If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

If M is a hyperbolic 3-manifold fibering over the circle, then the fundamental group of M acts faithfully by homeomorphisms on a circle—the circle at infinity of the universal cover of the fiber—preserving a pair of invariant (stable and unstable) laminations. Many different kinds of dynamical structures including taut foliations and quasigeodesic or pseudo-Anosov flows are known to give rise to universal circles—a circle with a faithful action of the fundamental group preserving a pair of invariant laminations—and those universal circles play a key role in relating the dynamical structure to the geometry of M. In my minicourse and in my colloquium talk, I will introduce the idea of *zippers*, which give a new and direct way to construct universal circles, streamlining the known constructions in many cases, and giving a host of new constructions in others. In particular, zippers—and their associated universal circles—may be constructed directly from homological objects (uniform quasimorphisms), causal structures (uniform left orders), and many other structures.

A knot bounds an oriented compact connected surface in the 3-sphere, and consequently in the 4-ball. The 4-genus of a knot is the minimal genus among all such surfaces in the 4-ball, and the 4-genus of a link is defined analogously. In this talk, I will discuss lower bounds on the 4-genus derived from Cheeger-Gromov-von Neumann rho-invariants. This is joint work with Jae Choon Cha and Min Hoon Kim.

In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).

In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).

A normal projective surface with the same Betti numbers of the projective plane CP2 is called a rational homology projective plane (briefly Q-homology CP2 or QHCP2). People working in algebraic geometry and topology have long studied a Q-homology CP2 with possibly quotient singularities. It has been known that it has at most five such singular points, but it is still mysterious so that there are many unsolved problems left. In this talk, I’ll review some known results and open problems in this field which might be solved and might not be solved in near future. In particular, I’d like to review the following two topics and to report some recent progress: 1. Algebraic Montgomery-Yang problem. 2. Classification of Q-homology CP2 with quotient singularities. This is a joint work with Woohyeok Jo and Kyungbae Park..

In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).

In this series of talks, I'll present the basics of combinatorial semigroup theory, starting with elementary results and ending in recent research using high-powered tools. I'll begin by giving an overview of the elements of semigroup theory, including the analogue of Cayley's theorem, eggbox diagrams, Green's relations, inverse semigroups, and a famous result due to Green & Penrose. In the subsequent talk, I'll present the elements of presentations of semigroups, free (inverse) semigroups, Munn trees, and rewriting systems, leading into the fundamental problem central to combinatorial semigroup theory: the word problem. In the next talk, I'll dive into a particular class of semigroups called "special" monoids, and give proofs via rewriting systems due to Zhang (1990s) of famous results due to Adian (1960s), giving a solution to the word problem in all monoids given by a single defining relation of the form w=1. In the final talk (if there is time) I will dip our toes into how rewriting systems can compute the (co)homology of a monoid, and give new proofs via the spectral sequence of certain rewriting systems (forthcoming) of homological results due to Gray & Steinberg (2023).

The Lipshitz-Ozsvath-Thurston correspondence is a combinatorial way to describe the bordered Floer homology of a knot complement from the UV=0 coefficient knot Floer homology of the given knot. This is then used to compute the knot Floer homology of satellite knots. In this talk, we show that there is a "relative" version of this correspondence, between homotopy classes of type D morphisms of bordered Floer homology and locally symmetric chain maps of knot Floer complexes, modulo the "canonical negative class". This gives us a fully combinatorial process to compute knot Floer cobordism maps of satellite concordances in the UV=0 knot Floer homology.