arXiv research

Constructs moduli stacks of quiver bundles and applies to 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 $\infty$-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.

Research characterizes intersection cohomology groups of gauge theories and cotangent bundles.

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.

Following \cite{citeSavelyevVirtualMorsetheoryon$Omega$Ham$(Momega)$.}, we develop here a connection between Morse theory for the (positive) Hofer length functional $L: Ω\text {Ham}(M, ω) \to \mathbb{R}$, 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.

Abstract: New geometric incarnation of isomonodromy functors.

Foundations of derived geometry in smooth settings.

Introduces linear K-systems for Hamiltonian Floer theory.

The paper aims to mathematically define and learn abstractions from data.

Study on moduli spaces of Higgs bundles over Abelian varieties, focusing on their topology and polynomials.

Generative models use latent abstractions to create images.

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.

The paper explores Higgs bundles and their moduli spaces on Riemann surfaces.

Develops moduli theory for Calabi-Yau pairs, constructing a projective space.

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.

Decouples moduli groups in heterotic string theory cohomology.

New proof for discrete Morse theory using combinatorial construction.

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 $\mathcal{O}(α')$ 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…

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…

Learning transferable knowledge across similar but different settings is a fundamental component of generalized intelligence. In this paper, we approach the transfer learning challenge from a causal theory perspective. Our agent is endowed with two basic yet general theories for transfer learning: (i) a task shares a c…

The paper proposes conjectures about moduli space rigidity.

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…

Can simple algorithms with a good representation solve challenging reinforcement learning problems? In this work, we answer this question in the affirmative, where we take "simple learning algorithm" to be tabular Q-Learning, the "good representations" to be a learned state abstraction, and "challenging problems" to be…

Constructs moduli stacks for quiver connections and extends non-Abelian Hodge theory.

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.

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.

Abstract cone operators prove scalar curvature comparisons on singular manifolds.

Study knot singularities in Bogomolny equation solutions.

Universal geometry organizes heterotic moduli spaces.

Math theory explains how neural networks learn abstract representations.

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…

Constructs a space for stable holomorphic submersions over a fixed base.

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.

Survey of methods for computing volumes of moduli spaces.