Public holiday
Speaker: Charlie Cifarelli
Title: Steady soliton deformations of Taub-NUT and higher-dimensional analogues
Abstract: I will discuss some joint work with V. Apostolov from the past few years in which we produce a deformation of the Taub-NUT gravitational instanton to a family of so-called steady Kähler-Ricci solitons. We also obtain a version of this result in higher dimensions, which has the surprising consequence of producing generalizations of Taub-NUT (as a Ricci-flat Kähler metric) to Euclidean space of dimension 2n for all n > 1.
Abstract: | ||
| Liouville quantum gravity (LQG) is the canonical one-parameter family of randomrntwo-dimensional Riemannian manifolds. Its metric is formally given by e γ h(z)(dx2 +dy2) where h is (some form of) the Gaussian free field (GFF) on a planar domain D and γ in [0, 2] is a parameter. The study of LQG has attracted a substantial amount of interest in probability theory due to itsrnconnections with random planar maps and Schramm-Loewner evolution (SLE). Many features of LQG surfaces are different from in ordinary Riemannian geometry as h is a distribution and not a function; in particular, LQG surfaces are rough and fractal. In this talk, I will discuss: (i) estimates for the associated heat kernel, (ii) the structure of geodesics, (iii) continuity properties in γ, and (iv) conformal welding of LQG surfaces and SLE. |
Speaker: Tristan Ozuch
Title: Regularity of Einstein 5-manifolds via 4-dimensional gap theorems
Abstract: (Joint work with Yiqi Huang) We refine the regularity of noncollapsed limits of Einstein 5-manifolds. In particular, we show uniqueness of tangent cones on the full top stratum, show that the structure of the singular set lies in countable unions of Lipschitz curves and points. We finally prove real-analytic orbifold regularity along curves of singularities, which are also proven to be geodesics, and establish uniqueness of tangent cone at infinity under Euclidean volume growth with a line split. The proofs rely on new 4-dimensional gap/isolation theorems for spherical and hyperbolic Einstein orbifolds.
Title: Toward an upper bound on the thermal mass gap at criticality
Abstract: An intrinsic quantity characterizing a scale-invariant quantum field theory (SFT) is the mass gap it acquires upon being compactified on a circle, alternatively referred to as the “thermal mass”. By scale-invariance, this is a pure number in units of the inverse circumference of the circle. In 2d conformal field theory (CFT), a theory-independent upper bound on the thermal mass follows straightforwardly from unitarity, locality, and conformality. In this talk I will discuss the possibility of a numerical bootstrap upper bound on the thermal mass for SFTs in 3d. I will propose a bootstrap framework to study the thermal free energy of circle-compactified 3d SFTs, which will invoke two constraints: an analog of modular invariance, and consistency with a formula expressing the thermal free energy in terms of the S matrix due to Dashen, Ma, and Bernstein. Accounting for the implications of the latter will motivate a positivity assumption that facilitates the use of numerical bootstrap techniques, which roughly restricts the analysis to theories with attractive interactions. I will show that, within the class of theories obeying the bootstrap constraints, there is a theory-independent upper bound on the thermal mass.
Aaron Landesman, MIT/Harvard
Malle's conjecture over function fields
The inverse Galois problem, a foundational question in number theory, asks whether every finite group G can be realized as the Galois group of a field extension of the rational numbers. Malle's conjecture is a refined version of the inverse Galois problem which predicts the asymptotic number of such extensions. In joint work with Ishan Levy, we prove a version of Malle's conjecture, computing the asymptotic growth of the number of Galois G extensions of Fq(t) for q sufficiently large and relatively prime to |G|. We use tools from algebraic geometry to relate this conjecture to a question in topology about the cohomology of certain Hurwitz spaces. We then complete the proof by solving the topological question using techniques from homotopy theory.
Mike Miller-Eismeier, University of Vermont
Surgery, ASD connections and bubbling
Surgery is a way of constructing and, in high dimensions, classifying manifolds. In low dimensions, however, questions about surgery remain poorly understood. I will discuss the problem of finding low-complexity surgery descriptions of 3-manifolds from the perspective of ASD connections on 4-manifolds, including new techniques for studying text"obstructed" 4-manifolds.
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Speaker: Mingyang Li
Title: Gravitational instantons and harmonic maps
Abstract: Gravitational instantons are by definition 4-dimensional complete Ricci-flat metrics with finite curvature energy. Previous studies mainly focused on gravitational instantons with special geometries, known as hyperkahler or conformally Kahler metrics. These special cases have been essentially classified in recent years.
In this series of two talks, we will explain a construction of an infinite family of new gravitational instantons, using axisymmetric harmonic maps from the 3-space into the hyper
Shuhao Li, Stony Brook University
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