Related papers: Some remarks on Thom's Transversality Theorem

We present some results and conjectures on a generalization to the noncommutative setup of the Brouwer fixed-point theorem from the Borsuk-Ulam theorem perspective.

We give a new proof of the converse theorem for Maass forms on ${\rm GL}(3)$ using a technique that is inspired by Langlands' philosophy of "beyond endoscopy", thereby implementing these ideas for the first time in a higher rank setting.

Kolmogorov's invariant torus theorem is proved using a simple fixed point theorem.

Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. After surveying and extending the literature on the subject, we prove a theorem that affords a presentation by generators and…

The Thom polynomial of a singularity $\eta$ expresses the cohomology class of the $\eta$-singularity locus of a map in terms of the map's simple invariants. In this informal survey -- based on two lectures given at the Isaac Newton…

We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A). To a map of spectra f: b ->…

We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete…

This is a slightly edited version of the transparencies for a seminar at UCL, May 7, 2003. It is intended to give a quick view of background, ideas, and some calculations, in the applicatioon of some non commutative methods to algebraic…

We discuss topological theories, arising from the general $\mathcal{N}=2$ twisted gauge theories. We initiate a program of their study in the Gromov-Witten paradigm. We re-examine the low-energy effective abelian theory in the presence of…

We discuss topological versions of the closed graph theorem, where continuity is inferred from near continuity in tandem with suitable conditions on source or target spaces. We seek internal characterizations of spaces satisfying a closed…

This note is based on Professor Vitali Kapovitch's comparison geometry course at the University of Toronto. It delves into various comparison theorems, including those by Rauch and Toponogov, focusing on their applications, such as…

This paper provides a preparatory introduction to torsors, written with a view toward later applications in the author's work. Rather than aiming at a comprehensive survey, the exposition focuses on those aspects of torsors that are most…

In this paper, we develop a new index theory for manifolds with polyhedral boundary. As an application, we prove Gromov's dihedral extremality conjecture regarding comparisons of scalar curvatures, mean curvatures and dihedral angles…

We review how log Gromov--Witten invariants of toric varieties can be used to express quiver Donaldson--Thomas invariants in terms of the simpler attractor Donaldson--Thomas invariants. This is an exposition of joint work with Pierrick…

We prove a converse theorem for the case of quasi-split non-split even special orthogonal groups over finite fields. There are two main difficulties which arise from the outer automorphism and non-split part of the torus. The outer…

We introduce the notion of abstract angle at a couple of points defined by two radial foliations of the closed annulus. We use this notion to give unified proofs of some classical results on area preserving positive twist maps of the…

We generalize Kuznetsov's theory of homological projective duality to the setting of noncommutative algebraic geometry. Simultaneously, we develop the theory over general base schemes, and remove the usual smoothness, properness, and…

We revisit the non-commutative Hodge-to-de Rham Degeneration Theorem of the first author, and present its proof in a somewhat streamlined and improved form that explicitly uses spectral algebraic geometry. We also try to explain why…

We give a combinatorial proof of a theorem of Gromov, which extends the scope of small cancellation theory to group presentations arising from labelled graphs.

We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.