Categories
Mini-sessions

Quantum invariants and low-dimensional topology: Hyperbolic volume and volume conjecture

Splice-unknotting and crosscap numbers.- Thomas Kindred (University of Nebraska at Lincoln)

Abstract: Ito-Takimura recently introduced the splice-unknotting number of a knot. This diagrammatic invariant provides an upper bound for a knot's crosscap number, with equality in the alternating case. Using results of Kalfagianni-Lee, this equality leads to corollaries regarding hyperbolic volume and the Jones polynomial.

 

Families of fundamental shadow links realized as links in $S^3$- Sanjay Kumar (Michigan State University)

Abstract: In 2015, Chen and Yang provided evidence that the asymptotics of the Turaev-Viro invariant of a hyperbolic $3$-manifold evaluated at the root of unity $\exp(\frac{2\pi i}{r})$ have growth rates given by the hyperbolic volume. This has been proven by Belletti, Detcherry, Kalfagianni, and Yang for an infinite family of hyperbolic links in connect sums of $S^1 \times S^2$ known as the fundamental shadow links. In this talk, I will present examples of links in $S^3$ satisfying the Turaev-Viro invariant volume conjecture through homeomorphisms with complements of fundamental shadow links along with an application towards the conjecture posed by Andersen, Masbaum, and Ueno (AMU conjecture).

 

Volume conjecture, geometric decomposition and deformation of hyperbolic structures (I) and (II)- Ka Ho Wong (Texas A & M University)

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Abstract: The Chen-Yang volume conjecture of the Turaev-Viro invariant is a new topic in quantum topology. It has been shown that the $(2N+1)$-th Turaev-Viro invariant for a link complement can be expressed as a sum of norm squared of the colored Jones polynomial of the link evaluated at $t=\exp\left(\frac{2\pi i}{ N+\frac{1}{2}}\right)$. This leads to the study of the asymptotics for the $M$-th colored Jones polynomials of links evaluated at $(N+\frac{1}{2})$-th root of unity, with a fixed limiting ratio of $M$ to $N+\frac{1}{2}$. In the first talk, I will recall the definition of the colored Jones polynomials and discuss how the asymptotics of the colored Jones polynomials of the Whitehead link is related to the (not necessarily complete) hyperbolic structures on its complement. Then, in the second talk, we will focus on some satellite links whose complements have more than one hyperbolic piece in the geometric decomposition, and relate the asymptotics of their colored Jones polynomials to the geometric structures on the geometric pieces.

Categories
Mini-sessions

Quantum invariants and low-dimensional topology: Link homologies and categorification

$\mathfrak{sl}_n$-homology theories obstruct ribbon concordance- Nicolle Gonzalez (University of California-Los Angeles)

Abstract: In a recent result, Zemke showed that a ribbon concordance between two knots induces an injective map between their corresponding knot Floer homology. Shortly after, Levine and Zemke proved the analogous result for ribbon concordances between links and their Khovanov homology. In this talk I will explain joint work with Caprau-Lee-Lowrance-Sazdanovic and Zhang where we generalize this construction further to show that a link ribbon concordance induces injective maps between $\mathfrak{sl}_n$-homology theories for all $n \geq 2$.

 

Khovanov-Rozansky homology of torus links (and beyond)- Matthew Hogancamp (Northeastern University)

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Abstract: Throughout the past decade, torus links and their homological invariants have been the subject of numerous fascinating conjectures connecting link homology to seemingly distant areas of algebraic geometry and and representation theory (see work of various subsets of Cherednik, Gorsky, Negut, Oblomkov, Rasmussen, Shende). Many of these conjectures are now proven by "computing both sides" (the link homology side this was done by myself and Anton Mellit, both independently and jointly). In this talk I will discuss the main technique for computing (by hand!) Khovanov-Rozansky's HOMFLY-PT homology introduced by myself and Ben Elias, and its application to link homology. The message I hope to convey is that this technique is strikingly simple to use, and is useful in a wide variety of settings.

