Constructs moduli stacks of quiver bundles and applies to Higgs bundles.
problem Classifying morphisms of vector bundles over a fixed base.
method General method for constructing moduli stacks of diagrams of vector bundles indexed by a simplicial set.
result Recovery of Nakajima quiver varieties and alternate construction of moduli stacks of Higgs bundles.
This article constructs the moduli stack of torsionfree G-jet-structures in homotopy type theory with one monadic modality. This yields a construction of this moduli stack for any ∞-topos equipped with any stable factorization systems. In the intended applications of this theory, the factorization systems are …
The constructions of the virtual Euler (or moduli) cycles and their properties are explained and developed systematically in the general abstract settings.
Abstract: Generalizes stability theories over toric varieties and Novikov type rings.
problem Stability theories over toric varieties and Novikov type base.
method Generalization of stability theories to families over toric varieties and their analytic analogues.
result Established properness of moduli of Calabi-Yau cones and Kahler-Ricci solitons.
Research characterizes intersection cohomology groups of gauge theories and cotangent bundles.
problem Characterizing intersection cohomology groups of Coulomb branch gauge theories.
method Uses geometric Satake correspondence for Kac-Moody settings.
result Sketches proof of conjecture in affine type A.
This is an expository article on the theory of Kuranishi structure and is based on a series of pdf files we uploaded for the discussion of the google group named `Kuranishi' (with its administrator H. Hofer). There we replied to several questions concerning Kuranishi structure raised by K. Wehrheim. At this stage we su…
The abstract discusses how Kähler-Einstein metrics relate to algebraic geometry.
problem Understanding the connection between Kähler-Einstein metrics and algebraic geometry.
method Exploring metric limits and rescalings of Kähler-Einstein metrics in relation to moduli spaces and singularities.
result Proposes tentative conjectural pictures connecting Kähler-Einstein metrics and algebraic geometry.
Following \cite{citeSavelyevVirtualMorsetheoryonOmegaHam(Momega).}, we develop here a connection between Morse theory for the (positive) Hofer length functional L:ΩHam(M,ω)→R, with Gromov-Witten/Floer theory, for monotone symplectic manifolds (M,ω). This gives some immediate restrictio…
Normal forms for equivariant maps in infinite dimensions established.
problem Establishing normal forms for equivariant maps in infinite-dimensional manifolds.
method Inspired by Lyapunov-Schmidt reduction and Kuranishi method, uses Slice Theorem for Fréchet manifolds.
result Abstract moduli spaces of equivariant maps are locally modeled on quotient by a compact group.
Abstract: New geometric incarnation of isomonodromy functors.
problem Classical isomonodromic deformations.
method Functorial upgrade of isomonodromic deformations using Lie groupoids.
result Geometric incarnation of isomonodromy functors as Morita equivalences.
Foundations of derived geometry in smooth settings.
problem Building tools for moduli spaces in differential geometry.
method Abstract structured spaces and universal properties in (∞,2)-categories. result Established derived flatness results for derived C∞-rings. Introduces linear K-systems for Hamiltonian Floer theory.
problem Constructing Floer cohomology for Hamiltonian systems on compact manifolds.
method Geometric realization of linear K-systems via pseudo-holomorphic curves and inductive construction of Kuranishi structures.
result Construction of Floer cohomology and isomorphism with singular cohomology.
The paper aims to mathematically define and learn abstractions from data.
problem Defining and learning abstractions from data.
method Characterize abstractions as summaries for answering queries, define leakiness as a loss function, and generalize classical statistics.
result A mathematical theory of abstraction can be learned from data.
Study on moduli spaces of Higgs bundles over Abelian varieties, focusing on their topology and polynomials.
problem Determine the topology and polynomials of moduli spaces of Higgs bundles over Abelian varieties.
method Analyzing the Poincaré polynomials and mixed Hodge polynomials of moduli spaces MAH(G) for various groups G and dimensions d. result Explicit formulas for Poincaré polynomials and mixed Hodge polynomials in specific cases, including rank 2 and 3 Higgs bundles.
