Abstract We introduce and study a fermionisation procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the $\unicode{x03BC}$ -deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.

A high resolution numerical method for solving atmospheric models

Abstract Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X. Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

We introduce special classes of irreducible representations of groups: thick representations and dense representations. Denseness implies thickness, and thickness implies irreducibility. We show that absolute thickness and absolute denseness are open conditions for representations. Thereby, we can construct the moduli schemes of absolutely thick representations and absolutely dense representations. We also describe several results and several examples on thick representations for developing a theory of thick representations.

Morphisms between the moduli functor of admissible semistable pairs and the Gieseker-Maruyama moduli functor (of semistable coherent torsion-free sheaves) with the same Hilbert polynomial on the surface are constructed. It is shown that these functors are isomorphic, and the moduli scheme for semistable admissible pairs is isomorphic to the Gieseker-Maruyama moduli scheme. All the components of moduli functors and corresponding moduli schemes which exist are looked at here. Bibliography: 16 titles.

We develop a theory of moduli of Galois representations. More generally, for an object in a rather general class \mathfrak A of noncommutative topological rings, we construct a moduli space of its absolutely irreducible representations of a fixed degree as a (so we call) "f- \mathfrak A scheme". Various problems on Galois representations can be reformulated in terms of such moduli schemes. As an application, we show that the "difference" between the strong and weak versions of the finiteness conjecture of Fontaine–Mazur is filled in by the finiteness conjecture of Khare–Moon.

We give a combinatorial description of all affine spherical varieties with prescribed weight monoid Г. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion’s moduli scheme MГ for such varieties. Moreover, we find several sufficient conditions for MГ to be irreducible and exhibit several examples where MГ is reducible. Finally, we provide examples of non-reduced MГ.

We show that, over every number field, the degree four del Pezzo surfaces that violate the Hasse principle are Zariski dense in the moduli scheme.

We determine, under a certain assumption, the Alexeev–Brion moduli scheme M of affine spherical G-varieties with a prescribed weight monoid . In Papadakis and Van Steirteghem (Ann. Inst. Fourier (Grenoble). 62(5) 1765–1809 19) we showed that if G is a connected complex reductive group of type A and is the weight monoid of a spherical G-module, then M is an affine space. Here we prove that this remains true without any restriction on the type of G.

We study p-adic integral models of certain PEL Shimura varieties with level subgroup at p related to the \({\Gamma_1(p)}\)-level subgroup in the case of modular curves. We will consider two cases: the case of Shimura varieties associated with unitary groups that split over an unramified extension of \({\mathbb{Q}_p}\) and the case of Siegel modular varieties. We construct local models, i.e. simpler schemes which are etale locally isomorphic to the integral models. Our integral models are defined by a moduli scheme using the notion of an Oort–Tate generator of a group scheme. We use these local models to find a resolution of the integral model in the case of the Siegel modular variety of genus 2. The resolution is regular with special fiber a nonreduced divisor with normal crossings.

We continue the study of the compactification of the moduli scheme for Gieseker-semistable vector bundles on a nonsingular irreducible projective algebraic surface S with polarization L, by locally free sheaves. The relation of main components of the moduli functor or admissible semistable pairs and main components of the Gieseker – Maruyama moduli functor (for semistable torsion-free coherent sheaves) with the same Hilbert polynomial on the surface S is investigated. The compactification of interest arises when families of Gieseker-semistable vector bundles E on the nonsingular polarized projective surface (S, L) are completed by vector bundles E on projective polarized schemes (S, L) of special form. The form of the scheme S, of its polarization L and of the vector bundle E is described in the text. The collection ((S, L), E) is called a semistable admissible pair. Vector bundles E on the surface (S, L) and E on schemes (S, L) are supposed to have equal ranks and Hilbert polynomials which are compute with respect to polarizations L and L, respectively. Pairs of the form ((S, L), E) named as S-pairs are also included into the class under the scope. Since the purpose is to study the compactification of moduli space for vector bundles, only families which contain S-pairs are considered. We build up the natural transformation of the moduli functor for admissible semistable pairs to the Gieseker – Maruyama moduli functor for semistable torsion-free coherent sheaves on the surface (S, L), with same rank and Hilbert polynomial. It is demonstrated that this natural transformation is inverse to the natural transformation built in the preceding paper and defined by the standard resolution of a family of torsion-free coherent sheaves with a possibly nonreduced base scheme. The functorial isomorphism constructed determines the scheme isomorphism of compactifications of moduli space for semistable vector bundles on the surface (S, L).