 

Khovanov homology detects T(2,6)- Gage Martin (Boston College)

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Abstract: Khovanov homology is a combinatorially defined link homology theory. Due to the combinatorial definition, many topological applications of Khovanov homology arise via connections to Floer theories. A specific topological application is the question of which links Khovanov homology detects. In this talk, we will give an overview of Khovanov homology and link detection, mention some of the connections to Floer theoretic data used in detection results, and show that Khovanov homology detects the torus link T(2,6).

 

Generalizing Rasmussen's $s$-invariant, and applications- Michael Willis (University of California-Los Angeles)

I will discuss a method to define Khovanov and Lee homology for links $L$ in connected sums of copies of $S^1\times S^2$. This allows us to define an $s$-invariant $s(L)$ that gives genus bounds on oriented cobordisms between links. I will discuss some applications to surfaces in certain 4-manifolds, including a proof that the $s$-invariant cannot detect exotic $B^4$'s coming from Gluck twists of the standard $B^4$. All of this is joint work with Ciprian Manolescu, Marco Marengon, and Sucharit Sarkar.

 

Categories
Mini-sessions

Quantum invariants and low-dimensional topology: New quantum invariants and other connections

Sequence of Undetectable Nonquasipositive Braids - Elaina Aceves (University of Iowa)

When $K$ is a quasipositive transverse knot, Hedden and Plamenevskaya proved that $2 \tau(K)-1=sl(K)$, where $\tau$ is the Ozsv\'ath-Szab\'o concordance invariant and $sl$ is the self-linking number of the knot. We construct a sequence of braids whose closures are nonquasipositive knots where we add more negative crossings as we progress through the sequence. Therefore, the knots formed from the sequence become more nonquasipositive as the sequence progresses. Furthermore, we conjecture that the knots obtained from this sequence behave like quasipositive knots in that they uphold the equality $2 \tau(K)-1=sl(K)$. Because of this property, the nonquasipositive knots obtained from our sequence are undetectable by transverse invariants like Ozsv\'ath and Szab\'o's $\hat{\theta}(K)$ from Heegaard Floer homology and Plamenevskaya's $\psi(K)$ from Khovanov homology.

 

On framing changes of links in $3$-manifolds- Dionne Ibarra (George Washington University)

Abstract: We show that the only way of changing the framing of a link by ambient isotopy in an oriented $3$-manifold is when the manifold admits a properly embedded non-separating $S^2$. We will illustrate the change in framing by the Dirac trick then relate the results to the framing skein module. Coauthors: Rhea Palak Bakshi, Gabriel Montoya-Vega, Jozef Przytycki, and Deborah Weeks.

 

Holonomy invariants of links and Holonomy invariants from quantum sl_2 and torsions- Calvin McPhail-Snyder (University of California-Berkeley)

Holonomy invariants of links

Holonomy invariants from quantum sl_2 and torsions

Abstract: To describe geometric information about a space X, we can equip it with a representation \rho: \pi_1(X) \to G, where G is a group. Quantum invariants using this extra data are called quantum holonomy invariants or homotopy quantum field theories. In this talk, I will give some motivation for this idea (including connections to volume conjectures) and discuss how to modify the usual Reshetikhin-Turaev construction to get holonomy invariants of links in S^3.

Blanchet, Geer, Patureau-Mirand, and Reshetikhin have constructed a holonomy invariant for G = SL_2(C) using the quantum group U_q(sl_2) at a root of unity. In this talk, I will give an overview of their construction, then discuss my recent work showing how to interpret their invariant in terms of twisted Reidemeister torsion.

 

How to make quantum groups easier? and Quantum knot invariants according to Alexander- Roland van der Veen (University of Groningen)

Joint work with Dror Bar-Natan (Toronto)

How to make quantum groups easier?

