Related papers: Some remarks on mirror symmmetry and noncommutativ…
These notes loosely follow an introductory course on graph complexes, held at Humboldt-Universit\"at zu Berlin in summer 23. Instead of simply typing up my lecture notes I decided to give here an overview over (parts of) the topic (lecture…
This version improves the old version entitled "On the modularity of elliptic curves with a residually irreducible representation". Let $E$ be an elliptic curve over an abelian totally real field $K$ unramified at 3,5, and 7. We prove that…
The goal of this paper is to present a number of problems about automorphism groups of nonpositively curved polyhedral complexes and their lattices, meant to highlight possible directions for future research.
These are (heavily revised) notes from lectures given at the AMS Algebraic Geometry meeting in Seattle, 2005. The main topic is symplectic homology seen from the point of view of Lefschetz fibrations. Most of the content is speculative, but…
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
These are (not updated) notes from the lectures I gave at the NATO ASI ``Symmetric Functions 2001'' at the Isaac Newton Institute in Cambridge (June 25 -- July 6, 2001). Their goal is an informal introduction to asymptotic combinatorics…
This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…
This is a reply to "Comment on 'Noncommutative gauge theories and Lorentz symmetry,'" Phys. Rev. D 77 (2008) 048701 by Alfredo Iorio.
We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern…
This paper studies $3d$ $\mathcal{N}=4$ supersymmetric gauge theories on an elliptic curve, with the aim to provide a physical realisation of recent constructions in equivariant elliptic cohomology of symplectic resolutions. We first study…
This is the companion article to the Bourbaki talk of the same name given in March 2009. The main theme of the talk and the article is to explain the interplay between homotopy theory and algebraic geometry through the Hopkins-Miller-Lurie…
3d mirror symmetry is a mysterious duality for certian pairs of hyperk\"ahler manifolds, or more generally complex symplectic manifolds/stacks. In this paper, we will describe its relationships with 2d mirror symmetry. This could be…
This article is based on a talk delivered at the RIMS--OCAMI Joint International Conference on Geometry Related to Integrable Systems in September, 2007. Its aim is to review a recent progress in the Hitchin integrable systems and character…
The aim of this communication is to correct inaccurate statements presented in a Commentary on the paper titled: "Radiation forces and torque on a rigid elliptical cylinder in acoustical plane progressive and (quasi)standing waves with…
The subject of this PhD thesis is noncommutative geometry - more specifically spectral triples - and how it can be generalized to semi-Riemannian manifolds generally, and Lorentzian manifolds in particular. The first half of this thesis…
After the appearance of my preprint [T3] (Special Lagrangian geometry and slightly deformed algebraic geometry (spLag and sdAG), Warwick preprint 22/1998, alg-geom/9806006, 54 pp.). I received an e-mail from Cumrun Vafa, who recognized that…
This is an essay to accompany the author's lecture at the introductory workshop on `Nonabelian fundamental groups in arithmetic geometry' at the Newton Institute, Cambridge in July, 2009.
Concave mirrors are fundamental optical elements, yet some easily observed behaviors are rarely addressed in standard textbooks, such as the formation of multiple reflected images. Here we investigate self-imaging -- where the observer is…
This is the extended abstract of the talk given at the workshop "Geometry and physics of Higgs bundles", held at the Mathematisches Forschungsinstitut Oberwolfach in May 2019.
Proceedings of the second edition of the international Workshop "Dynamics and Thermodynamics of Blackholes and Naked Singularities" (Department of Mathematics of the Politecnico of Milano from May 10-12, 2007.