In this talk I review the recent discovery of Yangian symmetry for massive Feynman integrals and how it can be used to set up a Yangian Bootstrap. I will provide elementary proofs of the symmetry at one and two loops, whereas at generic loop order I conjecture that all graphs cut from regular tilings of the plane with massive propagators on the boundary enjoy the symmetry. After demonstrating how the symmetry may be used to constrain the functional form of Feynman integrals on explicit examples, I comment on how a subset of the diagrams for which the symmetry is conjectured to hold is naturally embedded in a Massive Fishnet theory that descends from gamma-deformed Coulomb branch N=4 Super-Yang-Mills theory in a particular double scaling limit.

Saturated fusion systems capture and abstract conjugacy in $p$-subgroups of finite groups and have recently found application in finite group theory, representation theory and algebraic topology. In this talk, we describe a situation in which we may identify a rank $2$ amalgam within $\mathcal{F}$ and, using some local group theoretic techniques, completely determine $\mathcal{F}$ up to isomorphism.

We consider the random directed graph $D(n,p)$ with vertex set $\{1,2,…,n\}$ in which each of the $n(n-1)$ possible directed edges is present independently with probability $p$. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at $p = 1/n$, with critical window $p = 1/n + \lambda n-4/3$ for $\lambda \in \mathbb{R}$. We show that, within this critical window, the strongly connected components of $D(n,p)$, ranked in decreasing order of size and rescaled by $n-1/3$, converge in distribution to a sequence $(C_1,C_2,\ldots)$ of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs in the sense of an $L^1$ sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős-Rényi random graph $G(n,p)$, whose scaling limit is well understood. We show that the limiting sequence $(C_1,C_2,\ldots)$ contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.

I will discuss a class of generalized global symmetries, which we call “Chern-Weil global symmetries,” that arise ubiquitously in gauge theories. The Noether currents of these Chern-Weil global symmetries are given by wedge products of gauge field strengths and their conservation follows from Bianchi identities, so they are not easy to break. However, exact global symmetries should not be allowed in a consistent theory of quantum gravity. I will explain how these symmetries are typically gauged or broken in string theory. Interestingly, many familiar phenomena in string theory, such as axions, Chern-Simons terms, worldvolume degrees of freedom, and branes ending on or dissolving in other branes, can be interpreted as consequences of the absence of Chern-Weil symmetries in quantum gravity, suggesting that they might be general features of quantum gravity.

Chiral fermion anomalies in any spacetime dimension are computed by evaluating an eta-invariant on a closed manifold in one higher dimension. The APS index theorem then implies that both local and global gauge anomalies are detected by bordism invariants, each being classified by certain abelian groups that I will identify. Mathematically, these groups are connected via a short exact sequence that splits non-canonically. This enables one to relate global anomalies in one gauge theory to local anomalies in another, by which we revive (from the bordism perspective) an old idea of Elitzur and Nair for deriving global anomalies. As an example, I will show how the SU(2) anomaly in 4d can be derived from a local anomaly by embedding SU(2) in U(2).

We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties.

Our perspective furnishes a common generalisation, essentially solving the problem. We also give a geometric criterion for Weyl group elements to yield strictly transverse slices.

We will discuss several versions of Ito’s formula in the case where the function is path dependent and only concave or C1 in the sense of Dupire. In particular, we will show that it can be used to solve (super) hedging problems, in the context of market impact or under volatility uncertainty. 

I will discuss regularity properties for solutions of linear second order non-uniformly elliptic equations in divergence form. Assuming certain integrability conditions on the coefficient field, we obtain local boundedness and validity of Harnack inequality. The assumed integrability assumptions are sharp and improve upon classical results due to Trudinger from the 1970s.

As an application of the local boundedness result, we deduce a quenched invariance principle for random walks among random degenerate conductances. If time permits I will discuss further regularity results for nonlinear non-uniformly elliptic variational problems.