Abstract: Quantum groups such as U_q sl_2 often appear at the foundations of most quantum knot invariants. And usually this results in forbidding lists of generators, relations and formulas. Does it have to be this way? One way out is to pass to representations but then all computations will usually grow exponentially. (Try computing the Jones polynomial of a 50 crossing knot). In this talk I would like to propose another way out. Will modify the algebra itself and then work with it through Gaussian expressions. In fact one can also understand much of the way quantum groups are built from a purely topological standpoint but that is the subject of another talk.

Quantum knot invariants according to Alexander

Abstract: From the point of view of universal invariants the Alexander polynomial is the most fundamental quantum invariant. It appears as the one loop contribution in the perturbative expansion of the Chern-Simons integral regardless of what gauge group one starts with. This suggests that one should be able to use Alexander to gain insight into more complicated quantum invariants and conversely presents the challenge of generalizing the many nice properties of Alexander to a wider context. I will make these points concrete by reproving and extending the formula for the Alexander polynomial from a Seifert surface towards the universal quantum sl_2 invariant.

 

Categories
Mini-sessions

Floer theory and low-dimensional topology: Heegard Floer homology

Corks, Involutions, and Heegaard Floer Homology- Irving Dai (MIT)

Abstract: We introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution over any homology ball, rather than a particular contractible manifold. As an application, we define a modification of the homology cobordism group which takes into account an involution on each homology sphere, and prove that this admits an infinitely-generated subgroup of strongly non-extendable corks. We establish several new families of corks and prove that various known examples are strongly non-extendable. This is joint work with Matthew Hedden and Abhishek Mallick. This talk will be complementary to the talk given by Abhishek Mallick (in the "Knots, Surfaces, and 4-Manifolds" topic group), and will discuss the Heegaard Floer theory underlying various detection results.

 

Knot Floer homology and relative adjunction inequalities- Katherine Raoux (Michigan State University)

Abstract: In this talk, we present a relative adjunction inequality for 4-manifolds with boundary. We begin by constructing generalized Heegaard Floer tau-invariants associated to a knot in a 3-manifold and a nontrivial Floer class. Given a 4-manifold with boundary, the tau-invariant associated to a Floer class provides a lower bound for the genus of a properly embedded surface, provided that the Floer class is in the image of the cobordism map induced by the 4-manifold. We will also discuss several applications to links and contact manifolds. This is joint work with Matt Hedden.

 

 

Involutive Heegaard Floer homology and surgeries- Ian Zemke (Princeton University)

Abstract: In this talk, we investigate the behavior of involutive Heegaard Floer homology with respect to surgeries. We prove several surgery exact sequences, and also a mapping cone formula, for involutive Heegaard Floer homology. In this talk, we will describe a few elements of the proof, and also some applications to the homology cobordism group. This project is joint work in progress with Kristen Hendricks, Jen Hom and Matt Stoffregen.

 

 

Categories
Mini-sessions

Floer theory and low-dimensional topology: Gauge theory

Equivariant singular instanton homology, I: Applications to 4D clasp numbers​-Aliakbar Daemi (Washington University in St Louis)

Abstract: Any knot K is the boundary of a normally immersed disc in the 4-ball, and the 4D clasp number of K is the smallest number of double points of any such immersed disc. The 4D clasp number of K is bounded below by the slice genus of K. Motivated by the Caporaso-Harris-Mazur conjecture about algebraic curves in a quintic surface, Kronheimer and Mrowka asked whether the difference between the 4D clasp number and the slice genus can be arbitrarily large. In this talk I will introduce a knot invariant, called Gamma, and review some of its properties. Then I explain how this invariant can be used to answer Kronheimer and Mrowka's question. This is joint work with Chris Scaduto.​

Equivariant singular instanton homology, II: Introduction to the constructions- Chris Scaduto (University of Miami)

Abstract: The Gamma-invariant introduced in part I of this series is one of several outputs from equivariant singular instanton theory. This framework associates to a knot a suite of invariants which are morally derived from Morse theory of the Chern-Simons functional on an infinite-dimensional space with a circle action. After some general background, I will focus on the algebraic structures that are forefront to the theory. This is joint work with Ali Daemi.

