IA Scholar Query: Homomorphism Tensors and Linear Equations.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 01 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Isadore M. Singer (1924–2021) In Memoriam Part 1: Scientific Works
https://scholar.archive.org/work/aejx3oq2lvch5gdpwoqpzzlbqe
Robert Bryant, Jean-Michel Bismut, Jeff Cheeger, Phillip Griffiths, Simon Donaldson, Nigel Hitchin, H Blaine Lawson, Michail Gromov, Adam Marcus, Daniel Spielman, Nikhil Srivastava, Edward Wittenwork_aejx3oq2lvch5gdpwoqpzzlbqeSat, 01 Oct 2022 00:00:00 GMTK-stability of Gorenstein Fano group compactifications with rank two
https://scholar.archive.org/work/fqxgiqqtrfe23hmqdf6onzrwkq
We give a classification of Gorenstein Fano bi-equivariant compactifications of semisimple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kähler-Einstein metrics. As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) Kähler-Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties. This gives us three new examples on which each solution of the Kähler-Ricci flow is of type II.Jae-Hyouk Lee, Kyeong-Dong Park, Sungmin Yoowork_fqxgiqqtrfe23hmqdf6onzrwkqFri, 16 Sep 2022 00:00:00 GMTSpan(Graph): a Canonical Feedback Algebra of Open Transition Systems
https://scholar.archive.org/work/iklphrfi3zechgg3cc7h3rlmgq
We show that Span(Graph)*, an algebra for open transition systems introduced by Katis, Sabadini and Walters, satisfies a universal property. By itself, this is a justification of the canonicity of this model of concurrency. However, the universal property is itself of interest, being a formal demonstration of the relationship between feedback and state. Indeed, feedback categories, also originally proposed by Katis, Sabadini and Walters, are a weakening of traced monoidal categories, with various applications in computer science. A state bootstrapping technique, which has appeared in several different contexts, yields free such categories. We show that Span(Graph)* arises in this way, being the free feedback category over Span(Set). Given that the latter can be seen as an algebra of predicates, the algebra of open transition systems thus arises - roughly speaking - as the result of bootstrapping state to that algebra. Finally, we generalize feedback categories endowing state spaces with extra structure: this extends the framework from mere transition systems to automata with initial and final states.Elena Di Lavore, Alessandro Gianola, Mario Román, Nicoletta Sabadini, Paweł Sobocińskiwork_iklphrfi3zechgg3cc7h3rlmgqFri, 16 Sep 2022 00:00:00 GMTLearning the conditional law: signatures and conditional GANs in filtering and prediction of diffusion processes
https://scholar.archive.org/work/yfkvwoqy6rgztpw2wer65k7n64
We consider the filtering and prediction problem for a diffusion process. The signal and observation are modeled by stochastic differential equations (SDEs) driven by correlated Wiener processes. In classical estimation theory, measure-valued stochastic partial differential equations (SPDEs) are derived for the filtering and prediction measures. These equations can be hard to solve numerically. We provide an approximation algorithm using conditional generative adversarial networks (GANs) in combination with signatures, an object from rough path theory. The signature of a sufficiently smooth path determines the path completely. As a result, in some cases, GANs based on signatures have been shown to efficiently approximate the law of a stochastic process. For our algorithm we extend this method to sample from the conditional law, given noisy, partial observation. Our generator is constructed using neural differential equations (NDEs), relying on their universal approximator property. We show well-posedness in providing a rigorous mathematical framework. Numerical results show the efficiency of our algorithm.Fabian Germ, Marc Sabate-Vidaleswork_yfkvwoqy6rgztpw2wer65k7n64Thu, 15 Sep 2022 00:00:00 GMTMirror Symmetry for Quiver Algebroid Stacks
https://scholar.archive.org/work/443ubhpv7ja7hfk42bwjcuktey
