Triangulated surfaces in triangulated categories

For a triangulated category A with a 2-periodic dg-enhancement and a triangulated oriented marked surface S we introduce a dg-category F(S,A) parametrizing systems of exact triangles in A labelled by triangles of S. Our main result is that F(S,A) is independent on the choice of a triangulation of S up to essentially unique Morita equivalence. In particular, it admits a canonical action of the mapping class group. The proof is based on general properties of cyclic 2-Segal spaces. In the simplest case, where A is the category of 2-periodic complexes of vector spaces, F(S,A) turns out to be a purely topological model for the Fukaya category of the surface S. Therefore, our construction can be seen as implementing a 2-dimensional instance of Kontsevich's program on localizing the Fukaya category along a singular Lagrangian spine.Comment: 55 pages, v2: references added and typos corrected, v3: expanded version, comments welcom

Optimally Dense Packings for Fully Asymptotic Coxeter Tilings by Horoballs of Different Types

The goal of this paper to determine the optimal horoball packing arrangements and their densities for all four fully asymptotic Coxeter tilings (Coxeter honeycombs) in hyperbolic 3-space $\mathbb{H}^3$. Centers of horoballs are required to lie at vertices of the regular polyhedral cells constituting the tiling. We allow horoballs of different types at the various vertices. Our results are derived through a generalization of the projective methodology for hyperbolic spaces. The main result states that the known B\"or\"oczky--Florian density upper bound for "congruent horoball" packings of $\mathbb{H}^3$ remains valid for the class of fully asymptotic Coxeter tilings, even if packing conditions are relaxed by allowing for horoballs of different types under prescribed symmetry groups. The consequences of this remarkable result are discussed for various Coxeter tilings.Comment: 26 pages, 10 figure

Discrete Differential Geometry

This is the collection of extended abstracts for the 26 lectures and the open problem session at the fourth Oberwolfach workshop on Discrete Differential Geometry

란다우-긴즈버그 오비폴드의 푸카야 카테고리와 곡선 가역 특이점의 버글룬드-흅스 추측

학위논문 (박사) -- 서울대학교 대학원 : 자연과학대학 수리과학부, 2020. 8. 조철현.From a fixed cohomology class $\Gamma \in SH^\bullet(M)$ of a Liouville manifold $M$, we construct a new \AI category denoted by \CG on which the quantum cap action of $\Gamma: CW^\bullet(L,L) \to CW^\bullet(L,L)$ vanishes homotopically. With this construction on one hand, we consider a symplectic Landau - Ginzburg model $(W, G)$ defined by a weighted homogeneous polynomial $W$ and its symmetry group $G$. From wrapped Fukaya category and a monodromy information of the Milnor fiber, we construct a new Fukaya category \cF(W, G) for each pair $(W, G)$ on which the monodromy action vanishes. It is a symplectic analogue of the variation operator in singularity theory. We also show that the mirror of the monodromy action is a restriction of a mirror Landau-Ginzburg model to a certain hypersurface. As an application, we prove Berglund-H\"ubsch homological mirror symmetry for all invertible curve singularities.이 논문에서는 리우빌 다양체 $M$의 사교 코호몰로지 군의 원소 $\Gamma \in SH^\bullet(M)$ 가 주어져 있을 때, $\Gamma$ 의 양자 곱 작용 (quatum cap action) $\Gamma: CW^\bullet(L,L) \to CW^\bullet(L,L)$ 이 호모토피적으로 사라지는 새로운 호모토피 결합 범주 (\AI-category) \CG 를 건설하고자 한다. 이 새로운 건설법을 바탕으로 하여 가중 동차 다항식 $W$과 그것의 대칭군 $G$ 로 이루어진 사교 란다우-긴즈버그(Landau-Ginzburg) 모델 $(W, G)$ 을 만든다. 밀너 올(MIlnor fiber) 의 감긴 푸카야 범주 (wrapped Fukya category) 와 그것에 작용하는 모노드로미 작용 (monodromy action) 을 사용하여, 모노드로미 작용이 사라지는 새로운 범주 \cF(W, G)를 만든다. 이것은 고전적인 특이점 이론의 변분 연산자(variation operator)의 사교기하적 유추로 간주할 수 있다. 이에 더해, 모노드로미 작용의 거울 현상이 거울 란다우-긴즈버그 모델을 특정한 초곡면에 제한시키는 것임을 보인다. 그것의 응용으로, 모든 가역 곡선 특이점에 대해 버글룬드-흅스 추측을 증명한다.1 Introduction 1 2 Basic Floer theory 5 2.1 Liouville manifold with cylindrical end 5 2.2 Degree and index of Hamiltonian orbits and chords 7 2.3 Moduli space of pseudo-holomorphic curves 10 2.4 Wrapped Fukaya category 14 2.5 Symplectic cohomology and closed-open map 15 3 New A1 category C¡ 17 3.1 Popsicles with interior markings 17 3.2 Compactification 19 3.3 Cohomology category 28 3.4 Example: M2-operation 31 4 Algebro-geometric counterpart 34 4.1 Restricting to a hypersurface in DbCoh 34 4.2 Restricting to a graph hypersurface inMatrix factorizations 37 5 Equivariant topology ofMilnor fiber for invertible curve singularities 41 5.1 Topology of aMilnor fiber 41 5.2 Orbifold covering 44 5.3 Equivariant tessellation ofMilnor fibers 49 6 Equivariant Floer theory of aMilnor fiber 53 6.1 Hamiltonian 53 6.2 ­- and H1-grading 54 6.3 Orbifold wrapped Fukaya category 56 6.4 Orbifold symplectic cohomology 59 6.5 Floer algebra of Seidels immersed Lagrangian L and its deformation 62 6.6 Localized mirror functor toMatrix factorization category 64 7 Homological mirror symmetry forMilnor fibers of invertible curve singularity 66 7.1 Fermat cases 67 7.2 Chain cases 73 7.3 Loop cases 78 8 New Fukaya category for Landau-Ginzburg orbifolds 82 8.1 Preliminaries 82 8.2 Monodromy, Reeb orbit, and C¡W 84 9 Berglund-Hübsch HMS for curve singularity 88 9.1 Computation of ¡W 90 9.2 Mirror of the Monodromy action: Restriction of LG model to a hypersurface 93 9.3 Berglund-Hübsch mirror symmetry 98 Abstract (in Korean) 106Docto

Dimer models and the special McKay correspondence

We study the behavior of a dimer model under the operation of removing a corner from the lattice polygon and taking the convex hull of the rest. This refines an operation of Gulotta, and the special McKay correspondence plays an essential role in this refinement. As a corollary, we show that for any lattice polygon, there is a dimer model such that the derived category of finitely-generated modules over the path algebra of the corresponding quiver with relations is equivalent to the derived category of coherent sheaves on a toric Calabi-Yau 3-fold determined by the lattice polygon. Our proof is based on a detailed study of relationship between combinatorics of dimer models and geometry of moduli spaces, and does not depend on the result of math/9908027.Comment: 56 pages, v2: major revisio