 

 

Equivariant singular instanton homology, III: Singular Froyshov invariants and the Gamma-invariant​--Aliakbar Daemi (Washington University in St Louis)

​Abstract: A homology concordance is an embedded cylinder in a homology cobordism. Equivariant singular instanton homology is functorial with respect to homology concordances. We use this to produce a family of homology concordance invariants. The simplest elements of this family are integer valued invariants which are obtained by imitating the definition of the Froyshov homomorphisms in the context of three manifold invariants. The Gamma invariant used in the first talk is a refinement of these singular Froyshov invariants. This is joint work with Chris Scaduto.

 

 

Equivariant singular instanton homology, IV: Further applications​- Chris Scaduto (University of Miami)

Abstract: In this final talk I will present additional applications of the theory discussed in the previous parts of the series. One application is that certain topological assumptions imply the existence of non-trivial SU(2) representations for fundamental groups of knotted surface complements. Another application confirms a conjecture of Poudel-Saveliev and computes the irreducible mod 4 graded instanton homology of torus knots. This is joint work with Ali Daemi.​

 

 

Framed Instanton Floer homology revisited via sutures- Zhenkun Li (MIT)

Abstract: Framed Instanton Floer homology was introduced by Kronheimer and Mrowka for closed oriented 3-manifolds. It is conjectured to be isomorphic to the hat version of Heegaard Floer homology, and recently many computational results were achieved by several groups of people. In this talk, I will explain how the framed Instanton Floer homology of a closed oriented 3-manifold Y can be related to the sutured instanton Floer homology of the complement of a torsion knot inside Y with some suitable sutures. Then several applications follow. This is partially jointed with Sudipta Ghosh and C.-M. Michael Wong.

 

SL(2,C) Floer Homology, I: The 3-manifold invariant HP(Y)- Ikshu Neithalath (UCLA)

Abstract: We will construct HP(Y), the SL(2,C) Floer homology of a 3-manifold Y, as defined by Abouzaid and Manolescu. To do so, we will give a brief overview of the necessary algebro-geometric tools such as character varieties and perverse sheaves of vanishing cycles.

SL(2,C) Floer Homology, II: Invariants for Knots- Ikshu Neithalath (UCLA)

Abstract: We will sketch the construction of SL(2,C) Floer homology for knots as defined by Cote and Manolescu. We will then discuss some joint work with Cote on the properties of the knot invariant as well as some independent work computing the 3-manifold invariant for surgeries on knots.

 

 

Categories
Mini-sessions

Floer theory and low-dimensional topology: Connections to contact and symplectic geometry

Ascending surfaces, complex curves, and knot Floer homology- Kyle Hayden (Columbia University)

Abstract: Ascending surfaces are a natural class of surfaces in certain symplectic 4-manifolds that behave like complex curves, but they are much more flexible. I'll present a key structural result about these surfaces and highlight two applications: one to the study of complex curves, the other to recent results of Juhasz-Miller-Zemke regarding the transverse invariant in knot Floer homology. 

 

Right-veering open books and the Upsilon invariant- Diana Hubbard (Brooklyn College, CUNY)

Abstract: Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon. I will discuss an application of this work to the Slice-Ribbon conjecture.

 

GRID invariants obstruct decomposable Lagrangian cobordisms- C.-M. Michael Wong (Louisiana State Unversity) 

Abstract: Ozsvath, Szabo, and Thurston defined several combinatorial invariants of Legendrian links in the 3-sphere using grid homology, which is a combinatorial version of link Floer homology. These, collectively called the GRID invariants, are known to be effective in distinguishing some Legendrian knots that have the same classical invariants. In this talk, we show that the GRID invariants provide an obstruction to the existence of decomposable Lagrangian cobordisms between Legendrian links. This obstruction is stronger than the obstructions from the Thurston-Bennequin and rotation numbers, and is closely related to a recent result by Golla and Juhasz. This is joint work with John Baldwin and Tye Lidman.

