In this talk, I will talk about isotopy problems of Seifert surfaces pushed in to the 4-ball. In particular, I will prove that every Seifert surface of a non-split alternating link become isotopic in the 4-ball. This is a joint work with Maggie Miller and Jaehoon Yoo.

Let S be a simply-connected rational homology complex projective plane with quotient singularities. The algebraic Montgomery-Yang problem conjectures that the number of singular points of S is at most three. In this talk, we leverage results from the study of smooth 4-manifolds, such as the Donaldson diagonalization theorem, to establish additional conditions for S. As a result, we eliminate the possibility of a rational homology complex projective plane of specific types with four singularities. We also identify infinite families of singularities that satisfy properties in algebraic geometry, including the orbifold BMY inequality, but are obstructed from being a rational homology complex projective plane due to smooth conditions. Additionally, we discuss experimental results related to this problem. This is joint work with Jongil Park and Kyungbae Park.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball. Day 3: Recent developments of Khovanov homology and its applications to low-dimensional topology.

We introduce a version of Heegaard diagrams for 5-dimensional cobordisms with 2- and 3-handles, 5-dimensional 3-handlebodies, and closed 5-manifolds. We show that every such 5-manifold can be represented by a Heegaard diagram, and two Heegaard diagrams represent diffeomorphic 5-manifolds if and only if they are related by certain moves. As an application, we construct Heegaard diagrams for 5-dimensional cobordisms from the standard 4-sphere to the Gluck twists along knotted 2-spheres. This provides some equivalent statements regarding the Gluck twists being diffeomorphic to the standard 4-sphere.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball. Day 2: Numerical invariants from Khovanov homology and their applications.

Knot homology theories revolutionized the study of knots and links, much like (simplicial or singular) homology theory revolutionized the study of topological spaces. One of the major knot homology theories, Khovanov homology, was introduced by M. Khovanov in 2000 as a "categorification of the Jones polynomial." One notable feature of Khovanov homology is its ability to detect the unknot, a feature not known to be possessed by the Jones polynomial. Recently, it has found notable applications in low-dimensional topology, including the detection of exotic surfaces in the 4-ball. Day 1: Jones polynomial and its categorification.

In the 1980's Casson and Gordon produced the first non slice knots which are trivial in Levine's algebraic concordance group, and in 2003 Cochran-Orr-Teichner produced the first no slice knots undetectable by Casson and Gordon's invariants. They do so by producing a filtration of the concordance group by subgroups a knot in the 1.5th term of this filtration has vanishing Casson-Gordon invariants. Since then this work has been central to the study of knot concordance. We will introduce this filtration and review just enough of the theory of L^2 homology to prove that the successive quotients of this filtration are nontrivial.

In the 1970's J. Levine produced a surjection from the knot concordance group to the so called algebraic concordance group. This captured the known features of the knot concordance group to that point and classifies high dimensional concordance. During this survey talk we will explore the construction of the algebraic concordance group and explain some of its consequences.