부산대학교  기하학, 위상수학  세미나
(PNU Geometry and Topology seminar)

Title: A lower-bound of entropy from symplectic topology. 

Abstract: In this talk, I will introduce a powerful invariant of symplectic topology, called "Fukaya category", and its application to another area of mathematics. Specifically, we consider the following: Let $W$ be a manifold with a self-continuous map $f: W \to W$. Then, the iteration of $f$ forms a dynamical system, for example, sending a point $p \in W$ to $f(p)$ at time 1, $f^2(p)$ at time 2, and so forth.  When studying a dynamical system, the concept of entropy is a numerical invariant of the dynamical system, revealing the "complexity" of it. Usually, computing entropy is challenging, but if $W$ admits a symplectic structure, then the symplectic structure provides an insight. More precisely, the Fukaya category of $W$ gives us a lower-bound of entropy. In this talk, I will explain the basics of Fukaya category and how it is related to entropy.

The presentation will be given in English. 

Title: "Everyday I'm Shuffling" 

Abstract: We will generalize in 3 ways the shuffle of decks (of cards) to shuffles of a linear order and a poset. We then show 'cases in which' the combinatorics of shuffles of posets and the combinatorics of the tensor product of operads in dendroidal homotopy theory are equivalent.

The presentation will be given in English. 

Title. Quadratic Persistences and the Pythagoras number of projective curves 

Abstract. The Pythagoras number of a ring is the minimum number of elements required to represent any element within the ring. For instance, the Pythagoras number of the integer ring is four, due to the Lagrange’s Four Square Theorem. Blekherman, Sinn, Smith, and Velasco define the Pythagoras number of varieties by the Pythagoras number of their coordinate rings, inspired by the definition for rings. To investigate this semi-algebraic quantity, they introduce and analyze various algebraic invariants that establish bounds on the Pythagoras number. In this talk, we review their discoveries and present new results on projective curves. These results extend the scope of varieties for which Pythagoras numbers are known. This talk is based on joint work with Professor Euisung Park and Jongin Han.

Title. Deformations of singularities and Kollár conjecture

Abstract. Each isolated singularity has the versal deformation space. Therefore, the description of its base space is an important research topic in deformation theory. A basic approach is to find equations for the base but it is very complicated. On this subject, Janós Kollár conjectured that, for rational surface singularities, there is an one to one correspondence between the irreducible components of the deformation space and partial resolutions of the singularity, which is called 'P-modification’. I will talk about the background of the conjecture, its merits, known results and recent progress. This is a joint work with Dongsoo Shin.

Title: Introduction to Symplectic Cohomology and Applications

Abstract: In the late 1980's, Floer developed generalized Morse theories, where the generators of the complex are critical points of action functionals on various moduli spaces, solutions of certain ODEs, and the gradient flow lines are solutions of certain elliptic PDEs. Floer's breakthrough construction is based on that era's other breakthrough works done by Uhlenbeck, Gromov, Donaldson, Taubes, Witten and many others. In this talk, we learn one of his theories, called Hamiltonian Floer (co)homology and Symplectic (co)homology, that is a Morse (co)homology on the loop space of a symplectic manifold with Hamiltonian function on it. After reviewing definitions and some properties, we introduce criteria for affine varieties to admit uniruled subvarieties of certain dimensions. The measurements are from long exact sequences of versions of symplectic cohomology. We provide applications of the criteria in birational geometry of log pairs in the direction of the Minimal Model Program.

Title: Ancient mean curvature flows with finite total curvature

Abstract: Ancient flows, as singularity models of the mean curvature flow, have been intensively studied over the past decade. Particularly, in the spirit of the parabolic Liouville-type theorem for the non-compact case, flows with prescribed asymptotic behavior have been considered. In this context, we present $I$ family of ancient mean curvature flows that converge to a given two-sided complete embedded minimal hypersurface in $\mathbb{R}^{n+1}$ as $|x|^2-t\rightarrow \infty$, where $I$ is the Morse index of the given hypersurface. We establish that these flows possess finite total curvature and finite mass drop. Additionally, one family within these flows is mean convex.

Title: On toric Schubert varieties of simply-laced types. 

Abstract: Let $G$ be a simple Lie group of simply-laced type, let $B$ be a Borel subgroup, and let $T$ be a maximal torus contained in $B$. The homogeneous space $G/B$ becomes a smooth projective variety, called the flag variety. The left multiplication of $T$ induces an action of $T$ on $G/B$, providing a fruitful connection between the geometry and topology of the flag variety and the combinatorics. In this talk, we consider \emph{toric} Schubert varieties (with respect to the action of $T$) and their isomorphism classes in flag varieties. This talk is based on joint work with Mikiya Masuda and Seonjeong Park.

Title : Topological Fukaya categories of tagged arc systems

Abstract : Fukaya categories of general symplectic manifolds are very difficult to define, compute, and understand. However, for surfaces, Haiden, Katzarkov, and Kontsevich introduced a topological version of the Fukaya category which is defined combinatorially. In this talk, I will introduce this topological Fukaya category with examples and generalize it to \mathbb{Z}/2\mathbb{Z}-orbifold surface. Using this new category, I will give its algebraic application. This is based on a preprint arXiv:2404.10294.