Here we study the Brill–Noether theory of “extremal” Cornalba’s theta-characteristics on stable curves C of genus g, where “extremal” means that they are line bundles on a quasi-stable model of C with #(Sing(C)) exceptional components.

We study Alexeev and Brion's moduli scheme $M_\\Gamma$ of affine spherical\nvarieties with weight monoid $\\Gamma$ under the assumption that $\\Gamma$ is\nfree. We describe the tangent space to $M_\\Gamma$ at its `most degenerate\npoint' in terms of the combinatorial invariants of spherical varieties and\ndeduce that the irreducible components of $M_\\Gamma$, equipped with their\nreduced induced scheme structure, are affine spaces.\n

Abstract Here we study the Brill-Noether theory of “extremal” Cornalba’s theta-characteristics on stable curves C of genus g, where “extremal” means that they are line bundles on a quasi-stable model of C with #(Sing(C)) exceptional components

Evaluating the significance of hardening behavior and unloading modulus under strain reversal in sheet springback prediction

For connected reductive groups G over a finite extension F of Q_p and L the maximal unramified extension of F we study the sets H_{μ, N}(G) of elements b in G(L) with given Hodge points of (bσ), (bσ)^2, ..., (bσ)^N. We explain the relationship to stratifications of some moduli scheme of abelian varieties defined by Goren and Oort respectively Andreatta and Goren. We show that for sufficiently large N the Newton point is constant on the sets H_{μ, N}(G) and compute such N for certain classes of groups.

In this paper, for obtaining an overall size-dependent yield function for nanocomposites containing aligned cylindrical nanofibers, the effects of interface residual stress and interface elasticity are taken into account within a micromechanical framework. Toward this goal, the modified Hill’s condition is used, and then, in order to consider effects of the interface residual stress, strains are decomposed into two parts, a part due to the external loadings and the other due to the interface residual stress. Next, utilizing the field fluctuation method, an overall yield function containing effective elastic constants of the material is derived and then simplified for practical loading conditions. Moreover, a secant modulus scheme is adopted to examine the overall nonlinear behavior of the material in plastic deformation. Finally, by some numerical examples, it is shown that the interface stress, including the interface residual stress, makes the yield strength and plastic deformation of the metal matrix nanocomposites dependent on the nanofiber size, in contrast to the classical results.

In this paper we study the derivative at the center of symmetry of an incoherent Eisenstein series which is associated to an imaginary quadratic field. We show that each nonconstant Fourier coefficient of the derivative can be expressed as the degree of certain zero cycles on a moduli scheme. This result is a generalization of the work by Kudla-Rapoport-Yang.

This thesis develops a method (dimensional reduction) to compute motivic Donaldson--Thomas invariants. The method is then employed to compute these invariants in several different cases. Given a moduli scheme with a symmetric obstruction theory a Donaldson--Thomas type invariant can be defined by integrating Behrend's function over the scheme. Motivic Donaldson--Thomas theory aims to provide a more refined invariant associated to each such moduli space - a virtual motive. From the modern point of view motivic Donaldson-Thomas invariants should be defined for a three dimensional Calabi--Yau category. These categories often arise in a geometric context as the derived category of representations of a quiver with potential. Provided the potential has a linear factor we are able to reduce the problem of computing the corresponding virtual motives to a much simpler one. This includes geometric examples coming from local curves which we compute explicitly.

In the present study, a fast feed-forward blind equalizer with a two-stage generalized multilevel modulus algorithm (GMMA) and soft decision-directed (SDD) scheme was developed for high-order QAM cable receivers on broadcasting downstream wired cable channels. The proposed fast blind equalization algorithm uses a two-stage convergence strategy, and the modified SDD part applies an adaptively selected decision region. At the first convergence stage, joint GMMA and modified SDD equalization was applied for fast convergence. When the convergence process reached the steady state, the convergence detector changed the first equalization stage to the second stage. At the second stage, the modified SDD scheme reduced the mean square error (MSE) further. To prove the convergence, MSE analyses of two-stage GMMA+SDD /SDD blind equalization in the steady state were conducted. On the wired cable channel at 64, 256, and 1024QAM modes, the proposed blind algorithm had a faster convergence speed than previous blind methods. The proposed algorithm also achieved a smaller MSE than the other methods at the same signal-to-noise ratio (SNR). When the proposed method used the architecture of the decision-feedback equalizer (DFE) with a SDD-based feedback FIR filter (FBF), the steady-state MSE and bit-error-rate (BER) decreased further.