In this paper, we construct noncommutative algebroid stacks and the associated mirror functors for a symplectic manifold. First, we formulate a version of algebroid stack that is well adapted to gluing quiver algebras with different numbers of vertices. Second, we develop a representation theory of A_∞ categories by quiver stacks. A key step is constructing an extension of the A_∞ category over a quiver stack of a collection of nc-deformed objects. The extension involves non-trivial gerbe terms, which play an important role for quiver algebroid stacks. Third, we apply the theory to construct mirror quiver stacks of local Calabi-Yau manifolds. In this paper, we focus on nc local projective plane. This example has a compact divisor which gives rise to interesting monodromy and homotopy terms that can be found from mirror symmetry. Geometrically, we find a new method of 'gluing through a middle agent' for mirror construction using Floer theory. The method makes crucial use of the extension of Fukaya category over quiver stacks.Siu-Cheong Lau, Junzheng Nan, Ju Tanwork_443ubhpv7ja7hfk42bwjcukteyThu, 15 Sep 2022 00:00:00 GMTA Jacobian Criterion for Artin v-stacks
https://scholar.archive.org/work/s7w2ril5fbfjdl3k2wurd3ayte
We prove a generalization of the Jacobian criterion of Fargues-Scholze for spaces of sections of a smooth quasi-projective variety over the Fargues-Fontaine curve. Namely, we show how to use their criterion to deduce an analogue for spaces of sections of a smooth Artin stack over the (schematic) Fargues-Fontaine curve obtained by taking the stack quotient of a smooth quasi-projective variety by the action of a linear algebraic group. As an application, we show various moduli stacks appearing in the Fargues-Scholze geometric Langlands program are cohomologically smooth Artin v-stacks and compute their ℓ-dimensions.Linus Hamannwork_s7w2ril5fbfjdl3k2wurd3ayteThu, 15 Sep 2022 00:00:00 GMTHolomorphic Surface Defects in Four-Dimensional Chern-Simons Theory
https://scholar.archive.org/work/foc7wvtiwjc5pay45253mqmury
We derive the framing anomaly of four-dimensional holomorphic-topological Chern-Simons theory formulated on the product of a topological surface and the complex plane. We show that the presence of this anomaly allows one to couple four-dimensional Chern-Simons theory to holomorphic field theories with Kac-Moody symmetry, where the Kac-Moody level k is critical k=-h^∨. Applying this result to a holomorphic sigma model into a complex coadjoint orbit, we derive that four-dimensional Chern-Simons theory admits holomorphic monodromy defects.Ahsan Z. Khanwork_foc7wvtiwjc5pay45253mqmuryThu, 15 Sep 2022 00:00:00 GMTEquivalent definitions of Arthur packets for real unitary groups
https://scholar.archive.org/work/dnbo7dxp6jeujp4rx5ngg5vn5u
Mok and Moeglin-Renard have defined Arthur packets for unitary groups. Their definitions follow Arthur's work on classical groups and rely on harmonic analysis. For real groups there is an alternative definition of Arthur packets due to Adams-Barbasch-Vogan. It relies on sheaf-theoretic techniques instead of harmonic analysis. We prove that these two definitions of Arthur packets are equivalent in the case of real unitary groups.Nicolas Arancibia Robert, Paul Mezowork_dnbo7dxp6jeujp4rx5ngg5vn5uThu, 15 Sep 2022 00:00:00 GMTEquivariant infinite loop space theory, the space level story
https://scholar.archive.org/work/g6bbyzjb2jcndooldwpuvwbgya
We rework and generalize equivariant infinite loop space theory, which shows how to construct G-spectra from G-spaces with suitable structure. There is a classical version which gives classical Ω-G-spectra for any topological group G, but our focus is on the construction of genuine Ω-G-spectra when G is finite. We also show what is and is not true when G is a compact Lie group. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for classical G-spectra for general G but fails for genuine G-spectra. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving an illuminating direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving a number of corrections to results and proofs in the literature.J. Peter May, Mona Merling, Angélica M. Osornowork_g6bbyzjb2jcndooldwpuvwbgyaThu, 15 Sep 2022 00:00:00 GMTVafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification
https://scholar.archive.org/work/ebvujcbl7faq5p5uwpldamh5le