 

Categories
Mini-sessions

Geometric group theory: Homeomorphism and mapping class groups

Surjective homomorphisms from surface braid groups- Lei Chen (Caltech)

Abstract: In this talk, I will present that any surjective homomorphism from the surface braid group to a torsion-free hyperbolic group factors through a forgetful map. This extends and gives a new proof of an earlier result of the author which works only for free groups and surface groups.

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Inverse limits of covering spaces- Curtis Kent (Brigham Young University)

Abstract: Many techniques that have proved effective in the study of finitely generated groups at first approach seem insufficient for the study of uncountable groups, e.g. big mapping class groups or fundamental groups of complicated spaces. We will discuss how to use inverse limits to study such groups through approximation by finitely generated groups. Specifically we will show that inverse limits of covering spaces, when path-connected, satisfy a Galois type correspondence and how this leads to a splitting of the first homology for many spaces.

Video

 

Universal bounds for torsion generating sets of mapping class groups- Justin Lanier (Georgia Tech)

Abstract: I showed with Margalit that the conjugacy class of any periodic element is a generating set for Mod(S_g), as long as g is at least 3 and the element isn’t trivial or a hyperelliptic involution. Since Mod(S_g) is finitely generated, these infinite torsion generating sets can always be whittled down to finite generating sets. Is there a universal upper bound on the number of conjugates required, independent of g and the element chosen? In general, the answer is no. However, for elements with order at least 3 there is a universal upper bound; our proof shows that 60 conjugates always suffice. I will record two short talks: a first talk that describes the history and context of this result and that gives an overview of the proof strategy, and then a second talk that goes into some of the nuances of the proof.

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Analogs of the curve graph for infinite type surfaces- Alexander Rasmussen (Yale)

Abstract: The curve graph of a finite type surface is a crucial tool for understanding the algebra and geometry of the corresponding mapping class group. Many of the applications that arise from this relationship rely on the fact that the curve graph is hyperbolic. We will describe actions of mapping class groups of infinite type surfaces on various graphs analogous to the curve graph. In particular, we will discuss the hyperbolicity of these graphs, some of their quasiconvex subgraphs, properties of the corresponding actions, and applications to bounded cohomology.

Video

 

The 'what' and 'why' of framed mapping class groups- Nick Salter (Columbia)

Abstract: The 'what': a framed mapping class group is the stabilizer of an isotopy class of vector field on a surface. Despite being infinite-index subgroups, Aaron Calderon and I have shown that these are generated by simple collections of finitely many Dehn twists. The 'why': Many natural families of Riemann surfaces are also equipped with the extra data of a vector field. This is true of families of translation surfaces (strata), as well as the family of Milnor fibers of an isolated plane curve singularity. Understanding the behavior of these families thus requires an understanding of the framed mapping class group. Work on the framed mapping class group and on strata is joint with Aaron Calderon, and work on plane curve singularities is joint with Pablo Portilla Cuadrado.

Video

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Categories
Mini-sessions

Geometric group theory: Bounded cohomology and norms on groups

Addition of geometric volume classes- James Farre (Yale)

Abstract: We study the algebraic structure of three dimensional bounded cohomology generated by volume classes for infinite co-volume, finitely generated Kleinian groups. While bounded cohomology is generally unwieldy, we show that addition admits a natural geometric interpretation for the volume classes of tame hyperbolic manifolds of infinite volume and bounded geometry: the volume classes of singly degenerate manifolds sum to the volume classes for manifolds with many degenerate ends. It turns out that this generates the linear dependencies among volume classes, giving a complete description of the algebraic structure of some geometrically defined subspaces of bounded cohomology. We will indicate some problems left open by this discussion and give some suggestions for future directions. Definitions, background, and geometric aspects of hyperbolic manifolds homotopy equivalent to a closed surface will be reviewed, but we assume some familiarity with hyperbolic geometry.

