Abstract:   Given a smooth structure on a manifold M, one obtains a smooth structure on the product M x R. If M is closed and has dimension at least 5, then Kirby--Siebenmann's product structure theorem states that this assignment is a bijection from the set of smooth structures on M to the set of smooth structures on M x R. In the equivariant setting, the product structure theorem does not hold; the analogous map is neither injective nor surjective. In this talk, I will describe a way to construct infinitely many equivariant smooth structures of a G-manifold M which become equivalent after taking a product with R. This construction uses algebraic K-theory and a small amount of number theory. 

Abstract:   In 2019, Budney and Gabai introduced a construction of diffeomorphisms of a four-manifold called barbell diffeomorphisms. We will recount what these are and use them to construct interesting knotting in S^5 and S^4. This is joint work in progress with Seungwon Kim and Gheehyun Nahm.

Abstract:   Recently, Zemke proved that Heegaard Floer homology and lattice homology agree, for general plumbing trees, generalizing a theorem of Ozsváth-Szabó showing this equivalence for almost-rational plumbings.  In this talk, we'll give background on monopole Floer spectra, and give a calculation of the monopole Floer spectra of almost rational plumbings, based closely on Ozsváth-Szabó's proof, in terms of lattice homology.  We also include some obstructions to the existence of spin 4-manifolds with certain boundary that follow from these calculations.  This is joint work with Irving Dai and Hirofumi Sasahira.

Abstract:   We investigate the existence of non-orientable Lagrangian surfaces in symplectic S^2xS^2s. On one end of this question, it is known that a Klein bottle embedded in S^2xS^2 cannot be Lagrangian in almost half of the possible symplectic structures on S^2xS^2. Using trisection-inspired methods, we resolve the other end of this question: there is a homologically non-trivial non-orientable genus 12 surface which is Lagrangian in every symplectic S^2xS^2. This is joint work with Laura Wakelin.

Abstract:   Unknotting number is a fundamental measure of how complicated a knot is, measuring how `far' it is from the unknot via crossing changes. It is a challenging invariant to compute, with a vast array of tools applied to its calculation, and many conjectures have grown up around it. In this talk we will discuss three such conjectures, each aimed at simplifying the task of computing unknotting numbers. We will describe how work on them led to a resolution of the `oldest' one: the additivity of unknotting number under connected sum.

Abstract: I'll outline an elementary differential topology proof of the Narasimhan-Ramanan theorem that the SU(2) character variety of an oriented genus 2 surface is homeomorphic to CP^3, and give examples of Lagrangian immersions into this variety induced by 3-manifolds with genus 2 boundary.  Joint work with Boozer and Herald.

Abstract:   An aspherical space is one with vanishing higher homotopy groups; under mild assumptions this means that it is determined up to homotopy type by its fundamental group. A famous conjecture of Borel states that closed aspherical manifolds are in fact determined up to homeomorphism by their fundamental group. One could also ask (although Borel famously didn’t) whether smooth aspherical manifolds are determined up to diffeomorphism by their fundamental group; this is known to hold in dimensions at most 3 and to be false in dimension at least 5. We resolve the remaining case by exhibiting closed smooth aspherical 4-manifolds that are homeomorphic but not diffeomorphic. This is joint with Davis, Hayden, Huang, and Sunukjian. 

Abstract:   In this talk, we define and state properties about a homological skein invariant constructed from Bar-Natan's deformation of Khovanov homology. We extend the notion of H-torsion order for Bar-Natan homology and corresponding results about internal stabilization distances of exotically knotted surfaces to the 4-manifold setting through this construction. In particular, we examine gluing maps for this invariant corresponding to connect-sums of 4-manifolds, and we use these maps in conjunction with results of Hayden to produce exotic pairs of knotted surfaces in 4-manifolds other than the 4-ball. Furthermore, we show that these surfaces do not become smoothly isotopic rel boundary after a single internal stabilization.

