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The Barr-Beck-Lurie comonadicity theorem characterizes when an adjunction $\begin{tikzcd} \mathcal{C} \arrow[r,yshift=1ex,"L"] & \mathcal{D} \arrow[l,yshift=-1ex,"R"] \end{tikzcd}$ can be used to present $\mathcal{C}$ as the category of comodules in $\mathcal{D}$ for the coalgebra object \mbox{$L\circ R \in End(\mathcal{D})$}; loosely speaking, the comodule structure supplies the instructions for how to assemble object in $\mathcal{C}$ from an object in $\mathcal{D}$. This thesis explores the proofs and applications in a number of contexts of this method for endowing categories of interest $\mathcal{C}$ with this tautological algebraically-flavored description, in the hopes of being a kind of handbook and toolkit for someone looking to demonstrate a comandicity result.The toolkit grew out of an investigation of a fundamental comonadicity result: Zariski descent for quasicoherent sheaves. Our main effort, joint with Peng Zhou, is in (1) presenting comonadicity statements in the case where $mathcals{C}\xrightarrow{L}\mathcal{D}$ is a product of reflective localizations, and (2) applying it to deduce a descent statement for those closed covers of Lagrangian skeleta which are locally modeled on ones that arise in the coherent-constructible correspondence of Fang-Liu-Treumann-Zaslow.

Homological mirror symmetry (HMS) asserts that the Fukaya category of a symplectic manifoldis derived equivalent to the category of coherent sheaves on the mirror complex manifold.Without suitable enlargement (split closure) of the Fukaya category, certain objects of it aremissing to prevent HMS from being true. One possible solution is to include coisotropicbranes into the Fukaya category. This thesis gives a construction for linear symplectic tori ofa version of Fukaya category including coisotropic branes by using a doubling procedure, anddiscussing the relation between the Fukaya category of the doubling torus and the Fukayacategory of the original torus.

This dissertation is an exposition of the work conducted by the author in the later years of graduate school, when two main projects were completed. Both projects concern the application of sheaf-theoretic techniques to construct geometric structures on categories appearing in the mathematical description of the A-model, which are of interest to symplectic geometers and mathematicians working in mirror symmetry. This dissertation starts with an introduction to the aspects of the physics of mirror symmetry that will be needed for the exposition of the techniques and results of these two projects. The first project concerns the construction of Calabi-Yau structures on topological Fukaya categories, using the microlocal model of Nadler and others for these categories. The second project introduces and studies a similar local-to-global technique, this time used to construct Bridgeland stability conditions on Fukaya categories of marked surfaces, extending some results of Haiden, Katzarkov and Kontsevich on the relation between stability of Fukaya categories and geometry of holomorphic differentials.

Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs $X$ and $Y$ such that the complex geometry on $X$ mirrors the symplectic geometry on $Y$. It allows one to deduce information about $Y$ from known properties of $X$. Strominger-Yau-Zaslow (1996) described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. Kontsevich (1994) conjectured that a complex invariant on $X$ (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of $Y$ (the Fukaya category). This is known as homological mirror symmetry. In this project, we first use the construction of SYZ mirrors for hypersurfaces in abelian varieties following Abouzaid-Auroux-Katzarkov, in order to obtain $X$ and $Y$ as manifolds. The complex manifold comes from the genus 2 curve as a hypersurface in its Jacobian torus, and we equip the SYZ mirror manifold with a symplectic form. We then describe an embedding of the category on the complex side into a cohomological Fukaya-Seidel category of $Y$ as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations.

We prove the Landau-Ginzburg mirror symmetry conjecture between invertible quasi-homogeneous polynomial singularities at all genera. That is, we show that the FJRW theory (LG A-model) of such a polynomial is equivalent to the Saito-Givental theory (LG B-model) of the mirror polynomial.

