hilbert series and degree bounds for matrix semi invariants abstract we study the ring r n m of invariants for the left right action of sl n sl n on m tuples of n n complex matrices we show that r 3 m is generated by invariants of degree 309 for all m then we use a combinatorial description of the invariants to show that r n m cannot be generated by invariants of degree n 2 for large m we also compute the hilbert series for several cases

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

the quantization conjecture revisited a strong version of the quantization conjecture of guillemin and sternberg is proved for a reductive group action on a smooth compact polarized variety x l the cohomologies of l over the git quotient x g equal the invariant part of the cohomologies over x this generalizes the theorem of gs on global sections and strengthens its subsequent extensions jk li to riemannroch numbers remarkable by products are the invariance of cohomology of vector bundles over x g under a small change in the defining polarization or under shift desingularization as well as a new proof of boutot s theorem also studied are equivariant holomorphic forms and the equivariant hodgeto de rham spectral sequences for x and its strata whose collapse is shown one application is a new proof of the borel weil bott theorem of ti for the moduli stack of c bundles over a curve and of analogous statements for the moduli stacks and spaces of bundles with parabolic structures collapse of the hodge to de rham sequences for these stacks is also shown

degree bounds for syzygies of invariants abstract suppose that g is a linearly reductive group good degree bounds for generators of invariant rings were given in proc amer math soc 129 4 2001 955 here we study minimal free resolutions of invariant rings for finite linearly reductive groups g it was recently shown in adv math 156 1 2000 23 electron res announc amer math soc 7 2001 5 adv math 172 2002 151 that rings of invariants are generated in degree at most the group order g in characteristic 0 this degree bound is a classical result by emmy noether see math ann 77 1916 89 given an invariant ring of a finite linearly reductive group g we prove that the ideal of relations of a minimal set of generators is generated in degree at most 2 g

semi direct products of lie algebras and covariants abstract let q be a connected algebraic group with lie algebra q symmetric invariants of q i e the q invariants in the symmetric algebra s q of q is a first approximation to the understanding of the coadjoint action q q and coadjoint q orbits in this article we study a class of non reductive lie algebras where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant theoretic properties if g is a semisimple group with lie algebra g and v is g module then we define q to be the semi direct product of g and v then we are interested in the case where the generic isotropy group for the g action on v is reductive and commutative it turns out that in this case symmetric invariants of q can be constructed via certain g equivariant maps from g to v covariants

globally f regular type of moduli spaces we prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves with fixed determinant on a smooth projective curve are of globally f regular type

exploiting history dependent effects to infer network connectivity we present an approach to distinguish between causal connections and common input connections among nodes in a network by modeling how the activity of a node depends on its own recent history we demonstrate how this history dependence predicts different patterns of activity depending on the nature of the network connectivity in particular a causal connection between a pair of observed nodes can be distinguished from common input connections that originate from nodes whose activity remains unobserved this work builds on previous results where this same distinction was made based on modeling how the activity of a node depends on measured external variables such as stimuli the results have a potentially broad range of application as the analysis can be based on a fairly generic class of models

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

correlations without synchrony peaks in spike train correlograms are usually taken as indicative of spike timing synchronization between neurons strictly speaking however a peak merely indicates that the two spike trains were not independent two biologically plausible ways of departing from independence that are capable of generating peaks very similar to spike timing peaks are described here covariations over trials in response latency and covariations over trials in neuronal excitability since peaks due to these interactions can be similar to spike timing peaks interpreting a correlogram may be a problem with ambiguous solutions what peak shapes do latency or excitability interactions generate when are they similar to spike timing peaks when can they be ruled out from having caused an observed correlogram peak these are the questions addressed here the previous article in this issue proposes quantitative methods to tell cases apart when latency or excitability covariations cannot be ruled out

harmonic mean curvature flow and geometric inequalities abstract we employ the harmonic mean curvature flow of strictly convex closed hypersurfaces in hyperbolic space to prove alexandrov fenchel type inequalities relating quermassintegrals to the total curvature which is the integral of gaussian curvature on the hypersurface the resulting inequality allows us to use the inverse mean curvature flow to prove alexandrov fenchel inequalities between the total curvature and the area for strictly convex hypersurfaces finally we apply the harmonic mean curvature flow to prove a new class of geometric inequalities for h convex hypersurfaces in hyperbolic space

the geodesic x ray transform with matrix weights author s paternain gp salo m uhlmann g zhou h abstract 2019 by johns hopkins university press consider a compact riemannian manifold of dimension 3 with strictly convex boundary such that the manifold admits a strictly convex function we show that the attenuated ray transform in the presence of an arbitrary connection and higgs field is injective modulo the natural obstruction for functions and one forms we also show that the connection and the higgs field are uniquely determined by the scattering relation modulo gauge transformations the proofs involve a reduction to a local result showing that the geodesic x ray transform with a matrix weight can be inverted locally near a point of strict convexity at the boundary and a detailed analysis of layer stripping arguments based on strictly convex exhaustion functions as a somewhat striking corollary we show that these integral geometry problems can be solved on strictly convex manifolds of dimension 3 having nonnegative sectional curvature similar results were known earlier in negative sectional curvature we also apply our methods to solve some inverse problems in quantum state tomography and polarization tomography