We revisit Vafa-Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa-Witten equations. We physically derive (i) a novel Vafa-Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov-Witten invariants, (iii) a novel Vafa-Witten Floer homology assigned to three-manifold boundaries, (iv) a novel Vafa-Witten Atiyah-Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid-Manolescu in [1] about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homologies and hypercohomology, and (vii) a quantum geometric Langlands correspondence with purely imaginary parameter that specializes to the classical correspondence in the zero-coupling limit, where Higgs bundles feature in (ii), (iv), (vi) and (vii). We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification. We therefore relate differential and enumerative geometry, topology and geometric representation theory in mathematics, via a maximally-supersymmetric topological quantum field theory with electric-magnetic duality in physics.Zhi-Cong Ong, Meng-Chwan Tanwork_ebvujcbl7faq5p5uwpldamh5leThu, 15 Sep 2022 00:00:00 GMTOn a new geometric homology theory and an application in categorical Gromov-Witten theory
https://scholar.archive.org/work/ppkoqike2rh6xad4lxylkhnevi
The purpose of this paper is twofold: 1. we prove the triangulability of smooth orbifolds with corners, generalizing the same statement for orbifolds. 2. based on 1, we propose a new homology theory. We call it geometric homology theory (GHT for abbreviaty). GHT is a natural and flexible generalization of singular homology. It has some advantages overcoming the unpleasant combinatoric rigidity of singular homology, e.g. undefiness of pullbacks along fiber bundles. The method we use are mainly based on the celebrated stratification and triangulation theories of Lie groupoids and their orbit spaces, as well as the extension to Lie groupoids with corners by us. We illustrate a simple application of GHT in categorical Gromov-Witten theory, initiatied by Costello. We will develop further of this theory in our sequel paper.Hao Yuwork_ppkoqike2rh6xad4lxylkhneviThu, 15 Sep 2022 00:00:00 GMTPro-isomorphic zeta functions of nilpotent groups and Lie rings under base extension
https://scholar.archive.org/work/rr6bi5sc7jfjndsfcaikg6e4vi
We consider pro-isomorphic zeta functions of the groups Γ(𝒪_K), where Γ is a unipotent group scheme defined over ℤ and K varies over all number fields. Under certain conditions, we show that these functions have a fine Euler decomposition with factors indexed by primes 𝔭 of K and depending only on the structure of Γ, the degree [K : ℚ], and the cardinality of the residue field 𝒪_K / 𝔭. We show that the factors satisfy a certain uniform rationality and study their dependence on [K : ℚ]. Explicit computations are given for several families of unipotent groups. These include an apparently novel identity involving permutation statistics on the hyperoctahedral group.Mark N. Berman, Itay Glazer, Michael M. Scheinwork_rr6bi5sc7jfjndsfcaikg6e4viThu, 15 Sep 2022 00:00:00 GMTOn Amenable and Coamenable Coideals
https://scholar.archive.org/work/qey3cpla2ncmfatgnk4rkl3zgi
We study relative amenability and amenability of a right coideal N_P⊆ℓ^∞(𝔾) of a discrete quantum group in terms its group-like projection P. We establish a notion of a P-left invariant state and use it to characterize relative amenability. We also develop a notion of coamenability of a compact quasi-subgroup N_ω⊆ L^∞(𝔾) that generalizes coamenability of a quotient as defined by Kalantar, Kasprzak, Skalski, and Vergnioux, where 𝔾 is the compact dual of 𝔾. In particular, we establish that the coamenable compact quasi-subgroups of 𝔾 are in one-to-one correspondence with the idempotent states on the reduced C^*-algebra C_r(𝔾). We use this work to obtain results for the duality between relative amenability and amenability of coideals in ℓ^∞(𝔾) and coamenability of their codual coideals in L^∞(𝔾), making progress towards a question of Kalantar et al..Benjamin Anderson-Sackaneywork_qey3cpla2ncmfatgnk4rkl3zgiThu, 15 Sep 2022 00:00:00 GMTAlmost Ramanujan Expanders from Arbitrary Expanders via Operator Amplification
https://scholar.archive.org/work/5mrwukheqffgvfoykq7munqdhq
