Speaker: Ngo V. Trung
Title: Multiplicity sequence and integral dependence
Abstract: In 1961 Rees proved a criterion for integral dependence between two ideals of finite colengths by the equality of their Hilbert-Samuel multiplicity. This criterion plays an important role in Teissier’s work on the equisingularity of families of hypersurfaces with isolated singularities. For hypersurfaces with non-isolated singularities, one needs a similar numerical criterion for integral dependence of arbitrary ideals. For a long time, it was not clear how to extend Rees’ multiplicity theorem to arbitrary ideals because they do not have the Hilbert-Samuel multiplicity. A possibility is to replace the Hilbert-Samuel multiplicity by the multiplicity sequence which were introduced by Achilles and Manaresi in 1997. Independent works of Gaffney and Gassler in the analytic case have led to the conjecture that two arbitrary ideals in a local ring as in Rees' work have the same integral closure if and only if they have the same multiplicity sequence. This conjecture was recently solved by Claudia Polini, Ngo Viet Trung, Bernd Ulrich, and Javid Validashti. This talk will discuss the development leading to the solution and arising problems.
Speaker: Dosang Joe
Title: A glimpse of the early work of prof. June Huh
Abstract: 2022 fields medalist, prof. Jun Huh had published a paper on which he fixed the conjecture of Rota that concernes about the chromatic function of a graph in his first year of graduate study. In his paper, he proved the log-concavity property of the sequence of the alternating coefficients of the charcteristic function of given representable matroid, which generalize the Rota’s conjecture on graphs. His profound results was presumely inspired by the deep understanding of the resolution of sigularities of algebaric variety and personal communication with prof. Horonaka Heisuke who gave a lecture to him in Seoul National University. The discovery he had found was new cornerstone for algebraic geomery and combiantorics. He has been successfully developed his career as mathematician and believed to be leading researcher in his field. The detailed explanation of the his paper will be delieverd carefully.
Speaker: Hakho Choi
Title: On symplectic fillings of Seifert 3-manifolds
Abstract: One way to understand the topology of symplectic manifolds is to study techniques for constructing symplectic manifolds. When we try to get a new symplectic manifold by gluing together local pieces, the symplectic structure on each piece should be compatible with each other along the pasting region. This search for symplectic cut-and-paste techniques has led us to the study of symplectic 4-manifolds with convex boundaries, which we call symplectic fillings.
In this talk, we discuss minimal symplectic fillings of a Seifert 3-manifold $Y$ with a canonical contact structure. After a brief review of the classification scheme for minimal symplectic fillings of $Y$, I’ll explain surgery descriptions of the fillings together with relations between the fillings and Milnor fibers of the normal complex surface singularity corresponding to $Y$. This is a joint work with Jongil Park.
Speaker: Jong Hae Keum
Title: Mori dream surfaces
Abstract: The Cox ring of a variety is the total coordinate ring, i.e., the direct sum of all spaces of global sections of all divisors. When this ring is finitely generated, the variety is called Mori dream (MD). A necessary condition for being MD is the finite generation of Pic(X), i.e., the vanishing of the irregularity. Smooth rational surfaces with big anticanonical divisor are MD. Thus all (weak) del Pezzo surfaces are. A K3 surface or an Enriques surface is MD if and only if its automorphism group is finite.
In this talk I will consider the case of surfaces of general type with p_g=0, and provide several examples that are MD. I will also provide non-minimal examples that are not MD. If time permits, combinatorially minimal models of Mori dream surfaces will be discussed. This is a joint work with Kyoung-Seog Lee.
Speaker: Jong In Han
Title: Projective surfaces in P^4 that are counterexamples to the Eisenbud-Goto regularity conjecture
Abstract: It is well known that the Eisenbud-Goto regularity conjecture is true for projective curves, smooth surfaces, smooth threefolds in P^5, and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, a counterexample of a projective surface has not been found while counterexamples of dimension three or higher are known.
In this talk, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in P^4, and investigate some projective invariants and cohomological properties of them. These counterexamples are constructed via binomial maps between projective spaces. This is a joint work with Sijong Kwak.
Speaker: Sangjin Lee
Title: A construction of diffeomorphic but not symplectomorphic manifolds
Abstract: In the early '90s, Weinstein introduced the notion of symplectic handles. One can build a symplectic manifold by attaching symplectic handles. This talk will discuss how to construct diffeomorphic but not symplectomorphic manifolds using their symplectic handle decomposition. More precisely, to prove the "diffeomorphic" part, we will consider their Lefschetz fibrations. To show the "not symplectomorphic" part, we will discuss their wrapped Fukaya categories. It is joint work with Dongwook Choa and Dogancan Karabas.
Speaker: Woohyeok Jo
Title: A survey of the 3-dimensional homology cobordism group
Abstract: In this talk, we recall the 3-dimensional homology cobordism group and present various results about its algebraic structure. Further, we list related open problems about homology 3-spheres.
Speaker: Kyungbae Park
Title: Intersection forms of 4-manifolds with boundary
Abstract: The intersection pairing of the second homology classes provides a fundamental invariant for 4-manifolds. An interesting problem asks which symmetric, bilinear forms realizes the intersection form of closed 4-manifolds or 4-manifolds with a fixed boundary. In particular, the question reveals the difference between topological and smooth categories in dimension four. In this talk, we will review classical results relating to the question and introduce recent progress.
Speaker: Hyeonjun Park
Title: Counting surfaces on Calabi-Yau 4-folds
Abstract: Counting curves on Calabi-Yau 3-folds is one of the most central topics in modern enumerative geometry. In this poster presentation, we discuss counting surfaces on Calabi-Yau 4-folds. An interesting new feature of this topic is a deep connection to derived algebraic geometry and Hodge theory. In particular, the surface counting invariants can detect Grothendieck's variational Hodge conjecture. This is joint work with Younghan Bae and Martijn Kool.
Speaker: Jaewoo Jung
Title: A graph decomposition that bounds on regularity of graph ideals
Abstract: The minimal graded free resolution of square-free monomial ideals can be investigated combinatorially through the Stanley-Reisner correspondence between the ideals and simplicial complexes. Regarding of the correspondence, we study the bounds on Castelnuovo-Mumford regularity of (non-)edge ideals in terms of properties on the associated graphs. Note that the regularity of an ideal is an algebraic invariant that inscribes algebraic complexity of the ideal.
Our main result is a decomposition of the graph by its subgraphs that provides a bound on the regularity of corresponding ideal. We discuss some applications and improvements of bounds on regularity of monomial ideals.
Speaker: Le T. Hoa
Title: Upper bounds on two Hilbert coefficients
Abstract: In this talk we present some new upper bounds on the first and the second Hilbert coefficients of a Cohen-Macaulay module over a local ring. Characterizations are provided for some upper bounds to be attained. The characterizations are given in terms of Hilbert series as well as in terms of the Castelnuovo-Mumford regularity of the associated graded module. This is a joint work with L. X. Dung and J. Elias.