Generative models use latent abstractions to create images.
problem Understanding how generative models create high-dimensional data like images.
method Developed a theoretical framework using SDE and information theory.
result Diffusion models can be seen as a non-linear filter driven by latent abstractions.
A local normal form theorem for smooth equivariant maps between Fréchet manifolds is established. Moreover, an elliptic version of this theorem is obtained. The proof these normal form results is inspired by the Lyapunov-Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces, and uses a s…
Researchers compute differential K-theory for moduli stacks.
problem Computing differential K-theory for moduli stacks of principal G-bundles.
method Using homotopy theory of presheaves of spaces and spectra, they formulate results in terms of invariant polynomials and representation rings.
result They successfully compute the connective differential K-theory and differential cohomology of moduli stacks.
The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.
problem Developing the moduli theory of Higgs bundles and understanding their geometric properties.
method Establishing non-abelian Hodge correspondences and studying Hitchin fibration.
result Computing the Poincaré polynomial of rank 2 moduli space and verifying topological mirror symmetry.
Develops moduli theory for Calabi-Yau pairs, constructing a projective space.
problem Constructing a moduli space for Calabi-Yau pairs at the Calabi-Yau wall.
method Develops moduli theory, proving S-completeness and Θ-reductivity, constructing projective moduli space. result Constructs a projective moduli space for degenerate P2 pairs. The orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing an…
The variation of Hodge structure of a Calabi-Yau 3-fold induces a canonical Kähler metric on its Kuranishi moduli space, known as the Weil-Petersson metric. Similarly, special pseudo Kähler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classific…
New method constructs moduli spaces of Lagrangian surfaces in CP^2 from grid diagrams.
problem Constructing explicit examples of triple grid diagrams for Lagrangian surfaces in CP^2.
method Elegant geometric construction reducing to linear algebra.
result Explicit construction of moduli space of triple grid diagrams.
The paper proposes a method to transfer knowledge across different settings using causal theory.
problem Learning transfer across similar but different settings.
method Bayesian perspective of causal theory induction, integrating instance-level associative learning and abstract-level structural causal knowledge.
result The proposed model achieved transfer behavior across trials and learning situations, unlike RL algorithms.
Decouples moduli groups in heterotic string theory cohomology.
problem Decomposing moduli groups in heterotic string theory cohomology.
method Computing cohomology groups at the standard embedding and showing their direct sum decomposition.
result Decomposes moduli groups into a direct sum of cohomologies at the standard embedding.
New proof for discrete Morse theory using combinatorial construction.
problem Verifying the Morse differential in discrete Morse homology.
method Combinatorial construction of flowlines in discrete Morse theory.
result Morse differential squares to zero in discrete Morse homology.
We study the four-dimensional effective theory arising from ten-dimensional heterotic supergravity compactified on manifolds with torsion. In particular, given the heterotic superpotential appropriately corrected at O(α′) to account for the Green-Schwarz anomaly cancellation mechanism, we investigate proper…
A distinctive property of human and animal intelligence is the ability to form abstractions by neglecting irrelevant information which allows to separate structure from noise. From an information theoretic point of view abstractions are desirable because they allow for very efficient information processing. In artifici…
Tabular Q-Learning with learned state abstractions solves continuous control tasks.
problem Challenging reinforcement learning problems in continuous control.
method Learned state abstraction to transform continuous state-space into discrete.
result Tabular Q-Learning with learned abstractions achieves efficient learning in unseen tasks.
We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul an…
The paper proposes conjectures about moduli space rigidity.
problem Rigidity of moduli spaces in algebraic geometry.
method Group-theoretic, topological, and holomorphic approaches.
result Some conjectures have been proved.
Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperkähler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimens…
Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.
problem Extending non-Abelian Hodge theory to moduli stacks of quiver connections.
method Formalizes and constructs moduli stacks of bundles with λ-connections over prestacks.
result Shows moduli stacks are algebraic and locally of finite presentation when base is smooth and projective.
