Locally trivial bundles of [Formula: see text]-algebras with fiber [Formula: see text] for a strongly self-absorbing [Formula: see text]-algebra [Formula: see text] over a finite CW-complex [Formula: see text] form a group [Formula: see text] that is the first group of a cohomology theory [Formula: see text]. In this paper, we compute these groups by expressing them in terms of ordinary cohomology and connective [Formula: see text]-theory. To compare the [Formula: see text]-algebraic version of [Formula: see text] with its classical counterpart we also develop a uniqueness result for the unit spectrum of complex periodic topological [Formula: see text]-theory.
more »
« less
The companion section for classical groups
We use the companion matrix construction for [Formula: see text] to build canonical sections of the Chevalley map [Formula: see text] for classical groups [Formula: see text] as well as the group [Formula: see text]. To do so, we construct canonical tensors on the associated spectral covers. As an application, we make explicit lattice descriptions of affine Springer fibers and Hitchin fibers for classical groups and [Formula: see text].
more »
« less
- Award ID(s):
- 2201314
- PAR ID:
- 10637531
- Publisher / Repository:
- World Scientific Connect
- Date Published:
- Journal Name:
- International Journal of Mathematics
- Volume:
- 35
- Issue:
- 09
- ISSN:
- 0129-167X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra [Formula: see text] defined in Hartglass–Penneys’ article on canonical [Formula: see text]-algebras associated to a planar algebra. Under mild conditions, [Formula: see text] is a non-nuclear simple [Formula: see text]-algebra with unique tracial state. There is a canonical polynomial subalgebra [Formula: see text] together with a Dirac number operator [Formula: see text] such that [Formula: see text] is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify [Formula: see text] yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our [Formula: see text]-algebras are non-nuclear, we adjust the Lip-norm coming from [Formula: see text] to utilize the finite dimensional filtration of [Formula: see text]. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) [Formula: see text]-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS [Formula: see text]-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance.more » « less
-
Let [Formula: see text] be an integer and [Formula: see text] be a finite field with [Formula: see text] elements. We prove several results on the distribution in short intervals of polynomials in [Formula: see text] that are not divisible by the [Formula: see text]th power of any non-constant polynomial. Our main result generalizes a recent theorem by Carmon and Entin on the distribution of squarefree polynomials to all [Formula: see text]. We also develop polynomial versions of the classical techniques used to study gaps between [Formula: see text]-free integers in [Formula: see text]. We apply these techniques to obtain analogs in [Formula: see text] of some classical theorems on the distribution of [Formula: see text]-free integers. The latter results complement the main theorem in the case when the degrees of the polynomials are of moderate size.more » « less
-
In this paper, we study kernel ridge-less regression, including the case of interpolating solutions. We prove that maximizing the leave-one-out ([Formula: see text]) stability minimizes the expected error. Further, we also prove that the minimum norm solution — to which gradient algorithms are known to converge — is the most stable solution. More precisely, we show that the minimum norm interpolating solution minimizes a bound on [Formula: see text] stability, which in turn is controlled by the smallest singular value, hence the condition number, of the empirical kernel matrix. These quantities can be characterized in the asymptotic regime where both the dimension ([Formula: see text]) and cardinality ([Formula: see text]) of the data go to infinity (with [Formula: see text] as [Formula: see text]). Our results suggest that the property of [Formula: see text] stability of the learning algorithm with respect to perturbations of the training set may provide a more general framework than the classical theory of Empirical Risk Minimization (ERM). While ERM was developed to deal with the classical regime in which the architecture of the learning network is fixed and [Formula: see text], the modern regime focuses on interpolating regressors and overparameterized models, when both [Formula: see text] and [Formula: see text] go to infinity. Since the stability framework is known to be equivalent to the classical theory in the classical regime, our results here suggest that it may be interesting to extend it beyond kernel regression to other overparameterized algorithms such as deep networks.more » « less
-
A homology class [Formula: see text] of a complex flag variety [Formula: see text] is called a line degree if the moduli space [Formula: see text] of 0-pointed stable maps to X of degree d is also a flag variety [Formula: see text]. We prove a quantum equals classical formula stating that any n-pointed (equivariant, [Formula: see text]-theoretic, genus zero) Gromov–Witten invariant of line degree on X is equal to a classical intersection number computed on the flag variety [Formula: see text]. We also prove an n-pointed analogue of the Peterson comparison formula stating that these invariants coincide with Gromov–Witten invariants of the variety of complete flags [Formula: see text]. Our formulas make it straightforward to compute the big quantum [Formula: see text]-theory ring [Formula: see text] modulo the ideal [Formula: see text] generated by degrees d larger than line degrees.more » « less
An official website of the United States government