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The Spectrum of Simplicial Volume- Nicolaus Heuer (Cambridge)

Abstract: Simplicial volume was first introduced by Gromov to study the minimal volume of manifolds. Since then it has emerged as an active research field with a wide range of applications. In dimensions two and three, the set of possible values of simplicial volume may be fully computed using geometrization, but is hardly understood in higher dimensions. In joint work with Clara Löh (University of Regensburg), we show that the set of simplicial volumes in higher dimensions is dense in the non-negative reals. I will also discuss how the exact set of simplicial volumes in dimension four or higher may look like.

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Computability of the Minimal Genus on Second Homology- Thorben Kastenholz (University of Bonn)

Abstract: Surface representatives of second homology classes can be used to give geometric invariants for second homology classes, the most prominent examples are the genus and the Euler characteristic. In this talk I will explain why determining the minimal genus of a given homology class is in general undecidable, and how to compute it for a large class of "negatively-curved" spaces including 2-dimensional CAT(-1)-complexes. This will need a normal Form result proven by me and Mark Pedron, that extends a theorem by Edmonds on maps between surfaces.

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Quasimorphisms on diffeomorphism groups- Richard Webb (University of Manchester)

Abstract: I will explain how to construct an unbounded quasimorphism on the group of isotopically-trivial diffeomorphisms of a surface of positive genus. As a corollary the commutator length and fragmentation norm are both (stably) unbounded, which solves a problem of Burago--Ivanov--Polterovich. The proof uses a new hyperbolic graph on which these groups act by isometries, which is inspired by techniques from mapping class groups. This is joint work with Jonathan Bowden and Sebastian Hensel.

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Categories
Mini-sessions

Geometric group theory: Non-positively curved groups

Actions of big mapping class groups on the arc graph- Carolyn Abbott (Columbia)

Abstract: Given a finite-type surface (i.e. one with finitely generated fundamental group), there are two important objects naturally associated to it: a group, called the mapping class group, and an infinite-diameter hyperbolic graph, called the curve graph. The mapping class group acts by isometries on the curve graph, and this action has been extremely useful in understanding the algebraic and geometric properties of mapping class groups. One particularly important class of elements of the mapping class group are those which act as loxodromic isometries of the curve graph; these are called “pseudo Anosov” elements. Given an infinite-type surface with an isolated puncture, one can associate two analogous objects: the big mapping class group and the (relative) arc graph. In this talk, we will consider the action of big mapping class groups on the arc graph, and, in particular, we will construct an infinite family of “infinite-type” elements that act as loxodromic isometries of the arc graph, where an infinite-type element is (roughly) one which is not supported on any finite-type subsurface. This is joint work with Nick Miller and Priyam Patel.

Video

 

Morse Quasiflats- Jingyin Huang (Ohio State)

Abstract: We introduce the notion of Morse quasiflats, which is a common generalization of Morse quasigeodesics and quasiflats of top-rank. In the talk, we will provide motivations and examples for Morse quasiflats, as well as a number of equivalent definitions and quasi-isometric invariance (under mild assumptions). We will also show that Morse quasiflats are asymptotically conical, and provide several first applications. Based on joint work with B. Kleiner and S. Stadler.

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Effective ping-pong in CAT(0) cube complexes- Kasia Jankiewicz (University of Chicago)

Abstract: Ping-pong lemma is a useful tool for finding elements in a given group that generate free semigroups. We use it to prove uniform exponential growth of certain groups acting on CAT(0) cube complexes. We also construct examples of groups with arbitrarily large gap between their cohomological and cubical dimensions. This is partially joint work with Radhika Gupta and Thomas Ng.

Video

 

Mod p and torsion homology growth in nonpositive curvature- Kevin Schreve (University of Chicago)

Abstract: We compute the growth of mod p homology in finite index normal subgroups of right-angled Artin groups. We give examples where it differs from the rational homology growth, find such a group with exponential torsion growth, and for odd primes p construct closed locally CAT(0) manifolds with nontrivial mod p homology growth outside the middle dimension.

