Noncommutative geometry and conformal geometry. III. Vafa–Witten inequality and Poincaré duality

Ponge, Raphaë,l,Wang, Hang Elsevier 2015 Advances in mathematics Vol.272 No.-

<P>This paper is the third part of a series of papers whose aim is to use the framework of twisted spectral triples to study conformal geometry from a noncommutative geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten [42] in the setting of twisted spectral triples. This involves a notion of Poincare duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control. of the Vafa-Witten bound under conformal changes of metrics. This result has several noncommutative manifestations for conformal deformations of ordinary spectral triples, spectral triples associated with conformal weights on noncommutative tori, and spectral triples associated with duals of torsion-free discrete cocompact subgroups satisfying the Baum-Connes conjecture. (C) 2015 Elsevier Inc. All rights reserved.</P>

Cyclic homology and group actions

Ponge, Raphaë,l Elsevier 2018 Journal of geometry and physics Vol.123 No.-

<P><B>Abstract</B></P> <P>In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with algebraic crossed-products associated with group actions on unital algebras over any ring k ⊃ Q . In the second part, we extend the results to actions on locally convex algebras. We then deal with crossed-products associated with group actions on manifolds and smooth varieties. For the finite order components, the results are expressed in terms of what we call “mixed equivariant cohomology”. This “mixed” theory mediates between group homology and de Rham cohomology. It is naturally related to equivariant cohomology, and so we obtain explicit constructions of cyclic cycles out of equivariant characteristic classes. For the infinite order components, we simplify and correct the misidentification of Crainic (1999). An important new homological tool is the notion of “triangular S -module”. This is a natural generalization of the cylindrical complexes of Getzler–Jones. It combines the mixed complexes of Burghelea–Kassel and parachain complexes of Getzler–Jones with the S -modules of Kassel–Jones. There are spectral sequences naturally associated with triangular S -modules. In particular, this allows us to recover spectral sequences of Feigin–Tsygan and Getzler–Jones and leads us to a new spectral sequence.</P>

Index map, $\sigma$ -connections, and Connes-Chern character in the setting of twisted spectral triples

Ponge, Raphaë,l,Wang, Hang Duke University Press 2016 Kyoto Journal of Mathematics Vol.56 No.2

Noncommutative geometry and conformal geometry. I. Local index formula and conformal invariants

Ponge, Raphaë,l,Wang, Hang European Mathematical Society Publishing House 2018 Journal of noncommutative geometry Vol.12 No.4

Strain hardening by dynamic slip band refinement in a high-Mn lightweight steel

Welsch, E.,Ponge, D.,Hafez Haghighat, S.M.,Sandlö,bes, S.,Choi, P.,Herbig, M.,Zaefferer, S.,Raabe, D. Elsevier 2016 Acta materialia Vol.116 No.-

<P>The strain hardening mechanism of a high-Mn lightweight steel (Fe-30.4Mn-8A1-1.2C (wt%)) is investigated by electron channeling contrast imaging (ECCI) and transmission electron microscopy (TEM). The alloy is characterized by a constant high strain hardening rate accompanied by high strength and high ductility (ultimate tensile strength: 900 MPa, elongation to fracture: 68%). Deformation microstructures at different strain levels are studied in order to reveal and quantify the governing structural parameters at micro- and nanometer scales. As the material deforms mainly by planar dislocation slip causing the formation of slip bands, we quantitatively study the evolution of the slip band spacing during straining. The flow stress is calculated from the slip band spacing on the basis of the passing stress. The good agreement between the calculated values and the tensile test data shows dynamic slip band refinement as the main strain hardening mechanism, enabling the excellent mechanical properties. This novel strain hardening mechanism is based on the passing stress acting between co-planar slip bands in contrast to earlier attempts to explain the strain hardening in high-Mn lightweight steels that are based on grain subdivision by microbands. We discuss in detail the formation of the finely distributed slip bands and the gradual reduction of the spacing between them, leading to constantly high strain hardening. TEM investigations of the precipitation state in the as-quenched state show finely dispersed atomically ordered clusters (size < 2 nm). The influence of these zones on planar slip is discussed. (C) 2016 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.</P>

Tangent maps and tangent groupoid for Carnot manifolds

Choi, Woocheol,Ponge, Raphaë,l Elsevier 2019 Differential geometry and its applications Vol.62 No.-

<P><B>Abstract</B></P> <P>This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold ( M , H ) we get a smooth groupoid that encodes the smooth deformation of the pair M × M to the tangent group bundle <I>GM</I>. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche .</P>

Strengthening and strain hardening mechanisms in a precipitation-hardened high-Mn lightweight steel

Yao, M.J.,Welsch, E.,Ponge, D.,Haghighat, S.M.H.,Sandlö,bes, S.,Choi, P.,Herbig, M.,Bleskov, I.,Hickel, T.,Lipinska-Chwalek, M.,Shanthraj, P.,Scheu, C.,Zaefferer, S.,Gault, B.,Raabe, D. Elsevier 2017 ACTA MATERIALIA Vol.140 No.-