We give an efficient algorithm that transforms any bounded degree expander graph into another that achieves almost optimal (namely, near-quadratic, d ≤ 1/λ^2+o(1)) trade-off between (any desired) spectral expansion λ and degree d. Furthermore, the algorithm is local: every vertex can compute its new neighbors as a subset of its original neighborhood of radius O(log(1/λ)). The optimal quadratic trade-off is known as the Ramanujan bound, so our construction gives almost Ramanujan expanders from arbitrary expanders. The locality of the transformation preserves structural properties of the original graph, and thus has many consequences. Applied to Cayley graphs, our transformation shows that any expanding finite group has almost Ramanujan expanding generators. Similarly, one can obtain almost optimal explicit constructions of quantum expanders, dimension expanders, monotone expanders, etc., from existing (suboptimal) constructions of such objects. Another consequence is a "derandomized" random walk on the original (suboptimal) expander with almost optimal convergence rate. Our transformation also applies when the degree is not bounded or the expansion is not constant. We obtain our results by a generalization of Ta-Shma's technique in his breakthrough paper [STOC 2017], used to obtain explicit almost optimal binary codes. Specifically, our spectral amplification extends Ta-Shma's analysis of bias amplification from scalars to matrices of arbitrary dimension in a very natural way. Curiously, while Ta-Shma's explicit bias amplification derandomizes a well-known probabilistic argument (underlying the Gilbert–Varshamov bound), there seems to be no known probabilistic (or other existential) way of achieving our explicit ("high-dimensional") spectral amplification.Fernando Granha Jeronimo, Tushant Mittal, Sourya Roy, Avi Wigdersonwork_5mrwukheqffgvfoykq7munqdhqThu, 15 Sep 2022 00:00:00 GMTThe Dirac-Dolbeault Operator Approach to the Hodge Conjecture
https://scholar.archive.org/work/yoyaggt5yjaqdbel6qts3hb27i
The Dirac-Dolbeault operator for a compact oriented K\"ahler manifold is a special case of a Dirac operator. The Green function for the Dirac Laplacian over a manifold with boundary allows to express the values of the sections of the Dirac bundle in terms of the values on the boundary, extending the mean value theorem of harmonic analysis. Utilizing this representation and the Nash-Moser generalized inverse function theorem we prove the existence of complex submanifolds of a compact oriented variety satisfying globally a certain partial differential equation under a certain injectivity assumption. Next, we show the existence of complex submanifolds whose fundamental classes span the rational Hodge classes, proving the Hodge conjecture for non singular projective algebraic varieties.Simone Farinelliwork_yoyaggt5yjaqdbel6qts3hb27iThu, 15 Sep 2022 00:00:00 GMTCrypTen: Secure Multi-Party Computation Meets Machine Learning
https://scholar.archive.org/work/rwis75nn5bcljldbedj3v2pcea
Secure multi-party computation (MPC) allows parties to perform computations on data while keeping that data private. This capability has great potential for machine-learning applications: it facilitates training of machine-learning models on private data sets owned by different parties, evaluation of one party's private model using another party's private data, etc. Although a range of studies implement machine-learning models via secure MPC, such implementations are not yet mainstream. Adoption of secure MPC is hampered by the absence of flexible software frameworks that "speak the language" of machine-learning researchers and engineers. To foster adoption of secure MPC in machine learning, we present CrypTen: a software framework that exposes popular secure MPC primitives via abstractions that are common in modern machine-learning frameworks, such as tensor computations, automatic differentiation, and modular neural networks. This paper describes the design of CrypTen and measure its performance on state-of-the-art models for text classification, speech recognition, and image classification. Our benchmarks show that CrypTen's GPU support and high-performance communication between (an arbitrary number of) parties allows it to perform efficient private evaluation of modern machine-learning models under a semi-honest threat model. For example, two parties using CrypTen can securely predict phonemes in speech recordings using Wav2Letter faster than real-time. We hope that CrypTen will spur adoption of secure MPC in the machine-learning community.Brian Knott and Shobha Venkataraman and Awni Hannun and Shubho Sengupta and Mark Ibrahim and Laurens van der Maatenwork_rwis75nn5bcljldbedj3v2pceaThu, 15 Sep 2022 00:00:00 GMT6d SCFTs, Center-Flavor Symmetries, and Stiefel–Whitney Compactifications