locally constrained curvature flows and geometric inequalities in hyperbolic space in this paper we first study the locally constrained curvature flow of hypersurfaces in hyperbolic space which was introduced by brendle guan and li an inverse curvature type hypersurface flow in mathbb h n 1 preprint this flow preserves the mth quermassintegral and decreases m 1 th quermassintegral so the convergence of the flow yields sharp alexandrov fenchel type inequalities in hyperbolic space some special cases have been studied in brendle et al in the first part of this paper we show that h convexity of the hypersurface is preserved along the flow and then the smooth convergence of the flow for h convex hypersurfaces follows we then apply this result to establish some new sharp geometric inequalities comparing the integral of kth gauss bonnet curvature of a smooth h convex hypersurface to its mth quermassintegral for 0 le m le 2k 1 le n and comparing the weighted integral of kth mean curvature to its mth quermassintegral for 0 le m le k le n in particular we give an affirmative answer to a conjecture proposed by ge wang and wu math z 281 257 297 2015 in the second part of this paper we introduce a new locally constrained curvature flow using the shifted principal curvatures this is natural in the context of h convexity we prove the smooth convergence to a geodesic sphere of the flow for h convex hypersurfaces and provide a new proof of the geometric inequalities proved by andrews chen and the third author of this paper in 2018 we also prove a family of new sharp inequalities involving the weighted integral of kth shifted mean curvature for h convex hypersurfaces which as application implies a higher order analogue of brendle hung and wang s commun pure appl math 69 1 124 144 2016 inequality

der kanonische modul eines invariantenrings abstract we calculate the canonical module of a ring of invariants r of a reductive group acting on an affine variety from that we derive a criterion for the gorenstein property to hold for r in case r is the ring of invariants of a graded ring we obtain lower and upper bounds for the degree of the generating function of r

a fully nonlinear flow for two convex hypersurfaces in riemannian manifolds we consider a one parameter family of closed embedded hypersurfaces moving with normal velocity g kappa big sum i j frac 1 lambda i lambda j 2 kappa big 1 where lambda 1 le cdots le lambda n denote the curvature eigenvalues and kappa is a nonnegative constant this defines a fully nonlinear parabolic equation provided that lambda 1 lambda 2 2 kappa in contrast to mean curvature flow this flow preserves the condition lambda 1 lambda 2 2 kappa in a general ambient manifold our main goal in this paper is to extend the surgery algorithm of huisken sinestrari to this fully nonlinear flow this is the first construction of this kind for a fully nonlinear flow as a corollary we show that a compact riemannian manifold satisfying overline r 1313 overline r 2323 ge 2 kappa 2 with non empty boundary satisfying lambda 1 lambda 2 2 kappa is diffeomorphic to a 1 handlebody the main technical advance is the pointwise curvature derivative estimate the proof of this estimate requires a new argument as the existing techniques for mean curvature flow due to huisken sinestrari haslhofer kleiner and brian white cannot be generalized to the fully nonlinear setting to establish this estimate we employ an induction on scales argument this relies on a combination of several ingredients including the almost convexity estimate the inscribed radius estimate as well as a regularity result for radial graphs we expect that this technique will be useful in other situations as well

a presentation of the trace algebra of three 3 3 matrices abstract the trace algebra c n d is generated by all traces of products of d generic n n matrices minimal generating sets of c n d and their defining relations are known for n 3 and n 3 d 2 this paper states a minimal generating set and their defining relations for n d 3 furthermore the computations yield a description of c 33 as a free module over the ring generated by a homogeneous system of parameters

local convexity and nonnegative curvature gromov s proof of the sphere theorem an immersed hypersurface s in a riemannian manifold m will be called e convex for some e 0 if all principal curvatures have the same sign and absolute value at least e can one characterize all compact e convex immersed hypersurfaces of a complete manifold m if m is euclidean n space n 3 this problem is solved by a theorem of hadamard 14 generalized by hopf 15 s is embedded and bounds a convex n disk see 6 6 6 3 if m is a flat space form s is no longer necessarily embedded but it still bounds an immersed convex n disk the aim of this paper is to show the same fact for a manifold m of curvature k 0

moduli of vector bundles frobenius splitting and invariant theory let x be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic 2 3 in part i we prove adapting the degeneration arguments of n tr that moduli spaces of rank two parabolic bundles on x are frobenius split m r for generic smooth x a similar result holds for x nodal a consequence is the verlinde formula in positive characteristic in part ii we give a direct proof of the fact that the local rings of the moduli spaces are f split and further that they are cohen macaulay this involves showing that the ring of invariants of g copies of 2 x 2 matrices under the adjoint action of sl 2 is f split and cohen macaulay

a degree inequality for lie algebras with a regular poisson semi center for lie algebras whose poisson semi center is a polynomial ring we give a bound for the sum of the degrees of the generating semi invariants this bound was previously known in many special cases

revealing pairwise coupling in linear nonlinear networks through an asymptotic analysis of a simple network we derive an estimate of the coupling between a pair of units when all other units are unobservable the analysis is based on a model where the response of each unit is a linear nonlinear function of a white noise stimulus the results accurately determine the coupling when all unmeasured units respond to the stimulus differently than the measured pair to account for the possibility of unmeasured units similar to the measured pair we cast our results in the framework of subpopulations which are defined as a group of units who respond to the stimulus similarly we demonstrate that we can determine when correlations between two units are caused by a connection between their subpopulations although the precise identity of the units involved in the connection may remain ambiguous the result is rigorously valid only when the coupling is sufficiently weak to justify a second order approximation in the coupling strength we demonstrate through simulations that the results are still valid even with stronger coupling and in the presence of some deviations from the linear nonlinear model the analysis is presented in terms of neuronal networks although the general framework is more widely applicable

generating invariant rings of quivers in arbitrary characteristic abstract it is well known that the ring of polynomial invariants of a reductive group is finitely generated however it is difficult to give strong upper bounds on the degrees of the generators especially over fields of positive characteristic in this paper we make use of the theory of good filtrations along with recent results on the null cone to provide polynomial bounds for matrix semi invariants in arbitrary characteristic and consequently for matrix invariants our results generalize to invariants and semi invariants of quivers