Oxford Mathematician Yinan Wang talks about his and colleagues' work on classification of elliptic Calabi-Yau manifolds and geometric solutions of F-theory.
12:00
Mysteries of isolated horizons
Abstract
Mysteries of isolated horizons: the Near Horizon Geometry equation, geometric characterizations of the non-extremal Kerr horizon, spacetimes foliated by non-expanding horizons.
3-dimensional null surfaces that are Killing horizons to the second order are considered. They are embedded in 4-dimensional spacetimes that satisfy the vacuum Einstein equations with arbitrary cosmological constant. Internal geometry of 2-dimensional cross sections of the horizons consists of induced metric tensor and a rotation 1-form potential. It is subject to the type D equation. The equation is interesting from the both, mathematical and physical points of view. Mathematically it involves geometry, holomorphic structures and algebraic topology. Physically, the equation knows the secrete of black holes: the only axisymmetric solutions on topological sphere correspond to the the Kerr / Kerr-de Sitter / Kerr-anti-de-Sitter non-extremal black holes or to the near horizon limit of the extremal ones. In the case of bifurcated horizons the type D equation implies another spacial symmetry. In this way the axial symmetry may be ensured without the rigidity theorem. The type D equation does not allow rotating horizons of topology different then that of the sphere (or its quotient). That completes a new local non-her theorem. The type D equation is also an integrability condition for the Near Horizon Geometry equation and leads to new results on the solution existence issue.
16:00
The Structure and Dimension of Multiplicative Preprojective Algebras
Abstract
Multiplicative preprojective algebras (MPAs) were originally defined by Crawley-Boevey and Shaw to encode solutions of the Deligne-Simpson problem as irreducible representations.
MPAs have recently appeared in the literature from different perspectives including Fukaya categories of plumbed cotangent bundles (Etgü and Lekili) and, similarly, microlocal sheaves
on rational curves (Bezrukavnikov and Kapronov.) After some motivation, I'll suggest a purely algebraic approach to study these algebras. Namely, I'll outline a proof that MPAs are
2-Calabi-Yau if Q contains a cycle and an inductive argument to reduce to the case of the cycle itself.
GCD sums and sum-product estimates
Abstract
When S is a finite set of natural numbers, a GCD-sum is a particular kind of double-sum over the elements of S, and they arise naturally in several settings. In particular, these sums play a role when one studies the local statistics of point sequences on the unit circle. There are known upper bounds for the size of a GCD-sum in terms of the size of the set S, most recently due to de la Bretèche and Tenenbaum, and these bounds are sharp. Yet the known examples of sets S for which the GCD-sum over S provides a matching lower bound all possess strong multiplicative structure, whereas in applications the set S often comes with additive structure. In this talk I will describe recent joint work with Thomas Bloom in which we apply an estimate from sum-product theory to prove a much stronger upper bound on a GCD-sum over an additively structured set. I will also describe an application of this improvement to the study of the distribution of points on the unit circle, with a further application to arbitrary infinite subsets of squares.
16:00
Laplace eigenvalue bounds: the Korevaar method revisited
Abstract
I will give a short survey on classical inequalities for Laplace eigenvalues, tell about related history and questions. I will then discuss the so-called Korevaar method, and new results generalising to higher eigenvalues a number of classical inequalities known for the first Laplace eigenvalue only.
12:00
Unitarity bounds on charged/neutral state mass ratio.
Abstract
I will talk about the implications of UV completion of quantum gravity on the low energy spectrums. I will introduce the constraints on low-energy effective theory due to unitarity and analyticity of scattering amplitudes, in particular an infinite set of new unitarity constraints on the forward-limit limit of four-point scattering amplitudes due to the work of Arkani-Hamed et al. In three dimensions, we find the constraints imply that light states with charge-to-mass ratio z greater than 1 can only be consistent if there exists other light states, preferably neutral. Applied to the 3D Standard Model like spectrum, where the low energy couplings are dominated by the electron with z \sim 10^22, this provides a novel understanding of the need for light neutrinos.