Abstract: Recent developments demonstrate that invariants coming from real Seiberg-Witten gauge theory are fruitful for the detection of exotically knotted surfaces in the 4-sphere. Notably, Miyazawa introduces an invariant for small-genus knotted surfaces and proves the existence of an infinite family of (smoothly mutually distinct) exotic unknotted RP^2-knots. I will show how, by a shift in perspective, we can construct a larger (bi-)infinite family of exotic unknotted RP^2s distinguished by the same invariant, in a way that is relatively straightforward and procedural.

Abstract:   The linear independence of knots under satellite operation has been frequently studied in the field of knot theory. Most of the prior works focused on the independence of knots under a fixed satellite operation. In this talk, I will talk about my recent result which goes the other way: using the instanton filtration invariant r_s developed by Nozaki-Sato-Taniguchi, we show for a certain family of satellite patterns and fixed companion knot, such family of satellite knots are linear independent. We subsequently will show our criterion depends only on some classical and computable invariant for some specific cases.

Abstract:  Knot Floer homology is a powerful link invariant. In its most general version, it is a bigraded chain complex over a polynomial ring F[U,V]. In this talk, I will describe the structure theorem of such objects - they are a direct sum of snake complexes and local systems - and explain what information each summand carries about the knot.

Abstract: In this talk I will explain a connection between elliptic fibrations of complex surfaces, spherical braid groups, and SL_2 character varieties.  The only background assumed will be standard first-year graduate courses.

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Abstract:   I'll discuss some applications of Dowker duality in data analysis. I will then present three short, new proofs of Dowker duality using various poset fiber lemmas. I will introduce modifications of joins and products of simplicial complexes called relational join and relational product complexes. These relational complexes can be constructed whenever there is a relation between simplicial complexes, which includes the context of Dowker duality and covers of simplicial complexes. In this more general setting, I will show that the homologies of the simplicial complexes and the relational complexes fit together in a long exact sequence. If time permits, I'll discuss similar results for profunctors, which are generalizations of relations to categories.

Abstract:  In this talk we will discuss ideas in the proof of the following result. There exist finitely presented, residually finite groups that are profinitely rigid in the class of all finitely presented groups but not in the class of all finitely generated groups. These groups are of the form G x G where G is a profinitely rigid 3-manifold group (which can be taken to be the fundamental group of a certain kind of Seifert fibred space).

Abstract: A Lefschetz fibration M^4 \to S^2 is a generalization of a surface bundle which also allows finitely many nodal singular fibers. The Arakelov--Parshin rigidity theorem implies that nontrivial, holomorphic Lefschetz fibrations of genus g \geq 2 admit only finitely many holomorphic sections. In this talk, we will show that no such finiteness result holds for smooth or symplectic sections by giving examples of genus-g (g \geq 2) Lefschetz fibrations with infinitely many homologically distinct sections. We will also discuss examples with infinitely many orbits of sections under the action of fiberwise diffeomorphisms of M that preserves the set of fibers of M \to S^2. This is joint work with Carlos A. Serv\'an.

Abstract:  It is a classical result, due to Milnor, that every spin 3-manifold spin bounds a 4-manifold.  If we ask for this null-bordism to be of a prescribed normal 1-type, i.e. its stable normal bundle has a prescribed Postnikov-Moore approximation, then it may no longer exist.  We fix this data on a given 3-manifold and describe a three stage geometric obstruction theory for the existence of a filling which extends this structure, which we show coincide with the obstructions provided by the James spectral sequence.  If we assume the first two obstructions vanish, we can reduce the problem to a certain variation on a sphere embedding problem in a 4-manifold.  Our main contribution is a new `tertiary' obstruction, defined using Wall's equivariant self intersection number, that entirely governs this embedding problem.  This is joint work with Peter Teichner and Simona Veselá.