Symmetry perception in dot patterns is tolerant to temporal delays of up to 60 ms within and between element pairs. However, it is not known how factors effecting symmetry discrimination in static patterns might affect temporal integration in dynamic patterns. One such feature is luminance polarity. Using dynamic stimuli with increasing temporal delay (SOA) between the onset of the first and second element in a symmetric pair, we investigated how four different luminance-polarity conditions affected the temporal integration of symmetric patterns. All four luminance polarity conditions showed similar upper temporal limits of approximately 60 ms. However psychophysical performance over all delay durations showed significantly higher symmetry thresholds for unmatched-polarity patterns at short delays, but also significantly less sensitivity to increasing temporal delay relative to matched-polarity patterns. These varying temporal windows are consistent with the involvement of a fast, sensitive first-order mechanism for matched-polarity patterns, and a slower, more robust second-order mechanism for unmatched-polarity patterns. Temporal integration windows for unmatched-polarity patterns were not consistent with performance expected from attentional mechanisms alone, and instead supports the involvement of second-order mechanisms that combines information from ON and OFF channels.

For all integers k > 0 k>0 , we prove that the hypergeometric function \[ I ^ k ( α ) = ∑ j = 0 ∞ ( ( 8 k + 4 ) j ) ! j ! ( 2 j ) ! ( ( 2 k + 1 ) j ) ! 2 ( ( 4 k + 1 ) j ) ! α j \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of a pencil of curves of genus 3 k + 1 3k+1 . We prove that the function I ^ k \widehat {I}_k is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces X = X 8 k + 4 ⊂ P ( 2 , 2 k + 1 , 2 k + 1 , 4 k + 1 ) X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1) . Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces X X were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that | − K X | = | O X ( 1 ) | = ∅ |-K_X|=|\mathcal {O}_X (1)|=\varnothing . This means that there is no way to form a Calabi–Yau pair ( X , D ) (X,D) out of X X and hence there is no known mirror construction for X X other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.

For any triple of positive integers $A^{\prime} = (a_1^{\prime},a_2^{\prime},a_3^{\prime})$ and $c \in{{\mathbb{C}}}^*$, cusp polynomial ${ f_{A^\prime }} = x_1^{a_1^{\prime}}+x_2^{a_2^{\prime}}+x_3^{a_3^{\prime}}-c^{-1}x_1x_2x_3$ is known to be mirror to Geigle–Lenzing orbifold projective line ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$. More precisely, with a suitable choice of a primitive form, the Frobenius manifold of a cusp polynomial ${ f_{A^\prime }}$ turns out to be isomorphic to the Frobenius manifold of the Gromov–Witten theory of ${{\mathbb{P}}}^1_{a_1^{\prime},a_2^{\prime},a_3^{\prime}}$. In this paper we extend this mirror phenomenon to the equivariant case. Namely, for any $G$—a symmetry group of a cusp polynomial ${ f_{A^\prime }}$, we introduce the Frobenius manifold of a pair$({ f_{A^\prime }},G)$ and show that it is isomorphic to the Frobenius manifold of the Gromov–Witten theory of Geigle–Lenzing weighted projective line ${{\mathbb{P}}}^1_{A,\Lambda }$, indexed by another set $A$ and $\Lambda $, distinct points on ${{\mathbb{C}}}\setminus \{0,1\}$. For some special values of $A^{\prime}$ with the special choice of $G$ it happens that ${{\mathbb{P}}}^1_{A^{\prime}} \cong{{\mathbb{P}}}^1_{A,\Lambda }$. Combining our mirror symmetry isomorphism for the pair $(A,\Lambda )$, together with the “usual” one for $A^{\prime}$, we get certain identities of the coefficients of the Frobenius potentials. We show that these identities are equivalent to the identities between the Jacobi theta constants and Dedekind eta–function.

Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel, Kontsevich, and Siebert have given combinatorial constructions of canonical bases of "theta functions" on the coordinate rings of various log Calabi-Yau spaces, including cluster varieties. We prove that the theta bases for cluster varieties are determined by certain descendant log Gromov-Witten invariants of the symplectic leaves of the mirror/Langlands dual cluster variety, as predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We further show that these Gromov-Witten counts are often given by naive counts of rational curves satisfying certain geometric conditions. As a key new technical tool, we introduce the notion of "contractible" tropical curves when showing that the relevant log curves are torically transverse., Comment: 36 pages, 2 figures; published version

This is the first part of our project toward proving the Bershadsky–Cecotti–Ooguri–Vafa Feynman graph sum formula of all genera Gromov–Witten invariants of quintic Calabi–Yau threefolds. We introduce the notion of N–mixed-spin-P fields, construct their moduli spaces, their virtual cycles and their virtual localization formulas, and obtain a vanishing result associated with irregular graphs.

