The Gray Scott model is a set of reaction diffusion equations known to generate a wide variety of patterns. In this talk we consider a version of this model where diffusion is assumed to be nonlocal and can be described by convolution kernels that decay exponentially at infinity and have finite second moment. We prove the local well-posedness of the model on bounded one-dimensional domains with nonlocal Dirichlet and Neumann boundary constraints. We also present a numerical scheme that uses a quadrature-based finite difference to discretize the convolution operator. We show how the scheme allows us to approximate solutions to the nonlocal Gray Scott model both on bounded and unbounded domains.

We study unstable topological complex vector bundles over complex projective spaces. It is a classical problem in algebraic topology to count the number of rank $r$ bundles over $\mathbb{C}P^n$ (with $1 < r < n$) having fixed Chern class data. A particular case is when the Chern data is trivial, which we call the vanishing Chern enumeration. We apply a modern tool, Weiss calculus, to produce the vanishing Chern enumeration in the first two unstable cases (which belong to what we call the "metastable" range, following Mark Mahowald), namely rank $(n - 1)$ bundles over $\mathbb{C}P^n$ for $n > 2$, and rank $(n - 2)$ bundles over $\mathbb{C}P^n$ for $n > 3$.

The talk will describe some problems which arise in finding optimal quantum strategies for games.  In such problems one has a (noncommutative) algebra A which encodes quantum mechanical laws and a noncommutative polynomial b which corresponds to a particular game and tells its score.  The goal of the talk will be to give an idea of some  of the structure, methods and our results which arise in maximizing b.  To get more warning of what is in the talk see the last few years of my postings on arXiv with collaborators: Adam Bene Watts, Igor Klep and Vern Paulsen etal.

I will report on work in progress showing that the geodesic flow on any geometrically finite, rank one, locally symmetric space is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure of maximal entropy. The method is coding-free and is instead based on a spectral study of transfer operators on suitably constructed anisotropic Banach spaces, ala Gouezel-Liverani, to take advantage of the smoothness of the flow. As a consequence, we obtain more precise information on the size of the essential spectral gap as well as the meromorphic continuation properties of Laplace transforms of correlation functions.

The lecture will discuss recent joint work with Costante Bellettini at UCL. A main outcome of the work is a proof that for any closed Riemannian manifold $N$ of dimension $n \geq 3$ and any non-negative (or non-positive) Lipschitz function $g$ on $N$, there is a boundaryless $C^{2}$ hypersurface $M \subset N$ whose scalar mean curvature is prescribed by $g.$ More precisely, the hypersurface $M$ is the image of a quasi-embedding $\iota$ (of class $C^{2}$) admitting a global unit normal $\nu$ such that the mean curvature of $\iota$ at every point $x$ is $g(\iota(x))\nu(x)$. Here a 'quasi-embedding' is an immersion such that any point of its image near which the image is not embedded has an ambient neighborhood in which the image is the union of two $C^{2}$ embedded disks with each disk lying on one side of the other (so that any self-intersection is tangential). If $n \geq 7$, the singular set $\overline{M} \setminus M$ may be non-empty, but has Hausdorff dimension no greater than $n-7$. An important special case is the existence of a CMC hypersurface for any prescribed value of mean curvature. The method of proof is PDE theoretic. It utilises the elliptic and parabolic Allen-Cahn equations on $N$, and brings to bear on the question elementary, and yet very effective, variational and gradient flow principles in semi-linear elliptic and parabolic PDE theory--principles that serve as a conceptually and technically simpler alternative to the Geometric Measure Theory machinery pioneered by Almgren and Pitts to prove existence of a minimal hypersurface. For regularity conclusions the method relies on a new varifold regularity theory, a ''black-box'' tool of independent interest (also joint work with Bellettini). This theory provides multi-sheeted $C^{1, \alpha}$ regularity for mean-curvature-controlled codimension 1 integral varifolds $V$ near points where one tangent cone is a hyperplane of multiplicity $q \geq 2;$ this regularity holds whenever: (i) $V$ has no classical-singularities, i.e. no portion of $V$ is the union of three or more embedded hypersurfaces-with-boundary coming smoothly together along their common boundary, and (ii) the region where the mass density of $V$ is $< q$ is 'well-behaved' in a certain topological sense. A very important feature of this theory, crucial for its application to the Allen--Cahn method, is that $V$ is not assumed to be a critical point of a functional.

Learning operators between infinitely dimensional spaces is an important learning task arising in wide applications in machine learning, data science, mathematical modeling and simulations, etc. This talk introduces a new discretization-invariant operator learning approach based on data-driven kernels for sparsity via deep learning. Compared to existing methods, our approach achieves attractive accuracy in solving forward and inverse problems, prediction problems, and signal processing problems with zero-shot generalization, i.e., networks trained with a fixed data structure can be applied to heterogeneous data structures without expensive re-training. Under mild conditions, quantitative generalization error will be provided to understand discretization-invariant operator learning in the sense of non-parametric estimation.

Supersingular isogeny-based cryptosystems are strong contenders for post-quantum cryptography standardization. Such cryptosystems rely on the hardness of path-finding on supersingular isogeny graphs. The path-finding problem is known to reduce to the endomorphism ring problem. Can path-finding be reduced to knowing just one endomorphism? In this talk, we give explicit classical and quantum algorithms for path-finding to an initial curve using the knowledge of one endomorphism. An endomorphism gives an orientation of a supersingular elliptic curve. We use the theory of oriented supersingular isogeny graphs and algorithms for taking ascending/descending/horizontal steps on such graphs.

Interesting moduli spaces don't have many integral points. More precisely, if X is a variety over a number field, admitting a variation of Hodge structure whose associate period map is injective, then the number of S-integral points on X of height at most H grows more slowly than $H^ε$, for any positive ε.  This is a sort of weak generalization of the Shafarevich conjecture; it is a consequence of a point-counting theorem of Broberg, and the largeness of the fundamental group of X.  Joint with Ellenberg and Venkatesh.