf/2409.00178][https://arxiv.org/pdf/2401.17361]

For some values of degrees d=(d_1,...,d_c), we construct a compactification of a Hilbert scheme of complete intersections of type d. We present both a quotient and a direct construction. Then we work towards the construction of a quasiprojective coarse moduli space of smooth complete intersections via Geometric Invariant Theory.