We consider a waveguide composed of two not necessity equal semi-infinite strips (trunks) and a rectangle (resonator) connected by narrow openings in the shared walls. As their diameter vanishes, we construct asymptotics for the scattering coefficients, justifying them by the technique of weighted spaces with detached asymptotics. We establish a criterion for the possibility of observing, at a given frequency, almost complete transmission of waves through both perforated cross-walls. This effect is revealed due to a fine-tuning of the resonator height and the criterion involves an equation relating some geometrical characteristics of the waveguide to the wave numbers in the trunks, while any mirror symmetry turns the criterion trivial.

We consider a pair of quiver varieties $(X;X^{\prime})$ related by 3D mirror symmetry, where $X =T^*{Gr}(k,n)$ is the cotangent bundle of the Grassmannian of $k$-planes of $n$-dimensional space. We give formulas for the elliptic stable envelopes on both sides. We show an existence of an equivariant elliptic cohomology class on $X \times X^{\prime} $ (the mother function) whose restrictions to $X$ and $X^{\prime} $ are the elliptic stable envelopes of those varieties. This implies that the restriction matrices of the elliptic stable envelopes for $X$ and $X^{\prime}$ are equal after transposition and identification of the equivariant parameters on one side with the Kähler parameters on the dual side.

We review some classical and modern aspects of hypergeometric differential equations, including A-hypergeometric systems of Gel$$'$$ ′ fand, Graev, Kapranov and Zelevinsky. Some recent advances in this theory, such as Euler–Koszul homology, rank jump phenomena, irregularity questions and Hodge theoretic aspects are discussed with more details. We also give some applications of the theory of hypergeometric systems to toric mirror symmetry.

In this paper, we point out an interesting asymmetry in the rules of fundamental mathematics between positive and negative numbers. Further, we show an alternative numerical system identical to today’s system, but where positive numbers dominate over negative numbers. This is like a mirror symmetry of the existing number system. The asymmetry in both systems leads to imaginary and complex numbers. We also suggest an alternative number system with perfectly symmetrical rules—that is, where there is no dominance of negative numbers over positive numbers or vice versa, and where imaginary and complex numbers are no longer needed. This number system seems to be superior to other numerical systems, as it brings simplicity and logic back to areas that complex rules have dominated for much of the history of mathematics. Finally, we also briefly discuss how the Riemann hypothesis may be linked to the asymmetry in the current number system. The foundation rules of a number system can, in general, not be proven incorrect or correct inside the number system itself. However, the ultimate goal of a number system is, in our view, to describe nature accurately. The optimal number system should therefore be developed with feedback from nature. If nature, at a very fundamental level, is ruled by symmetry, then a symmetric number system should make it easier to understand nature than an asymmetric number system would. We hypothesize that a symmetric number system may thus be better suited to describing nature. Further, such a number system should eliminate imaginary numbers in space-time and quantum mechanics; for example, two areas of physics that are clouded in mystery to this day.

These notes were born out of a five-hour lecture series for graduate students at the May 2018 Snowbird workshop Crossing the Walls in Enumerative Geometry. After a short primer on equivariant cohomology and localization, we provide proofs of the genus-zero mirror theorems for the quintic threefold, first in Fan-Jarvis-Ruan-Witten theory and then in Gromov-Witten theory. We make no claim to originality, except in exposition, where special emphasis is placed on peeling away the standard technical machinery and viewing the mirror theorems as closed-formula manifestations of elementary localization recursions., Comment: 24 pages, comments welcome

We study mirror symmetry of families of elliptic K3 surfaces with elliptic fibers of type $E_6,~E_7$ and $E_8$. We consider a moduli space $\mathsf{T}$ of the mirror K3 surfaces enhanced with the choice of differential forms. We show that coordinates on $\mathsf{T}$ are given by the ring of quasi modular forms in two variables, with modular groups adapted to the fiber type. We furthermore introduce an algebraic group $\mathsf{G}$ which acts on $\mathsf{T}$ from the right and construct its Lie algebra $\mathrm{Lie}(\mathsf{G})$. We prove that the extended Lie algebra generated by $\mathrm{Lie}(\mathsf{G})$ together with modular vector fields on $\mathsf{T}$ is isomorphic to $\mathrm{sl}_2(\mathbb{C})\oplus\mathrm{sl}_2(\mathbb{C})$., 19 pages

Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy Kahler moduli space. A few basic examples are studied. In particular, we identify our Weil-Petersson metric with the Bergman metric on a Siegel modular variety in the case of the self-product of an elliptic curve.