<P>We report on the strengthening and strain hardening mechanisms in an aged high-Mn lightweight steel (Fe-30.4Mn-8Al-1.2C, wt.%) studied by electron channeling contrast imaging (ECCI), transmission electron microscopy (TEM), atom probe tomography (APT) and correlative TEM/APT. Upon isothermal annealing at 600 degrees C, nano-sized kappa-carbides form,, as characterized by TEM arid APT. The resultant alloy exhibits high strength and excellent ductility accompanied by a high constant strain hardening rate.& para;& para;In comparison to the as-quenched kappa-free state, the precipitation of kappa-carbides leads to a significant increase in yield strength (similar to 480 MPa) without sacrificing much tensile elongation. To study the strengthening and strain hardening behavior of the precipitation-hardened material, deformation microstructures were analyzed at different strain levels. TEM and correlative TEM/APT results show that the kappa-carbides are primarily sheared by lattice dislocations, gliding on the typical face-centered-cubic (fcc) slip system {111 }<110>, leading to particle dissolution and solute segregation. Ordering strengthening is the predominant strengthening mechanism. As the deformation substructure is characterized by planar slip bands, we quantitatively studied the evolution of the slip band spacing during straining to under stand the strain hardening behavior. A good agreement between the calculated flow stresses and the experimental data suggests that dynamic slip band refinement is the main strain hardening mechanism. The influence of kappa-carbides on mechanical properties is discussed by comparing the results with that of the same alloy in the as-quenched, kappa-free state. (C) 2017 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved</P>

The effects of prior austenite grain boundaries and microstructural morphology on the impact toughness of intercritically annealed medium Mn steel

Han, Jeongho,da Silva, Alisson Kwiatkowski,Ponge, Dirk,Raabe, Dierk,Lee, Sang-Min,Lee, Young-Kook,Lee, Sang-In,Hwang, Byoungchul Elsevier 2017 Acta materialia Vol.122 No.-

<P><B>Abstract</B></P> <P>The effects of prior austenite (γ) grain boundaries and microstructural morphology on the impact toughness of an annealed Fe-7Mn-0.1C-0.5Si medium Mn steel were investigated for two different microstructure states, namely, hot-rolled and annealed (HRA) specimens and cold-rolled and annealed (CRA) specimens. Both types of specimens had a dual-phase microstructure consisting of retained austenite (γ<SUB>R</SUB>) and ferrite (α) after intercritical annealing at 640 °C for 30 min. The phase fractions and the chemical composition of γ<SUB>R</SUB> were almost identical in both types of specimens. However, their microstructural morphology was different. The HRA specimens had lath-shaped morphology and the CRA specimens had globular-shaped morphology. We find that both types of specimens showed transition in fracture mode from ductile and partly quasi-cleavage fracture to intergranular fracture with decreasing impact test temperature from room temperature to −196 °C. The HRA specimen had higher ductile to brittle transition temperature and lower low-temperature impact toughness compared to the CRA specimen. This was due to intergranular cracking in the HRA specimens along prior γ grain boundaries decorated by C, Mn and P. In the CRA specimen intergranular cracking occurred along the boundaries of the very fine α and α′ martensite grains. The results reveal that cold working prior to intercritical annealing promotes the elimination of the solute-decorated boundaries of coarse prior γ grains through the recrystallization of αʹ martensite prior to reverse transformation, hence improving the low-temperature impact toughness of medium Mn steel.</P> <P><B>Graphical abstract</B></P> <P>[DISPLAY OMISSION]</P>

Conformal Invariants from Nodal Sets. I. Negative Eigenvalues and Curvature Prescription

Canzani, Yaiza,Gover, Rod,Jakobson, Dmitry,Ponge, Raphaë,l Oxford University Press 2014 International mathematics research notices Vol.2014 No.9

<P>In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the Graham, Jenne, Mason, and Sparling (GJMS) operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant’s Nodal Domain Theorem. We also show that on any manifold of dimension <I>n</I>≥3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension <I>n</I>≥3. We obtain similar results for some higher order GJMS operators on some Einstein and Heisenberg manifolds. We describe the invariants arising from the Yamabe and Paneitz operators associated to left-invariant metrics on Heisenberg manifolds. Finally, in Appendix, the second named author and Andrea Malchiodi study the <I>Q</I>-curvature prescription problems for noncritical <I>Q</I>-curvatures.</P>

Reversion to Ultrafine-Grained Austenite in a Medium-Mn AHSS

Benzing, J T,Bentley, J,da Silva, A Kwiatkowski,Morsdorf, L,McBride, J R,Ponge, D,Gault, B,Han, J,Raabe, D,Wittig, J E Cambridge University Press 2018 Microscopy and Microanalysis Vol.24 No.suppl1