University of Birmingham

We introduce a non-probabilistic, path-by-path framework for continuous-time, path-dependent portfolio allocation. Extending the self-financing concept recently introduced in Chiu & Cont (2023), we characterize self-financing portfolio allocation strategies through a path-dependent PDE and provide explicit solutions for the portfolio value in generic markets, including price paths that are not necessarily continuous or exhibit variation of any order.

As an application, we extend an aggregating algorithm of Vovk and the universal algorithm of Cover to continuous-time meta-algorithms that combine multiple strategies into a single strategy, respectively tracking the best individual and the best convex combination of strategies. This work extends Cover’s theorem to continuous-time without probability.

Plethysms lie at the intersection of representation theory and algebraic combinatorics. We give a recursive formula for a family of plethysm coefficients encompassing those involved in Foulkes' Conjecture. We also describe some applications, such as to the stability of plethysm coefficients and Sylow branching coefficients for symmetric groups. This is joint work with Y. Okitani.

In this talk I will discuss the representation theory of truncated current Lie algebras in prime characteristic. I will first give an introduction to modular representation theory for general restricted Lie algebras and introduce the Kac-Weisfeiler conjectures. Then I will introduce a family of Lie algebras known as truncated current Lie algebras, and discuss their representation theory and its relationship with the representation theory of reductive Lie algebras in positive characteristic.

Magnant and Martin conjectured that every $d$-regular graph on $n$ vertices can be covered by $n/(d+1)$ vertex-disjoint paths. Gruslys and Letzter verified this conjecture in the dense case, even for cycles rather than paths. We prove the analogous result for directed graphs and oriented graphs, that is, for all $\alpha>0$, there exists $n_0=n_0(\alpha)$ such that every $d$-regular digraph on $n$ vertices with $d \ge \alpha n $ can be covered by at most $n/(d+1)$ vertex-disjoint cycles. Moreover if $G$ is an oriented graph, then $n/(2d+1)$ cycles suffice. This also establishes Jackson's long standing conjecture for large $n$ that every $d$-regular oriented graph on $n$ vertices with $n\leq 4d+1$ is Hamiltonian.
This is joint work with Viresh Patel and  Mehmet Akif Yıldız.

In the last few years, major advances have been made in our understanding of the representation theory of reductive algebraic groups over algebraically closed fields of positive characteristic. Four key tools which are central to this progress are highest weight theory, reduction to the principal block, wall-crossing functors, and tilting modules. When considering instead the representation theory of the Lie algebras of these algebraic groups, more subtleties arise. If we look at those modules whose p-character is in so-called standard Levi form we are able to recover the four tools mentioned above, but they have been less well-studied in this setting. In this talk, we will explore the similarities and differences which arise when employing these tools for the Lie algebras rather than the algebraic groups. This research is funded by a research fellowship from the Royal Commission for the Exhibition of 1851.

Isogeny-based cryptography is a candidate for post-quantum cryptography. The underlying hardness of isogeny-based protocols is the problem of computing endomorphism rings of supersingular elliptic curves, which is equivalent to the path-finding problem on the supersingular isogeny graph. Can path-finding be reduced to knowing just one endomorphism? An endomorphism gives an explicit orientation of a supersingular elliptic curve. In this talk, we use the volcano structure of the oriented supersingular isogeny graph to take ascending/descending/horizontal steps on the graph and deduce path-finding algorithms to an initial curve. This is joint work with Sarah Arpin, Kristin E. Lauter, Renate Scheidler, Katherine E. Stange and Ha T. N. Tran.

The so called Drinfeld conjecture states that the complement to very stable bundles has pure codimension one in the moduli space of vector bundles. In this talk I will explain a constructive proof in rank three, and discuss if/how it generalises to wobbly fixed points of the nilpotent cone as defined by Hausel and Hitchin. This is joint work with Pauly (Nice).

Let $K$ be an algebraically closed field. Given three elements of some Lie algebra over $K$, we say that these elements form an $sl_2$-triple if they generate a subalgebra which is a homomorphic image of $sl_2(K).$ In characteristic 0, the Jacobson-Morozov theorem provides a bijection between the orbits of nilpotent elements of the Lie algebra and the orbits of $sl_2$-triples. In this talk I will discuss the progress made in extending this result to fields of characteristic $p$. In particular, I will focus on the results in classical Lie algebras, which can be found as subsets of $gl_n(K)$.