arXiv subjects

Benjamin Aslan

Publications and source records attributed to Benjamin Aslan.

5 recordsLinked to original sources

arXiv2023

Calibrated Geometry in Hyperkahler Cones, 3-Sasakian Manifolds, and Twistor Spaces

We systematically study calibrated geometry in hyperk\"ahler cones $C^{4n+4}$, their 3-Sasakian links $M^{4n+3}$, and the corresponding twistor spaces $Z^{4n+2}$, emphasizing the relationships between submanifold geometries in various spaces. Our analysis emphasizes the role played by a canonical $\mathrm{Sp}(n)\mathrm{U}(1)$-structure $\gamma$ on the twistor space $Z$. We observe that $\mathrm{Re}(e^{- i \theta} \gamma)$ is an $S^1$-family of semi-calibrations, and make a detailed study of their associated calibrated geometries. As an application, we obtain new characterizations of complex Lagrangian and complex isotropic cones in hyperk\"{a}hler cones, generalizing a result of Ejiri and Tsukada. We also generalize a theorem of Storm on submanifolds of twistor spaces that are Lagrangian with respect to both the K\"{a}hler-Einstein and nearly-K\"{a}hler structures.

math.DG↗

arXiv2023

On $G_2$ manifolds with cohomogeneity two symmetry

We consider $G_2$ manifolds with a cohomogeneity two $\mathbb{T}^2\times \mathrm{SU}(2)$ symmetry group. We give a local characterization of these manifolds and we describe the geometry, including regularity and singularity analysis, of cohomogeneity one calibrated submanifolds in them. We apply these results to the manifolds recently constructed by Foscolo--Haskins--Nordstr\"om and to the Bryant--Salamon manifold of topology $S^3\times \mathbb{R}^4$. In particular, we describe new large families of complete $\mathbb{T}^2$-invariant associative submanifolds in them.

math.DG↗

arXiv2022

Special Lagrangians in nearly K\"ahler $\mathbb{CP}^3$

This article explores special Lagrangian submanifolds in $\mathbb{CP}^3$, viewed as a nearly K\"ahler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using $\mathrm{SU}(2)$ moment-type maps. We describe new homogeneous examples, from both perspectives, and classify totally geodesic special Lagrangian submanifolds. We show that every special Lagrangian in $\mathbb{CP}^3$, or the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$ admitting a symmetry of an $\mathrm{SU}(2)$ subgroup of nearly K\"ahler automorphisms is automatically homogeneous.

math.DG↗

arXiv2022

Group invariant machine learning by fundamental domain projections

We approach the well-studied problem of supervised group invariant and equivariant machine learning from the point of view of geometric topology. We propose a novel approach using a pre-processing step, which involves projecting the input data into a geometric space which parametrises the orbits of the symmetry group. This new data can then be the input for an arbitrary machine learning model (neural network, random forest, support-vector machine etc). We give an algorithm to compute the geometric projection, which is efficient to implement, and we illustrate our approach on some example machine learning problems (including the well-studied problem of predicting Hodge numbers of CICY matrices), in each case finding an improvement in accuracy versus others in the literature. The geometric topology viewpoint also allows us to give a unified description of so-called intrinsic approaches to group equivariant machine learning, which encompasses many other approaches in the literature.

cs.LG↗

arXiv2021

Transverse $J$-holomorphic curves in nearly K\"ahler $\mathbb{CP}^3$

$J$-holomorphic curves in nearly K\"ahler $\mathbb{CP}^3$ are related to minimal surfaces in $S^4$ as well as associative submanifolds in $\Lambda^2_-(S^4)$. We introduce the class of transverse $J$-holomorphic curves and establish a Bonnet-type theorem for them. We classify flat tori in $S^4$ and construct moment-type maps from $\mathbb{CP}^3$ to relate them to the theory of $\mathrm{U}(1)$-invariant minimal surfaces on $S^4$.

math.DG↗