정지혜 (이화여대 수학과 부호론)
Title : Construction of binary few-weight optimal linear codes and its applications
Jong Yoon Hyun*, Jihye Jeong**, Yoonjin Lee**
*College of Liberal Arts, Konkuk University, Glocal Campus, 268, Chungwon-daero Chungcheongbuk-do, Chungju-si 27478, Republic of Korea
**Department of Mathematics, Ewha Womans University, 52, Ewhayeodae-gil, Seodaemun-gu, Seoul 03760, Republic of Korea
Abstract
Error-correcting codes were invented to correct errors on noisy communication channels since a paper by Shannon was published in 1948. A major issue in Coding theory is to construct good error-correcting codes (e.g., few-weight codes and optimal codes). We construct new infinite families of binary optimal few-weight codes by using the shortening method. Furthermore, we completely determine the weight distributions of our shortened codes. To achieve our goal, we use certain families of multivariable functions, and we interpret a shortening method followed by puncturing in terms of multivariable functions. According to this interpretation, we find explicit criterion for the shortened codes to have fewer weight or to have fewer weights and larger minimum weights after the shortening process. We emphasize that some infinite families of few-weight optimal linear codes have new parameters. As applications, we produce support t-designs (t = 2 or 3) and find many optimal quantum codes via our code families.
Keywords: Optimal code, few-weight code, weight distribution, multivariable function
이은정 (이화여대 수학과 기하학)
Title : Polygon space and topology
Abstract
Polygon space arise as the spaces of closed linkages formed by vectors in -dimensional Euclidean space, considered up to similarity transformations such as dilation and rotation. These spaces hold deep connections to diverse fields, including Hamiltonian geometry, mathematical robotics and application in physics.
A canonical map, known as the length map, projects the polygon space to a geometric object called the rectified simplex, obtained by truncation of the regular simplex. Further truncation of the rectified simplex reveals interesting cell structures, which, in turn, provide insights into the connectedness of spaces of polygons with fixed side-lengths. (Kapovich and Millson)
In this talk, I will provide an introduction to polygon spaces and explore their intriguing cells structure, shedding light on their geometric and topological properties.
성혜원 (이화여대 수학과 암호론)
Title : Efficient Scalar multiplications over Twisted Edwards cuves of Isogeny-based PQC
Abstract
In the realm of Post-Quantum Cryptography (PQC), recent research has exposed vulnerabilities in SIDH (Supersingular Isogeny Diffie-Hellman) based on isogeny, necessitating the development of robust countermeasures. A surge of diverse research methods is underway to effectively address these vulnerabilities, with a predominant focus on protocols centered around scalar multiplication operations.
This paper serves a dual purpose by not only spotlighting vulnerabilities within SIDH but also introducing novel curve settings meticulously tailored for PQC. Our principal contribution lies in the introduction of enhanced curve settings, a tailored solution designed explicitly to counteract SIDH vulnerabilities and elevate the efficiency of scalar multiplication. This comprehensive approach is poised to significantly enhance the overall applicability of PQC.
The practical implementation of our proposed scalar multiplication method is carried out within isogeny-based cryptographic systems, harnessing the capabilities of twisted Edwards curves. Noteworthy is the application of our modified GLV method to twisted Edwards curves, resulting in a remarkable efficiency boost ranging from approximately 45% to 58% at each bit. This achievement translates into a tangible enhancement of the overall performance of isogeny-based cryptographic systems, reinforcing their resilience in the face of emerging threats in the PQC landscape.
서재현 (연세대학교 수학과 조합론)
Title : Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials
Abstract
For graphs H and G, a homomorphism from H to G is a map between their vertex sets which preserves the edges. We focus on the case when G is antiferromagnetic, meaning it has nonnegative weights, at most one positive eigenvalue, and possibly contains loops. The number of graph homomorphisms with an antiferromagnetic target G generalizes various important parameters in graph theory, including the number of independent sets and proper vertex colorings. We prove inequalities on the number of homomorphisms when a target graph is antiferromagnetic. Our method uses the emerging theory of Lorentzian polynomials due to Brändén and Huh, which may be of independent interest. This is a joint work with Joonkyung Lee and Jaeseong Oh.
진형준 (연세대학교 수학과 사교기하)
Title : Cohomological Invariants in Symplectic Geometry
Abstract
Symplectic structure on geometric objects allows one to investigate more richer algebraic structures on geometric instances on that object. As such, we shall introduce two cohomological theories, one called an open string invariant and the other called a closed string invariant. We shall seek for a relation between them, and will be acquainted with a theorem of S. Ganatra(Theorem 1.1), which says that the latter invariant can be transformed to the deformation invariants of the former via closed-open and open-closed string maps.
The moduli of closed-open maps can also be applied in the context of closed string mirror symmetry. This aspect may be covered if the time permits.
김동현 (연세대학교 수학과 대수기하)
Title : On Running Anti-canonical MMP Using Valuative Approach
with Sungrak Choi, Dae-won Lee and Sungwook Jang
Abstract
The aim of the minimal model program is to construct “minimal model”, or “Mori fibre space” for any given smooth projective variety (or a projective variety with mild singularities).
One can consider a variant of MMP, the “anti-canonical minimal model program” for a smooth projective variety X with −K_X pseudo-effective. However, in general, anti-canonical MMP is obstructed by various pathologies. To address this, we define an invariant, called the potential discrepancy, for a smooth projective varieties X with −K_X pseudo-effective. We expect that X is pklt if and only if X admits an anti-canonical model.
In this work, we aim to prove this conjecture in the case where κ(−K_X ) ≥ 0 assuming termination of flips for anti-canonical MMP. Our approach relies heavily on valuation-theoretic techniques, particularly the theory of log canonical thresholds developed by Jonsson and Mustață.