Abstract: I will describe a new proof, joint with Adam Piggott (UQ), that groups presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 are exactly the plain groups (free products of finite and infinite cyclic groups). Our proof is essentially graph-theoretic: we prove a new result about properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.
Title: Gerbes, unbounded spectral flow, and physics
Friday, 02 October 2020 at 12:10pm, Eng Sth S112 Lecture
Theatre
Abstract: The old concept of spectral flow applies
directly to sufficiently continuous loops of un- bounded
differential operators. Dirac operator families on noncompact
manifolds-with-boundary provide the basic new examples, and
arise naturally in real physics, where the spectral flow is a
detectable boundary “anomaly”. For higher-dimensional and/or
equivariant families, the “higher spectral flow” data is
encoded in a “Fermi gerbe”, and a mod-2 version is realised in
actual topological insulators. Joint work with A. Carey and K.
Gomi.
Title: Tautological classes of definite 4-manifolds
Friday, 18 September 2020 at 12:10pm at Eng Sth
S112 Lecture Theatre
Abstract: Tautological classes are characteristic classes of manifold bundles. They have been extensively studied for bundles of surfaces, where they were first introduced by Mumford in the setting of moduli spaces of curves. In higher dimensions there are not many examples of manifolds for which the tautological ring, the ring generated by tautological classes, is known. We will use gauge theory study tautological classes of 4-manifolds with positive definite intersection form. Amongst other things, this allows us to compute the tautological ring for CP^2 and the connected sum of CP^2 with itself.
Abstract: The double ramification cycle in the moduli space of smooth algebraic curves with marked points is the locus of curves whose marked points form the support of a principal divisor. A natural Chow class representing the compactification of such locus is the push-forward of the virtual fundamental class of the space of rubber maps to P1 relative to 0 and infinity. In a series of papers with A. Buryak (and, in part, with B. Dubrovin and J. Guéré), we have studied the intersection theory of such cycle with other tautological classes. In particular, inspired by ideas from Eliashberg, Givental and Hofer's Symplectic Field Theory, we have developed a construction using the DR cycle to associate to any cohomological field theory on the moduli space of stable curves an integrable system (of Hamiltonian PDEs) and its quantization, called the DR hierarchy. This turns out to have a deep relation, via a more classical construction of Dubrovin and Zhang, with Gromov-Witten theory, mirror symmetry and the structure of the tautological ring. I will try to introduce in a simple way the main ideas and features of this construction and showcase some of its applications.
Abstract: Gauged linear sigma models (GLSMs) can be used to analyse Calabi-Yaus and their moduli spaces. Recent results in supersymmetric localisation have made it possible to compute exact, i.e. fully quantum corrected, quantities that are relevant in string compactifications directly in the GLSM. After a review of the general framework, I will present some recent applications with focus on the sphere and hemisphere partition function of the GLSM.
Abstract: Closing lemmas and connecting lemmas are of great importance in smooth dynamics. I will explain what they are. Then I will describe new joint work with Leandro Arosio. We have proved new closing lemmas and connecting lemmas in holomorphic dynamics.
Abstract: We consider the alpha invariant of any smooth complex projective spin complete intersection of dimension 1 mod 4. We prove that the alpha invariant depends only on the total degree and Pontryagin classes. This is in agreement with a long-standing conjecture which states that two complete intersections (in any dimension greater than 2) with the same total degrees, Pontryagin and Euler classes are diffeomorphic.
Abstract: The study of hyperbolic knot complements has a long history leading to many exciting results in the field of 3-manifold topology. In this talk, I will present a 4-dimensional analogue of this study. Namely, I will consider when a closed smooth simply connected 4-manifold can contain a collection of 2-tori, whose complement can admit a complete finite volume hyperbolic structure. I will start by presenting some necessary conditions, based on a classification theorem of S. Donaldson and M. Freedman, and then move on to outline how one can try to build such complements.
Abstract: The concept of holonomy originated in the study of problems in mechanics - roughly speaking, holonomy keeps track of the ``path-dependence" of the final state of a time-evolving system. In differential geometry, holonomy has found rigorous expression in the theory of fibre bundles, connections and curvature, with respect to which holonomy is determined by the solutions of first order differential equations. There is another notion of holonomy which arises in the theory of foliations, whose interpretations have for several decades taken a less systematic and geometrically intuitive form. In this talk, I will discuss recent work which unifies the foliation notion of holonomy with the fibre bundle notion using the tool of diffeology. The talk will be accessible to anyone with a working understanding of differential geometry.