To a one-parameter family of Calabi-Yau threefolds, we can associate the extended period map by the log Hodge theory of Kato and Usui. In the present paper, we study the image of a maximally unipotent monodromy point under the extended period map. As an application, we prove the generic Torelli theorem for a large class of one-parameter families of Calabi-Yau threefolds.

We consider the phenomenon of the complete coincidence of key properties of Calabi–Yau manifolds realized as hypersurfaces in two different weighted projective spaces. More precisely, the first manifold in such a pair is realized as a hypersurface in a weighted projective space, and the second is realized as a hypersurface in an orbifold of another weighted projective space. The two manifolds in each pair have the same Hodge numbers and the same geometry on the complex structure moduli space and are also associated with the same $$N{=}2$$ gauged linear sigma model. We explain these coincidences using the correspondence between Calabi–Yau manifolds and the Batyrev reflexive polyhedra.

In this paper, we describe recent work towards the mirror $$\mathrm P=\mathrm W$$ conjecture, which relates the weight filtration on the cohomology of a log Calabi–Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi–Yau manifold taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi–Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini [1] and de Cataldo and Migliorini [2]. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror $$\mathrm P=\mathrm W$$ conjecture provides a possible bridge between the mirror $$\mathrm P=\mathrm W$$ conjecture and the well-known $$\mathrm P=\mathrm W$$ conjecture in non-Abelian Hodge theory.

We compute the $g=1$, $n=1$ B-model Gromov–Witten invariant of an elliptic curve $E$ directly from the derived category $\mathsf{D}_{\mathsf{coh}}^{b}(E)$. More precisely, we carry out the computation of the categorical Gromov–Witten invariant defined by Costello using as target a cyclic $\mathscr{A}_{\infty }$ model of $\mathsf{D}_{\mathsf{coh}}^{b}(E)$ described by Polishchuk. This is the first non-trivial computation of a positive-genus categorical Gromov–Witten invariant, and the result agrees with the prediction of mirror symmetry: it matches the classical (non-categorical) Gromov–Witten invariants of a symplectic 2-torus computed by Dijkgraaf.

By mirror symmetry, the quantum connection of a weighted projective line is closely related to the localized Fourier–Laplace transform of some Gauß–Manin system. Following an article of D’Agnolo, Hien, Morando, and Sabbah, we compute the Stokes matrices for the latter at $$\infty $$ ∞ for the cases $${\mathbb {P}}(1,3)$$ P ( 1 , 3 ) and $${\mathbb {P}}(2,2)$$ P ( 2 , 2 ) by purely topological methods. We compare them to the Gram matrix of the Euler–Poincaré pairing on $$D^b(\mathrm{Coh}({\mathbb {P}}(1,3)))$$ D b ( Coh ( P ( 1 , 3 ) ) ) and $$D^b(\mathrm{Coh}({\mathbb {P}}(2,2)))$$ D b ( Coh ( P ( 2 , 2 ) ) ) , respectively. This article is based on the doctoral thesis of the author.

Theoretical physics is taking an increasing part in the universe of mathematics. After calculus, vector and tensorial analysis, topological theories make their entry into quantum field theories. More precisely, in this domain, topological theories are the most relevant. A fundamental theorem of the Atiyah has important repercussions in several branches of quantum physics in the geometric approach. We can cite the work of Alain Connes on non-commutative geometry, but also all the developments due to Donaldson, E. Witten around Gauge theories, superstring and Mirror Symmetry. We present here an historical survey of some topological field theories, especially Mirror Symmetry to understand the interpenetration between quantum physics and topology.

