A5055139632- Homotopy and Cohomology in Algebraic Topology
- Geometric and Algebraic Topology
- Algebraic structures and combinatorial models
- Algebraic Geometry and Number Theory
- Geometry and complex manifolds
- Mathematical Dynamics and Fractals
- Topological and Geometric Data Analysis
- Spondyloarthritis Studies and Treatments
- Spine and Intervertebral Disc Pathology
- Scoliosis diagnosis and treatment
- Advanced Combinatorial Mathematics
- semigroups and automata theory
- Advanced Algebra and Geometry
- Nonlinear Waves and Solitons
- Black Holes and Theoretical Physics
Seoul National University
2019-2024
Abstract Let W be a symplectic manifold, and let $\phi :W \to W$ automorphism. This automorphism induces an auto-equivalence $\Phi $ defined on the Fukaya category of . In this paper, we prove that categorical entropy provides lower bound for topological , where is Weinstein manifold compactly supported. Furthermore, motivated by [cCGG24], propose conjecture generalizes result [New88, Prz80, Yom87].
Abstract In this paper, we prove that the derived Rabinowitz Fukaya category of a Liouville domain dimension is ‐Calabi–Yau, assuming wrapped admits an at most countable set Lagrangians generate it and satisfy some finiteness condition on morphism spaces between them.
In this paper, we prove that the derived Rabinowitz Fukaya category of a Liouville domain $M$ dimension $2n$ is $(n-1)$-Calabi--Yau assuming wrapped admits an at most countable set Lagrangians generate it and satisfy some finiteness condition on morphism spaces between them.
For a simply-connected compact semisimple Lie group $G$ and its maximal torus $T$, we study the $A_{\infty}$-functor associated to moment Lagrangian correspondence from cotangent bundle $T^*G$ square $G/T^{-} \times G/T$. In particular, compute leading term of $A_{\infty}$-homomorphism wrapped Floer cohomology $HW^*(T^*_e G, T^*_e G)$ fiber $T_e^*G$ $HF^*(Δ, Δ)$ diagonal $Δ$ in G/T$ by determining count certain pseudo-holomorphic quilts. As consequence, prove that cohomologies $HF^*(Δ,Δ)$...
We quantize the problem considered by Bott-Samelson who applied Morse theory to any compact symmetric space $G/K$ and associated real flag manifold $G_{\mathbb{R}}/B$ which is a locus of complex partial variety $G_{\mathbb{C}}/P_σ$. prove that Pontryagin ring $H_{-*}(Ω(G/K))$ based loop $Ω(G/K)$ isomorphic Floer cohomology $HF^*(G_{\mathbb{R}}/B,G_{\mathbb{R}}/B)$ after localization. When Lie group, this conjecture Peterson, proved combinatorially Lam-Shimozono, in context quantum...
The variation operator in singularity theory maps relative homology cycles to compact the Milnor fiber using monodromy. We construct its symplectic analogue for an isolated singularity. define a new Floer cohomology, called monodromy Lagrangian which provides categorifications of standard theorems on and Seifert form. key ingredients are special class $\Gamma$ cohomology inverse closed-open images. For plane curve singularities whose A'Campo divide has depth zero, we find exceptional...
Within $N$-Calabi-Yau categories associated with quivers whose base graphs form trees, we delve into the study of asymptotic behaviors autoequivalences a specific type. These autoequivalences, which call "Penner type," exhibit straightforward characteristics, making them noteworthy exemplars "pseudo-Anosov" in sense \cite{Fan-Filip-Haiden-Katzarkov-Liu21}, and also stronger that define present paper. In addition, provide practical methodology for calculating stretching factors Penner type...
We give an explicit computation of the ring structure in wrapped Floer homology a class real Lagrangians $A_k$-type Milnor fibers. In plumbing description, those correspond to cotangent fibers or diagonal Lagrangians. The main ingredient is apply version Seidel representation. For technical reason, we first carry out computations v-shaped homology, and this turn gives desired via Viterbo transfer map.
In this paper, motivated by symplectic topology, we explore categorical entropy and present two main results. The first result establishes a relation between entropies of functors on category its localization. Additionally, it demonstrates analogies the notions topological entropy. This is then applied to where provide method for calculating functor (partially) wrapped Fukaya category, assuming that induced compactly supported automorphism. For second observe existence natural examples...
Let $W$ be a symplectic manifold, and let $\phi:W \to W$ automorphism. Then, $\phi$ induces an auto-equivalence $\Phi$ defined on the Fukaya category of $W$. In this paper, we prove that categorical entropy bounds topological from below where is Weinstein manifold compactly supported. Moreover, being motivated by work Cineli, Ginzburg, Gurel, propose conjecture which generalizes result in dynamical system.
We show that the derived wrapped Fukaya category $D^\pi\mathcal{W}(X_{Q}^{d+1})$, compact $D^\pi\mathcal{F}(X_{Q}^{d+1})$ and cocore disks $L_{Q}$ of plumbing space $X_{Q}^{d+1}$ form a Calabi--Yau triple. As consequence, quotient $D^\pi\mathcal{W}(X_{Q}^{d+1})/D^\pi\mathcal{F}(X_{Q}^{d+1})$ becomes cluster associated to $Q$. One its properties is structure. Also it known this quasi-equivalent Rabinowitz due work Ganatra--Gao--Venkatesh. compute morphism in using structure, which isomorphic...
A narrowed sacroiliac joint (SIJ) space has been considered to be a major morphologic parameter of ankylosing spondylitis (AS). Previous studies revealed that the thickness (SIJT) correlated with AS in patients. However, irregular narrowing is different from thickness. Thus, we devised method using cross-sectional area (SIJA) as new morphological for use evaluating AS. We hypothesized SIJA key diagnosing SIJ samples were collected 107 patients AS, and 85 control subjects who underwent...