Abstract: In 4-manifold topology, differences between the smooth and topological categories often “dissolve” after stabilisation by connected sum with enough copies of S^2xS^2. I will discuss recent joint work exploring whether this holds for the mapping class group of a 4-manifold: if two self-diffeomorphisms are topologically isotopic, are they always smoothly isotopic, after stabilisation? We produce general conditions on the fundamental group that guarantee the answer is indeed “yes”. On the other hand, by weakening the initial hypothesis to topologically pseudo-isotopic, we produce examples where the answer is “no”. This is joint with Mark Powell and Oscar Randal-Williams.

Abstract: Dehn surgery is a fundamental construction in topology where one removes a neighborhood of a knot from the three-sphere and reglues to obtain a new three-manifold. The Cosmetic Surgery Conjecture predicts two different surgeries on the same non-trivial knot always gives different three-manifolds. We discuss how gauge theory, in particular, the Chern-Simons functional, can help approach this problem. This technique allows us to solve the conjecture in essentially all but one case. This is joint work with Ali Daemi and Mike Miller Eismeier.

Abstract: Seminal work of Thurston shows that one can construct a plethora of finite volume, hyperbolic 3-manifolds using mapping tori of surface homeomorphisms. In fact, Thurston shows that such a mapping torus is hyperbolizable precisely when the corresponding bundle is atoroidal which is equivalent to the associated mapping class being pseudo-Anosov.  Motivated by the desire to push this construction into dimension 4, it has been an open question as to whether one can similarly construct atoroidal surface bundles over surfaces. This was finally resolved this year by Kent and Leininger, who produced infinitely many such bundles. It is a folklore conjecture that, despite the analogy, such manifolds should not be hyperbolizable and one hope was to prove this by showing the non-vanishing of a certain obstruction to hyperbolicity called the signature. In this talk we will show that, on the contrary, the signature of these bundles vanishes and hence they retain the potential to be hyperbolic 4-manifolds. This is joint work with Jean-Francois Lafont and Lorenzo Ruffoni.

Abstract: This is joint work with Lisa Traynor. I'll introduce the notion of generating families, which produce many interesting examples of Legendrians in jet bundles (for example, Legendrian knots in R^3). Then I'll introduce an invariant of generating families, called generating family spectra, which lift classically known (co)homological invariants. I'll present a proof that there are infinitely many connected components of generating families that can be detected using spectra that cannot be detected classically -- so generating family spectra are stronger invariants than previously known invariants. If time allows, I may explain how generating families form an infinity-category enriched in spectra, and the invariants in this talk are (dual to) the endomorphisms -- so, in particular, admit a (co)multiplication.

Abstract: The Khovanov homology groups of the torus knots T(n,m) are known to stabilize as m goes to infinity with n fixed. In this talk, we make the observation that when n = 2, the stable limit happens to be isomorphic to the homology of the free loop space of the 2-sphere. Our main result suggests that this is not merely a coincidence: we prove that the k-colored sl(N) homology of T(2,m) stabilizes to the homology of the free loop space of the complex Grassmannian Gr(k,N).

Abstract: Seiberg-Witten theory is developed from physics and it can detect smooth structures of 4-manifolds. In this talk I will introduce a surgery formula for the ordinary Seiberg-Witten invariants, and surgery formulas for the families Seiberg-Witten invariants of families of 4-manifolds obtained through fiberwise surgery. Our formula expresses the Seiberg-Witten invariants of the manifold after the surgery, in terms of the original Seiberg-Witten moduli space cut down by a cohomology class in the configuration space. We use these surgery formulas to study how a surgery can preserve or produce exotic phenomena. 

Abstract: In this talk we will review definitions and properties of double-slice genus and equivariant 4-genus. Combining these ideas, we will define a notion of equivariant double-slice genus for strongly invertible knots and cover a variety of results differentiating the equivariant double-slice genus from other genera. Using the equivariant double-slice genus we begin to study symmetric surfaces in S^4 up to equivariant isotopy, constructing interesting examples of symmetric surfaces in S^4 and introducing notions of equivariant stabilization distance for symmetric surfaces.