The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti [arXiv:0709.1453, arXiv:0807.0597] relates all genus open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifolds/3-orbifolds to the Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold. In this paper, we present a proof of the Remodeling Conjecture for open-closed orbifold Gromov-Witten invariants of an arbitrary affine toric Calabi-Yau 3-orbifold relative to a framed Aganagic-Vafa Lagrangian brane. This can be viewed as an all genus open-closed mirror symmetry for affine toric Calabi-Yau 3-orbifolds.

We give a short new computation of the quantum cohomology of an arbitrary smooth toric variety $X$, by showing directly that the Kodaira-Spencer map of Fukaya-Oh-Ohta-Ono defines an isomorphism onto a suitable Jacobian ring. The proof is based on the purely algebraic fact that a class of generalised Jacobian rings associated to $X$ are free as modules over the Novikov ring. In contrast to previous results of this kind, $X$ need not be compact. When $X$ is monotone the presentation we obtain is completely explicit, using only well-known computations with the standard complex structure., This article has been accepted for publication in The Quarterly Journal of Mathematics Published by Oxford University Press. The accepted version is available on my webpage

В настоящей работе описаны последние достижения в направлении зеркальной гипотезы $\mathrm P=\mathrm W$, которая связывает весовую фильтрацию на когомологиях многообразия лог-Калаби-Яу с превратной фильтрацией Лере на когомологиях гомологически зеркально двойственного многообразия Калаби-Яу, взятой для отображения аффинизации. Эта гипотеза обобщает классическую связь межу числами Ходжа зеркально двойственных компактных многообразий Калаби-Яу, включая в нее идеи, появившиеся в новаторских работах де Катальдо, Мильгорини и Хаузеля [1] и де Катальдо и Мильгорини [2]. Мы даем обзор мотивировки этой гипотезы и последних связанных с ней результатов и описываем, как эти результаты возникают из формулировки SYZ зеркальной симметрии. Такая интерпретация зеркальной гипотезы $\mathrm P=\mathrm W$ дает ее возможную связь с хорошо известной в неабелевой теории Ходжа гипотезой $\mathrm P=\mathrm W$. Библиография: 22 названия.

Given a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-H\"ubsch transpose. This was previously shown for Brieskorn-Pham and $D$-type singularities by Futaki-Ueda. The proof involves explicit construction of a tilting object on the B-side, and comparison with a specific basis of Lefschetz thimbles on the A-side.

We develop an approach to calculating the cup and cap products on Hochschild cohomology and homology of curved algebras associated with polynomials and their finite abelian symmetry groups. For polynomials with isolated critical points, the approach yields a complete description of the products. We also reformulate the result for the corresponding categories of equivariant matrix factorizations. In an Appendix written jointly with Alexey Basalaev, we apply the formulas to calculate the Hochschild cohomology of a simple but non-trivial class of so-called invertible LG orbifold models. The resulting algebras turn out to be isomorphic to what has already appeared in the literature on LG mirror symmetry under the name of twisted or orbifolded Milnor/Jacobian algebras. We conjecture that this holds true for all invertible LG models. In the second part of the Appendix, the formulas are applied to a different class of LG orbifolds which have appeared in the context of homological mirror symmetry for varieties of general type as mirror partners of surfaces of genus 2 and higher. In combination with a homological mirror symmetry theorem for the surfaces, our calculation yields a new proof of the fact that the Hochschild cohomology of the Fukaya category of a surface is isomorphic, as an algebra, to the cohomology of the surface.

There are three parts in this thesis. First, we generalize the class of tropicalcurves from trivalent to 3-colorable which can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by $\Gamma$.Second, we prove the equality of two canonical bases of a rank 2 clusteralgebra, the greedy basis of Lee-Li-Zelevinsky and the theta basis of Gross-Hacking-Keel-Kontsevich.Third, we link up scattering diagrams D with quiver representations ofcorresponding quivers Q. We define a notion of good crossing of broken lines $\gamma$ on D. Then we show if $\gamma$ has good crossing over D, then it goes in the opposite direction of the Auslander-Reiten quiver of Q. Then we give a stratification of quiver representations by the bendinga of $\gamma$..

