General coherent systems are defined as pairs $(ℰ\mathrm{,}S)$, where $ℰ$ is a rank-n holomorphic vector bundle and S is a vector subspace of $H^0({ℰ}_{\rho })$, ${ℰ}_{\rho }$ is the vector bundle induced by $ℰ$ and a representation $\rho :GL(n)\stackrel{}{\to }GL(W)$. In this work we will prove a Kobayashi-Hitchin correspondence for a simple general coherent system. This is done using well known gauge theoretic techniques which were developed by Bradlow, Garcia-Prada and Mundet i Riera. The focus of this paper is proving that the stability condition that arises is equivalent to the one defined by Alexander Schmitt.

We give an overview of the goals and recent progress in the development of an enumerative geometry valued in quadratic forms.

The notion of linear stability of a variety in projective space was introduced by Mumford in the context of GIT. It has subsequently been applied by Mistretta and others to Butler’s conjecture on stability of the dual span bundle (DSB) $M_{V\mathrm{,}E}$ of a general generated coherent system $(E\mathrm{,}V)$. We survey recent progress in this direction on rank one coherent systems, prove a new result for hyperelliptic curves, and state some open questions. We then extend the definition of linear stability to generated coherent systems of higher rank. We show that various coherent systems with unstable DSB studied in \cite{bmno} are also linearly unstable. We show that linearly stable coherent systems of type (2, d, 4) for low enough d have stable DSB, and use this to prove a particular case of Butler’s conjecture. We then exhibit a linearly stable generated coherent system with unstable DSB, confirming that linear stability of $(E\mathrm{,}V)$ in general remains weaker than semistability of $M_{V\mathrm{,}E}$ in higher rank. We end with a list of open questions on the higher rank case.

Many examples of zeta functions in number theory, combinatorics and algebraic geometry are special cases of a construction in homotopy theory known as a decomposition space. This article aims to introduce readers to the relevant concepts in homotopy theory and lays some foundations for future applications of decomposition spaces in the theory of zeta and L-functions.

Let N be the normalizer of the diagonal torus $T_1\cong {\mathbb{G}}_m$ in $S{L}_2$. We prove localization theorems for $S{L}_2^n$ and $N^n$ for equivariant cohomology with coefficients in the (twisted) Witt sheaf, along the lines of the classical localization theorems for equivariant cohomology for a torus action. We also have an analog of the Bott residue formula for $S{L}_2^n$ and N. In the case of an $S{L}_2^n$-action, there is a rather serious restriction on the orbit type. For an N-action, there is no restriction for the localization result, but for the Bott residue theorem, one requires a certain type of decomposition of the fixed points for the $T_1$-action, which is always available if the subscheme of $T_1$ fixed points has dimension zero.

Let X be a quasiprojective scheme. In this expository note we collect a series of useful structural results on the stack $\mathcal{C}o{h}^n(X)$ parametrising 0-dimensional coherent sheaves of length n over X. For instance, we discuss its functoriality (in particular its behaviour along \’etale maps), the support morphism to ${}^n(X)$, and its relationship with the Quot scheme of points $Quo{t}_X(ℰ\mathrm{,}n)$ for fixed $ℰ\in (X)$.

We introduce a “hybrid” conjecture which is a common generalisation of the André-Oort conjecture and the André-Pink-Zannier conjecture and we prove that it is a consequence of the Zilber-Pink conjecture. This extends a previous work of V. Aslanyan and C. Daw. We also show that our hybrid conjecture implies the Zilber-Pink conjecture for hypersurfaces contained in weakly special subvarieties.

The goal of this paper is to construct universal cohomology classes on the moduli space of stable bundles over a curve when it is not a fine moduli space, that is, when the rank and degree are not coprime. More precisely, we show that certain Chern classes of the universal bundle on the product of the curve with the moduli stack of bundles lift to the product of the curve with the moduli space of stable bundles.

This text presents an overview of recent developments on compactifications of moduli stacks of shtukas. The aim is to explain how to tackle the problem of compactifying stacks of shtukas by two different methods: the Langton semistable reduction and the Geometric Invariant Theory.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that.

Topics covered include correlation inequalities, Lee–Yang theorems, the Peierls method, the Hohenberg–Mermin–Wagner method, infrared bounds, random cluster methods, random current methods, and BKT transition. The final section outlines major open problems to inspire future work.

This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

As noted in the Introduction, in this chapter we consider running the Toda algorithm only until time $T^{(1)}$, the deflation time with block decomposition k = 1 fixed, when the norm of the off-diagonal elements in the first row, and hence the first column, is $\mathcal{O}(ϵ)$. Define $E(t)={\displaystyle \sum _{n=2}^N|}{X}_{1n}(t){|}^2$ so that if $E(t)=0$ then $X_{11}(t)$ is an eigenvalue of H. Thus, with $E(t)$ as in (6.1), the halting time (or 1-deflation time) for the Toda algorithm is given by $T^{(1)}(H)=\inf \{t:E(t)\le {ϵ}^2\}$.