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The 7-shere S^7 has a squashed Einstein metric that is related to both G_2 and Spin(7) holonomy. Like its cousin, the Berger space SO(5)/SO(3), it admits an action by SO(4) of cohomogeneity one. This motivates the search for new such "nearly parallel" 7-manifolds, in analogy to the situation for nearly Kähler 6-manifolds. 

Small eigenvalues of hyperbolic surfaces have attracted the attention of mathematicians from a wide variety of research areas, including number theory and spectral geometry. In this talk, I will give a brief history of this subject, leading to some very recent results on the stability of these eigenvalues under finite-sheeted Riemannian coverings. 

For any lattice polarised elliptic K3 surface, van Geemen's Brauer twist construction associates to any order 2 element in its Brauer group another elliptic K3 surface, where the original K3 surface can be recovered by taking the relative Jacobian fibration. We will give explicit geometric constructions of some of the Brauer twists of a very general K3 surface that admits a van Geemen-Sarti involution, as well as their birational models. We also observe the same construction works to geometrically realise the Brauer twists of K3 surfaces in some families of higher Picard ranks. This is an ongoing work with Adrian Clingher and Andreas Malmendier. 

In the 18th century, Euler introduced the number p(n) of 
partitions of an integer n into positive parts (e.g. p(4)=5 because 
4 can be partitioned in five ways: 1+1+1+1, 1+1+2, 1+3, 2+2, or 4) 
and its generating function  p(0)+p(1)x+p(2)x^2+..., whose reciprocal 
later turned out to be the simplest example of a "modular form". 
A question arising in the theory of gravity and integrable systems 
required studying the numbers of partitions of a number into positive 
squares or higher powers. The corresponding generating fuctions are 
no longer modular, but satisfy certain surprising "modular-type" 
transformation equations. There are also analogues of the famous 
Hardy-Ramanujan approximate partition formula, but far more subtle. 
Both results were discovered by numerical computations together with 
some nice tricks.  The only prerequisites for the talk are basic 
mathematical background and some interest in numerical techniques. 

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A Hessian manifold is a Riemannian space where the infinitesimal metric is locally determined by the second derivatives of a convex potential function. This talk reveals how the local geometric structure of a Hessian manifold severely restricts its global shape and topology. We establish fundamental constraints on these spaces and introduce new invariants to study them. Our results provide a powerful unifying theorem for a special class of Hessian manifolds, connecting their existence to deep properties in Riemannian, symplectic, and complex geometry. As an application, this framework proves that any compact, curved example must be a mapping torus, and finally leads to a complete classification in low dimensions.

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Necessary and sufficient conditions for the existence of polyhedral Kähler metrics whose singular set is a hyperplane arrangement and whose cone angles are in (0,2π) will be discussed. These conditions take the form of linear and quadratic constraints on the cone angles and are entirely determined by the intersection poset of the arrangement. The proof of existence relies on a parabolic version of the Kobayashi-Hitchin correspondence, due to T. Mochizuki. Based on joint work with Dmitry Panov. 

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The notion of shadow sequences was introduced by Ovsienko, motivated by the search (with Tabachnikov) for super cluster algebras. In the case of supersymmetry defined by a Grassmann algebra with two odd anticommuting generators, the even part of the algebra is commutative and isomorphic to the dual numbers, being equivalent to an extension of the ground field by a nilpotent element $\epsilon$ with $\epsilon^2 =0$. Ovsienko observed that various recurrence relations exhibiting the Laurent phenomenon admit extensions to the dual numbers, and the associated integer sequences have shadows in the form of order $\epsilon$ components. In this talk we discuss shadows and noncommutative extensions of integrable maps associated with Stieltjes continued fractions, which correspond to finite genus solutions of the Volterra lattice. In particular, we construct the explicit analytic solution of the shadow Somos-5 recurrence, and an associated dual QRT map which we show to be Liouville integrable in 4D. We then move onto the so-called Kontsevich map, which is a discrete symmetry of a noncommutative vector field whose Lax pair was derived by Efimovskaya & Wolf. We explain how the latter corresponds to special solutions of the noncommutative Volterra lattice, and describe some new integrable maps in 4D & 5D which arise from the realization of this system in GL_2 x GL_2. This is based on results obtained with Joe Harrow, and work in progress with Sasha Mikhailov, Pol Vanhaecke and Jing Ping Wang.   

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