https://scholar.archive.org/work/pppjl7wvgjcb7mkh2emb7s647q
The center-flavor symmetry of a gauge theory specifies the global form of consistent gauge and flavor bundle background field configurations. For 6d gauge theories which arise from a tensor branch deformation of a superconformal field theory (SCFT), we determine the global structure of such background field configurations, including possible continuous Abelian symmetry and R-symmetry bundles. Proceeding to the conformal fixed point, this provides a prescription for reading off the global form of the continuous factors of the zero-form symmetry, including possible non-trivial mixing between flavor and R-symmetry. As an application, we show that this global structure leads to a large class of 4d 𝒩 = 2 SCFTs obtained by compactifying on a T^2 in the presence of a topologically non-trivial flat flavor bundle characterized by a 't Hooft magnetic flux. The resulting "Stiefel–Whitney twisted" compactifications realize several new infinite families of 4d 𝒩 = 2 SCFTs, and also furnish a 6d origin for a number of recently discovered rank one and two 4d 𝒩 = 2 SCFTs.Jonathan J. Heckman, Craig Lawrie, Ling Lin, Hao Y. Zhang, Gianluca Zoccaratowork_pppjl7wvgjcb7mkh2emb7s647qWed, 14 Sep 2022 00:00:00 GMTThe geometric distribution of Selmer groups of elliptic curves over function fields
https://scholar.archive.org/work/cxzsq3b6urgjbjwyq6qhwqxwkq
Fix a positive integer n and a finite field 𝔽_q. We study the joint distribution of the rank of E, the n-Selmer group of E, and the n-torsion in the Tate-Shafarevich group of E as E varies over elliptic curves of fixed height d ≥ 2 over 𝔽_q(t). We compute this joint distribution in the large q limit. We also show that the "large q, then large height" limit of this distribution agrees with the one predicted by Bhargava-Kane-Lenstra-Poonen-Rains.Tony Feng, Aaron Landesman, Eric M. Rainswork_cxzsq3b6urgjbjwyq6qhwqxwkqWed, 14 Sep 2022 00:00:00 GMTHorn conditions for quiver subrepresentations and the moment map
https://scholar.archive.org/work/hi5h6sxw2ngfhhylt2ug3lv5lq
We give inductive conditions that characterize the Schubert positions of subrepresentations of a general quiver representation. Our results generalize Belkale's criterion for the intersection of Schubert varieties in Grassmannians and refine Schofield's characterization of the dimension vectors of general subrepresentations. This implies Horn type inequalities for the moment cone associated to the linear representation of the group G=∏_x GL(n_x) associated to a quiver and a dimension vector 𝐧=(n_x).Velleda Baldoni and Michèle Vergne and Michael Walterwork_hi5h6sxw2ngfhhylt2ug3lv5lqWed, 14 Sep 2022 00:00:00 GMTClassification of K-forms in nilpotent Lie algebras associated to graphs
https://scholar.archive.org/work/ypnzkretyzewnl5kqv7ebaehma
Given a simple undirected graph, one can construct from it a c-step nilpotent Lie algebra for every c ≥ 2 and over any field K, in particular also over the real and complex numbers. These Lie algebras form an important class of examples in geometry and algebra, and it is interesting to link their properties to the defining graph. In this paper, we classify the isomorphism classes of K-forms in these real and complex Lie algebras for any subfield K ⊂ℂ from the structure of the graph. As an application, we show that the number of rational forms up to isomorphism is always one or infinite, with the former being true if and only if the group of graph automorphisms is generated by transpositions.Jonas Deré, Thomas Witdouckwork_ypnzkretyzewnl5kqv7ebaehmaWed, 14 Sep 2022 00:00:00 GMT