Related papers: $\tau$-tilting theory: a self-contained introducti…
The possibility of noncommutative topological gravity arising in the same manner as Yang-Mills theory is explored. We use the Seiberg-Witten map to construct such a theory based on a SL(2,C) complex connection, from which the Euler…
Let R be an associative ring with identity, and let T be a tilting right R-module, with S=End(T). It is known that if R is a Noetherian algebra that satisfies the Auslander-Reiten conjecture, then so is S. In this paper, we consider the…
We give a characterization of $\tau$-rigid modules over Auslander algebras in terms of projective dimension of modules. Moreover, we show that for an Auslander algebra $\Lambda$ admitting finite number of non-isomorphic basic tilting…
The class of support $\tau$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $\tau$-tilting finite algebras, i.e. finite dimensional…
A descent conjecture of Wittenberg [Wit24, Conjecture 3.7.4] predicts that if all the twists of a rationally connected torsor over a smooth base satisfy weak approximation with Brauer-Manin obstruction, then so does the base. We give an…
An exposition of the basic geometry of twistor integrals, intended for mathematicians.
This is lecture notes for a course given at the PCMI Summer School "Quantum Field Theory and Manifold Invariants" (July 1 -- July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main…
For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of…
Let $\Lambda$ be an Artin algebra and $K^b(proj(\Lambda))$ be the triangulated category of bounded co-chain complexes in $proj(\Lambda).$ It is well known that two-terms silting complexes in $K^b(proj(\Lambda))$ are described by the…
We introduce ultradiscrete tau functions associated with rigged configurations for A^{(1)}_n. They satisfy an ultradiscrete version of the Hirota bilinear equation and play a role analogous to a corner transfer matrix for the box-ball…
The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical tau-functions, depending on two different sets of variables: the parameters of the UV gauge theory or the bare compexified couplings, and the…
In this paper, we study tau-tilting modules over Nakayama algebras. We establish bijections between tau-tilting modules, triangulations of a polygon with a puncture and certain integer sequences. Moreover, we give an algorithm to construct…
These notes are a self-contained introduction to Galois theory, designed for the student who has done a first course in abstract algebra.
We show that any accordion complex associated to a dissection of a convex polygon is isomorphic to the support $\tau$-tilting simplicial complex of an explicit finite dimensional algebra. To this end, we prove a property of some induced…
For Brauer graph algebras, tilting mutation is compatible with flip of Brauer graphs. The aim of this paper is to generalize this result to the class of Brauer configuration algebras introduced by Green and Schroll recently. More precisely,…
We construct a representation of the blob algebra over a ring allowing base change to every interesting (i.e. non--semisimple) specialisation which, in quasihereditary specialisations, passes to a full tilting module.
We review new aspects of integrable systems discovered recently in N=2 supersymmetric gauge theories and their topologically twisted versions. The main topics are (i) an explicit construction of Whitham deformations of the Seiberg-Witten…
The torus and the Klein bottle amplitude coefficients are computed in permutation orbifolds of RCFT-s in terms of the same quantities in the original theory and the twist group. An explicit expression is presented for the number of self…
In this small note we ask several questions which are relevant to the construction of the self-consistent neutrino theory of light. The previous confusions in such attempts are explained in the more detailed publication.
These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.