Abstracts

이소연

Title : Regular Schur labeled skew shape posets and their $0$-Hecke modules

In 1972, Stanley proposed a conjecture that provides a necessary and sufficient condition for the $(P, \omega)$-partition generating function $K_{(P, \omega)}$ to be symmetric. Specifically, the conjecture states that $K_{(P, \omega)}$ is symmetric if and only if $(P, \omega)$ is a Schur labeled skew shape poset. To solve this conjecture, it is natural to study Schur labled skew shape posets. In this talk, we study some of them from the perspective of the representation theory of the $0$-Hecke algebra. More precisely, we study $0$-Hecke modules associated to regular Schur labeled skew shape posets.

 To achieve our purpose, the first half of this talk will be devoted to the representation theory of the $0$-Hecke algebra and combinatorics related to it. More precisely, we introduce its close connection with quasisymmetric functions. It was discovered by Duchamp-Krob-Leclerc-Thibon, who constructed an isomorphism called the quasisymmetric characteristic between the Grothendieck ring associated to $0$-Hecke algebras and the ring  of quasisymmetric functions. Then, we introduce the $0$-Hecke modules $M_{(P, \omega)}$ associated with labeled posets $(P, \omega)$, which are constructed by Duchamp-Hivert-Thibon.

 The second half will be devoted to my work on $0$-Hecke modules associated with regular Schur labeled skew shape posets. We first describe their bases in terms of standard Young tableaux. Then, we classify $0$-Hecke modules associated with regular Schur labeled skew shape posets upto isomorphism. Finally, we give a characterization for regular Schur labeled skew shape posets. Using it, we give interesting filtrations of $0$-Hecke modules associated with Schur labeled skew shape posets. This work is joint with Young-Hun Kim and Young-Tak Oh.

성혜원

Title : Group Signatures from Lattices

A digital signature is a cryptographic technique used to verify the authenticity and integrity of digital documents or messages. It serves as a digital equivalent of a handwritten signature and provides assurance that the document or message has not been tampered with since it was signed. A group signature, introduced by Chaum and van Heyest, is a type of digital signature scheme that allows a group member to sign messages anonymously on behalf of the group. It provides a way for an individual to sign a message in a manner that hides their identity within a specified group, while still allowing authorized entities to verify the signature. Numerous group signature schemes based on various settings have been proposed in the last quarter-century and depending on the specific setting, the size of the signature can vary very significantly. In this talk, we will introduce the first lattice-based constant-size group signature. 

송호영

Title : General version of Bellow and Furstenberg problem

Let $d, k \in \mathbb{Z}_{+}$be given and let $(X, \mathcal{B}(X), \mu)$ be a probability measure space endowed with a family of invertible commuting measure-preserving transformations $T_1, \ldots, T_d: X \rightarrow$ $X$. Assume that $P_1, \ldots, P_d \in \mathbb{Z}\left[\mathrm{m}_1, \ldots, \mathrm{m}_k\right]$. Then for any $f \in L^{\infty}(X)$ the multi-parameter linear polynomial averages

$$A_{M_1, \ldots, M_k ; X, T_1, \ldots, T_d}^{P_1, \ldots, P_d} f(x)=\mathbb{E}_{m \in \prod_{j=1}^k\left[M_j\right]} f\left(T_1^{P_1(m)} \cdots T_d^{P_d(m)} x\right)$$

converge for $\mu$-almost every $x \in X$, as $\min \left\{M_1, \ldots, M_k\right\} \rightarrow \infty$.

이호형

Title :  Partitions which are simultaneously s- and t-core

 For a natural number t, a partition is t-core if each of the hook length from its Ferrers-Young diagram is not divisivle by t. For two relatively prime numbers s and t, there are only finitely many partitions which are both s-core and t-core. Moreover, the number of partitions has a combinatorial approach which is different from the original partition theory.

서시은

Title : Homomorphic encryption and CKKS

Homomorphic encryption (HE) is a class of encryption schemes which supports homomorphic operations over encrypted data. After HE first constructed in 2009, various kinds of schemes came out, and among them the Cheon-Kim-Kim-Song (CKKS) scheme is the first HE scheme that enables homomorphic computation over encrypted real-value data. CKKS scheme is levelled homomorphic encryption scheme, so it consumes one level for each homomorphic multiplication or rescale step. After spending all available levels, a special decryption circuit called bootstrapping should be executed in order to conduct further homomorphic operations.