Video

 

Categories
Mini-sessions

Knots, surfaces, and 4-manifolds: 4-manifolds

Corks, involutions, and Heegaard Floer homology- Abhishek Mallick (Michigan State)

Abstract: We introduce and study a set of Floer-theoretic invariants aimed at detecting corks. Our invariants obstruct the extension of a given involution of a 3-manifold over any homology ball that it may bound (rather than a particular contractible manifold). We utilize the formalism of local equivalence coming from involutive Heegaard Floer homology. As an application, we define a modification of the integer homology cobordism group which takes into account involutions acting on homology spheres, and prove that this group admits an infinite rank subgroup generated by corks. Using our invariants, we also establish several new families of (strong) corks (This is joint work with Irving Dai and Matthew Hedden). This talk will be complementary to the talk given by Irving Dai (in the "Floer theory and low-dimensional topology" topic group).

A relative invariant of smooth 4-manifolds- Hyunki Min (Georgia Tech)

Abstract: We define a polynomial invariant of a smooth and compact 4-manifold with connected boundary by modifying an invariant of closed 4-manifolds from Heegaard Floer homology. Using this invariant, we show that there exist infinitely many exotic fillings of 3-manifolds with non-vanishing contact invariant. This is a joint work with John Etnyre and Anubhav Mukherjee.

Standardizing Some Low Genus Trisections- Jesse Moeller (University of Nebraska-Lincoln)

Abstract: Trisections are a novel way to study smooth 4-manifold topology reminiscent of Heegaard splittings of a 3-manifolds; a surface, together with families of embedded curves, determines a 4-manifold up to diffeomorphism. Given a specific genus, we would like to know which 4-manifolds have trisection diagrams inhabiting a surface with this genus. For genus one and two, this is known. In this talk, we will introduce a family of seemingly complicated genus three trisection diagrams and demonstrate that they are, in fact, connected sums of well understood diagrams.

Symplectic 4-Manifolds on the Noether Line and between the Noether and Half Noether Lines - Sümeyra Sakalli (MPIM Bonn)

Abstract: It is known that all minimal complex surfaces of general type have exactly one (Seiberg-Witten) basic class, up to sign. Thus, it is natural to ask if one can construct smooth 4-manifolds with one basic class. First, Fintushel and Stern built simply connected, spin, smooth, nonsymplectic 4-manifolds with one basic class. Next, Fintushel, Park and Stern constructed simply connected, noncomplex, symplectic 4-manifolds with one basic class. Later Akhmedov constructed infinitely many simply connected, nonsymplectic and pairwise nondiffeomorphic 4-manifolds with nontrivial Seiberg-Witten invariants. Park and Yun also gave a construction of simply connected, nonspin, smooth, nonsymplectic 4-manifolds with one basic class. All these manifolds were obtained via knot surgeries, blow-ups and rational blow-downs.  In this talk, we will first review some main concepts and recent techniques in symplectic 4-manifolds theory. Then we will construct minimal, simply connected and symplectic 4-manifolds on the Noether line and between the Noether and half Noether lines by the so-called star surgeries, and by using complex singularities. We will show that our manifolds have exotic smooth structures and each of them has one basic class. We will also present a completely geometric way of constructing certain configurations of Kodaira’s singularities in the rational elliptic surfaces, without using any monodromy arguments. 

Ribbon homology cobordisms- Mike Wong (Louisiana State University)

Abstract: A cobordism between 3-manifolds is ribbon if it has no 3-handles. Such cobordisms arise naturally from several different topological and geometric contexts. In this talk, we describe a few obstructions to their existence, from Thurston geometries, character varieties, and instanton and Heegaard Floer homologies, and some applications. This is joint work with Aliakbar Daemi, Tye Lidman, and Shea Vela-Vick.