This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of 'quantum invariants' of manifolds inspired by ideas originating from physics.…
This is a commentary on Teichmüllers' paper "Veränderliche Riemannsche Flächen" (Variable Riemann Surfaces), published in 1944. This paper is the last one that Teichmüller wrote on the problem of moduli. At most places the paper contains ideas and no technical details. The author presents a completely new approach to T…
Study classifies moduli spaces of spin connections on 3D homogeneous spaces.
problem Classifying moduli spaces of spin connections on 3D homogeneous spaces.
method Analysis of the topology of moduli spaces, focusing on finite-dimensional topological manifolds with trivial homotopy groups.
result Moduli spaces are finite-dimensional topological manifolds with trivial homotopy groups, essential for consistent cosmological models.
These are notes on the theory of super Riemann surfaces and their moduli spaces, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.
The paper develops a framework for abstracting causal models using category theory.
problem Difficulties in changing the variables used to describe a system, especially from fine-grained to coarse-grained.
method Introduces a category of interventional causal models and uses enriched category theory to prove compositionality properties.
result Compositionality of model transformations is established, with bounded errors for each step.
Abstract cone operators prove scalar curvature comparisons on singular manifolds.
problem Proving scalar curvature inequalities on manifolds with cone singularities.
method Using index theory for twisted Dirac operators on Lipschitz bundles.
result Lipschitz rigidity for scalar curvature on odd-dimensional manifolds.
Study knot singularities in Bogomolny equation solutions.
problem Understanding solutions with knot singularities.
method Analyzes the moduli space of solutions on R^3 with specific asymptotic conditions.
result Potential applications in low-dimensional topology and knot theory.
Universal geometry organizes heterotic moduli spaces.
problem Understanding the structure of heterotic compactifications.
method Fibering compactification data over moduli space and analyzing universal curvatures.
result Deformations are components of universal curvatures, incorporating α′2 corrections. Math theory explains how neural networks learn abstract representations.
problem Understanding how neural networks learn abstract representations.
method Mathematical theory reformulating network optimization into mean field optimization over neural preactivations.
result Abstract representations of latent variables are guaranteed to appear in neural networks trained on tasks that depend on these variables.
Constructs a space for stable holomorphic submersions over a fixed base.
problem Stability of holomorphic submersions over a compact Kaehler base.
method Geometric invariant theory combined with geometric PDEs.
result Moduli space is a Hausdorff complex space with a Weil-Petersson type Kaehler metric.
We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkahler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton. W…
The goal of this paper is to develop some aspects of the deformation theory of piecewise flat structures on surfaces and use this theory to construct new geometric structures on the moduli space of Riemann surfaces.
moment maps arise as a generalization of genuine moment maps on symplectic manifolds when the symplectic structure is discarded, but the relation between the mapping and the action is kept. Particular examples of abstract moment maps had been used in Hamiltonian mechanics for some time, but the abstract notion originat…
The Bogomol'nyi-Prasad-Sommerfield (BPS) multi-wall solutions are constructed in supersymmetric U(N_C) gauge theories in five dimensions with N_F(>N_C) hypermultiplets in the fundamental representation. Exact solutions are obtained with full generic moduli for infinite gauge coupling and with partial moduli for finite …
The paper studies the local structure of a moduli space for a specific string theory system.
problem Investigating the local structure of the moduli space of solutions to the Hull--Strominger system.
method Using a vector bundle and studying the deformation complex associated with a differential operator $ar{D}$, establishing an isomorphism between cohomology groups.
result The moduli space has an expected dimension of zero.
Survey of methods for computing volumes of moduli spaces.
problem Computing volumes of moduli spaces for Riemann surfaces with different metrics.
method Combinatorial enumeration, intersection theory, recursion relations.
result Review of key results and methods in computing both Weil-Petersson and Masur-Veech volumes.