Symmetry (and group theory) is a fundamental principle of theoretical physics. Finite symmetries, continuous symmetries of compact groups, and infinite-dimensional representations of noncompact Lie groups are at the core of solid physics, particle physics, and quantum physics, respectively. The latter groups now play an important role in many branches of mathematics. In more recent years, we have been faced with the impact of topological quantum field theory (TQFT). Topology and symmetry have deep connections, but topology is inherently broader and more complex. While the presence of symmetry in physical phenomena imposes strong constraints, topology seems to be related to low-energy states and is very likely to provide information about the different dynamical trajectories and patterns that particles can follow. For example, regarding the relationship of topology to low-energy states, Hodge’s theory of harmonic forms shows that the zero-energy states (for differential forms) correspond to the cohomology. Regarding the relationship of topology to particle trajectories, a topological knot can be seen as an orbit with complex properties in spacetime. The various deformations or embeddings of the knot, performed in low or high dimensions, allow defining different equivalence classes or topological types, and interestingly, it is possible from these types to study the symmetries associated with the deformations and their changes. More specifically, in the present work, we address two issues: first, that quantum geometry deforms classical geometry, and that this topological deformation may produce physical effects that are specific to the quantum physics scale, and second, that mirror symmetry and the phenomenon of topological change are closely related. This paper was aimed at understanding the conceptual and physical significance of this connection.

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), that we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del~Pezzo surfaces. Furthermore we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anticanonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP., Comment: 21 pages, plus a 321 page appendix; 7 figures; 100 tables

We define a new class of symplectic spaces called ``pumpkin domains'', which roughly speaking comprise a Liouville domain and a Liouville hypersurface of its boundary. To such an object we assign an $A_\infty$-category called its partially wrapped Fukaya category. An exact Landau-Ginzburg model gives rise to a pumpkin domain, and the partially wrapped Fukaya category of this pumpkin domain is meant to agree with the Fukaya category one is supposed to assign to the Landau-Ginzburg model. As evidence, we prove a formula that relates the partially wrapped Fukaya category of a pumpkin domain to the wrapped Fukaya category of its underlying Liouville domain. This operation is mirror to removing a divisor.

The wedge product on de Rham complex of a Riemannian manifold $M$ can be pulled back to $H^\ast (M)$ via explicit homotopy constructed by using Green’s operator which gives higher product structures. We prove Fukaya’s conjecture which suggests that Witten deformation of these higher product structures has semiclassical limits as operators defined by counting gradient flow trees with respect to Morse functions, which generalizes the remarkable Witten deformation of de Rham differential from a statement concerning homology to one concerning real homotopy type of $M$. Various applications of this conjecture to mirror symmetry are also suggested by Fukaya in [6].

A natural example of evolution can be described by a time-dependent two degrees-of-freedom Hamiltonian. We choose the case where initially the Hamiltonian derives from a general cubic potential, the linearised system has frequencies 1 and ω&gt, 0. The time-dependence produces slow evolution to discrete (mirror) symmetry in one of the degrees-of-freedom. This changes the dynamics drastically depending on the frequency ratio ω and the timescale of evolution. We analyse the cases ω=1,2,3 where the ratio’s 1,2 turn out to be the most interesting. In an initial phase we find 2 adiabatic invariants with changes near the end of evolution. A remarkable feature is the vanishing and emergence of normal modes, stability changes and strong changes of the velocity distribution in phase-space. The problem is inspired by the dynamics of axisymmetric, rotating galaxies that evolve slowly to mirror symmetry with respect to the galactic plane, the model formulation is quite general.

Статья посвящена исследованию линиаризованных соподчинительных структур, которые в теории зависимостей называются шириной куста. Исследование имеет квантитативный характер. В нем используются ранговые распределения фигур, в которых фиксируется не только число зависимых членов, но и варианты их расположения справа (в постпозиции) или слева (в препозиции) относительно вершины дерева предложения. Установлен факт сходства ранговых структур у разных авторов при существенном различии частотных характеристик. В работе рассматриваются симметрийные свойства кустовых фигур, их тяготения к зеркальной и золотой симметрии. Установлено примерное равновесие левой и правой ширины в русском рассказе первой трети XX века. Протестирован на предмет адекватности русскому языку и скорректирован закон нарастающих членов предложения немецкого филолога Отто Бехагеля.