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The SYZ Conjecture explains Mirror Symmetry between mirror Calabi-Yau 3-folds M,M' in terms of special Lagrangian fibrations f : M --> B and f' : M' --> B over the same base B, whose fibres are dual 3-tori, except for singular fibres. One…
We study the relation between various notions of exterior convexity introduced in Bandyopadhyay-Dacorogna-Sil \cite{BDS1} with the classical notions of rank one convexity, quasiconvexity and polyconvexity. To this end, we introduce a…
In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous;…
In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the…
The geometrical theory of partial differential equations in the absolute sense, without any additional structures, is developed. In particular the symmetries need not preserve the hierarchy of independent and dependent variables. The order…
Motivated by works on extension sets in standard domains we introduce a notion of the Carath\'eodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a…
This paper is a sequel to [3]. We formulate a natural algebraic geometry conjecture, give some of its number theoretic and analytical consequences, and show that those can be used to get further advances in wave turbulence theory.
The coupling of topological matter to topological Yang-Mills theory in four dimensions is considered and a model is presented. It is shown that, contrary to the two-dimensional case, this coupling leads to a breaking of the topological…
In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also…
For 34 years it has been known that chiral supersymmetry (SUSY) in 3+1 dimensions has a very large Becchi-Rouet-Stora (BRS) cohomology space at ghost charge one. This suggests that there might be corresponding SUSY anomalies coming from…
The relationship between mathematics and physics has long been an area of interest and speculation. Subscribing to the recent definition by Tegmark, we present a mathematical structure involving the only division rings - the real,…
This paper analyses the convergence and degeneration of sequences of metrics on a 3-manifold, and relations of such with Thurston's geometrization conjecture. The sequences are minimizing sequences for a certain (optimal) scalar-curvature…
In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We…
A speculative discussion on the role of various symmetries in physics.
We give an overview of the constrained Willmore problem and address some conjectures arising from partial results and numerical experiments. Ramifications of these conjectures would lead to a deeper understanding of the Willmore functional…
Given a finite CW complex $K$, we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embedding $K$ into a Euclidean space $\mathbb{R}^d$. For $2$-dimensional complexes in $\mathbb{R}^4$, a geometric analogue…
We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive…
Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of…
We consider the generalization of classical Blaschke's Problem to higher codimension case, characterizing Darboux pair of isothermic surfaces and dual S-Willmore surfaces as the only non-trivial surface pairs that envelop a 2-sphere…
There are many open problems and some mysteries connected to the realizations of the associahedra as convex polytopes. In this note, we describe three -- concerning special realizations with the vertices on a sphere, the space of all…