5,364 research outputs found
An Alexandrov topology for maximal Cohen-Macaulay modules
Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies
Classical and quantum algorithms for scaling problems
This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size
Slopes of modular forms and geometry of eigencurves
Under a stronger genericity condition, we prove the local analogue of ghost
conjecture of Bergdall and Pollack. As applications, we deduce in this case (a)
a folklore conjecture of Breuil--Buzzard--Emerton on the crystalline slopes of
Kisin's crystabelian deformation spaces, (b) Gouvea's
-conjecture on slopes of modular forms, and (c)
the finiteness of irreducible components of the eigencurve. In addition,
applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce
as corollaries in the reducible and strongly generic case, (d) Gouvea--Mazur
conjecture, (e) a variant of Gouvea's conjecture on slope distributions, and
(f) a refined version of Coleman's spectral halo conjecture.Comment: 97 pages; comments are welcom
Constructible sheaves on schemes
We present a uniform theory of constructible sheaves on arbitrary schemes
with coefficients in topological or even condensed rings. This is accomplished
by defining lisse sheaves to be the dualizable objects in the derived
infinity-category of pro\'etale sheaves, while constructible sheaves are those
that are lisse on a stratification. We show that constructible sheaves satisfy
pro\'etale descent. We also establish a t-structure on constructible sheaves in
a wide range of cases. We finally provide a toolset to manipulate categories of
constructible sheaves with respect to the choices of coefficient rings, and use
this to prove that our notions reproduce and extend the various approaches to,
say, constructible ell-adic sheaves in the literature.Comment: This paper has been split off arXiv:2012.02853. Comments welcome
Deformation theory of G-valued pseudocharacters and symplectic determinant laws
We give an introduction to the theory of pseudorepresentations of Taylor, Rouquier, Chenevier and
Lafforgue. We refer to Taylorâs and Rouquierâs pseudorepresentations as pseudocharacters. They are
very closely related, the main difference being that Taylorâs pseudocharacters are defined for a group,
where as Rouquierâs pseudocharacters are defined for algebras. Chenevierâs pseudorepresentations are
so-called polynomial laws and will be called determinant laws. Lafforgueâs pseudorepresentations are a
generalization of Taylorâs pseudocharacters to other reductive groups G, in that the corresponding notion
of representation is that of a G-valued representation of a group. We refer to them as G-pseudocharacters.
We survey the known comparison theorems, notably Emersonâs bijection between Chenevierâs determinant
laws and Lafforgueâs GL(n)-pseudocharacters and the bijection with Taylorâs pseudocharacters away from
small characteristics.
We show, that duals of determinant laws exist and are compatible with duals of representations. Analogously,
we obtain that tensor products of determinant laws exist and are compatible with tensor products
of representations. Further the tensor product of Lafforgueâs pseudocharacters agrees with the tensor
product of Taylorâs pseudocharacters.
We generalize some of the results of [Che14] to general reductive groups, in particular we show that
the (pseudo)deformation space of a continuous Lafforgue G-pseudocharacter of a topologically finitely
generated profinite group Î with values in a finite field (of characteristic p) is noetherian. We also show,
that for specific groups G it is sufficient, that Î satisfies Mazurâs condition Ί_p.
One further goal of this thesis was to generalize parts of [BIP21] to other reductive groups. Let F/Qp
be a finite extension. In order to carry this out for the symplectic groups Sp2d, we obtain a simple and
concrete stratification of the special fiber of the pseudodeformation space of a residual G-pseudocharater
of Gal(F) into obstructed subloci Xdec(Î), Xpair(Î), Xspcl(Î) of dimension smaller than the expected dimension
n(2n + 1)[F : Qp].
We also prove that Lafforgueâs G-pseudocharacters over algebraically closed fields for possibly nonconnected
reductive groups G come from a semisimple representation. We introduce a formal scheme
and a rigid analytic space of all G-pseudocharacters by a functorial description and show, building on
our results of noetherianity of pseudodeformation spaces, that both are representable and admit a decomposition
as a disjoint sum indexed by continuous pseudocharacters with values in a finite field up to
conjugacy and Frobenius automorphisms.
At last, in joint work with Mohamed Moakher, we give a new definition of determinant laws for symplectic
groups, which is based on adding a âPfaffian polynomial lawâ to a determinant law which is invariant under
an involution. We prove the expected basic properties in that we show that symplectic determinant laws
over algebraically closed fields are in bijection with conjugacy classes of semisimple representation and
that Cayley-Hamilton lifts of absolutely irreducible symplectic determinant laws to henselian local rings
are in bijection with conjugacy classes of representations. We also give a comparison map with Lafforgueâs
pseudocharacters and show that it is an isomorphism over reduced rings
Periodicity of ideals of minors in free resolutions
We study the asymptotic behavior of the ideals of minors in minimal free
resolutions over local rings. In particular, we prove that such ideals are
eventually 2-periodic over complete intersections and Golod rings. We also
establish general results on the stable behavior of ideals of minors in any
infinite minimal free resolution. These ideals have intimate connections to
trace ideals and cohomology annihilators. Constraints on the stable values
attained by the ideals of minors in many situations are obtained, and they can
be explicitly computed in certain cases.Comment: 28 page
The local Picard group of a ring extension
Given an integral domain and a -algebra , we introduce the local
Picard group as the quotient between the Picard group
and the canonical image of in
, and its subgroup generated by the the
integral ideals of that are unitary with respect to . We show that, when
is a ring extension that satisfies certain properties (for
example, when is the ring of polynomial or the ring of
integer-valued polynomials ), it is possible to decompose
as the direct sum , where
ranges in a Jaffard family of . We also study under what hypothesis this
isomorphism holds for pre-Jaffard families of
Diagonal F-splitting and Symbolic Powers of Ideals
Let be any ideal in a strongly -regular, diagonally -split ring
essentially of finite type over an -finite field. We show that for all for which the formula makes sense. We use this to show a number of novel
containments between symbolic and ordinary powers of prime ideals in this
setting, which includes all determinantal rings and a large class of toric
rings in positive characteristic. In particular, we show that for all prime ideals of height in such rings.Comment: Many small changes. Notably, I added missing Noetherianity and
reducedness assumptions to section 2 and corrected an error in lemma 2.2.
Upon reflection, the assumptions on and in prop 5.3 were just
slightly more general than assuming and were fields, so I went ahead
and said and should be field
Interpolation Problems and the Characterization of the Hilbert Function
In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi â 1 derivative for each point. A natural question would then be to consider the case in higher dimensions corresponding to polynomials in several variables. For this thesis, we will classify polynomials in three variables passing through a set of discrete points using an abstract algebraic structure known as an ideal. Then, we will analyze these ideals and specifically provide structural and numerical information. That is, we characterize their Hilbert Functions, which in our setting, are functions describing the number of linearly independent polynomials passing through the set of up to six points with a given multiplicity. Specifically, we will also see that in these cases, there is an expected âmaximalâ Hilbert Function value, and the main goal is to determine whether these ideals have the âmaximalâ Hilbert Function or not
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