Abstract: Seiberg-Witten theory has an analogue for 3- and 4-manifolds with involutions called real Seiberg-Witten theory. This theory can be used to construct invariants of links and embedded surfaces by passing to double branched covers. This talk will focus on a framed version of real Seiberg-Witten-Floer homology. It turns out this invariant of links has rather surprising properties not seen in ordinary Seiberg-Witten theory. I will explain why it is special and how it is related to some recent developments.

Abstract: We consider certain decision problems about PL, locally flat, closed, connected, orientable surfaces in S^4. In particular we determine explicit integers g_1 and g_2 such that if g is at least g_1 (resp. g_2) then there is no algorithm to decide whether or not a surface in S^4 of genus g is PL (resp. TOP) unknotted. The proofs depend on results about surface groups, i.e. the fundamental groups of complements of surfaces in S^4. For example we determine explicit integers g_3 and g_4 such that there is a surface in S^4 of genus g_3 whose group has unsolvable word problem, and a surface in S^4 of genus g_4 whose group contains an isomorphic copy of every finitely presented group.  

Abstract: I will present a cork theorem for diffeomorphisms of simply connected 4-manifolds, showing that one can sometimes localise a diffeomorphism to a contractible submanifold. I will sketch the proof and describe some applications. Joint work with Slava Krushkal, Anubhav Mukherjee, and Terrin Warren.  

Abstract:  In a recent note Francesco Lin showed that if a rational homology sphere Y admits a taut foliation then the Heegaard Floer module $HF^-(Y)& contains a copy of $F[U]/U$ as a summand. This implies that either the L-space conjecture is false or that Heegaard Floer homology satisfies a geography restriction. In a recent paper in collaboration with Fraser Binns we verified that Lin's geography restriction holds for a wide class of rational homology spheres. I shall discuss our argument, and advance the hypothesis that the Heegaard Floer module $HF^-(Y)$ may satisfy a stronger geography restriction than the one suggested by Lin’s theorem.

Abstract:  Kreck’s modified surgery gives an approach to classify 2n-manifolds up to stable diffeomorphism, i.e., up to a connected sum with copies of S^n x S^n. In dimension 4, we use a combination of modified and classical surgery to compare the stable diffeomorphism classification with other stable equivalence relations. Most importantly, we consider homotopy equivalence up to connected sum with copies of S^2 x S^2. This is joint work with John Nicholson and Simona Veselá. 

Abstract:  The cosmetic crossing conjecture posits that switching a non-trivial crossing in a knot diagram always changes the knot type. This question is closely related to cosmetic surgery problems for three-manifolds, and has seen significant progress in recent years. We will discuss the conjecture, and present new obstructions to cosmetic crossing changes for a family of links that includes all alternating knots. 

Abstract:  The Khovanov skein lasagna module S(X;L) is a smooth invariant of a 4-manifold X with link L in its boundary.  In this talk I will outline the construction of Khovanov skein lasagna modules, as well as new computations and applications including the detection of some exotic 4-manifolds.  This work is joint with Qiuyu R. 

Abstract:  A Dehn surgery slope p/q is said to be characterizing for a knot if the homeomorphism type of the p/q-Dehn surgery along the knot determines the knot up to isotopy. I will discus the first and only examples to date of a complete and non-empty list of non-characterizing slopes for a knot. 

Abstract:  In this joint work with Wenzhao Chen, we study the different types of symmetric crossing changes possible on strongly invertible knots. We will discuss which strongly invertible knots can be unknotted with a sequence of each type of symmetric crossing change. For knots that can be unknotted this way, we discuss methods to obtain lower bounds on the minimum number of symmetric crossing changes needed. I will also explain why some natural equivariant versions of the conjecture that the unknotting number is additive under connected sum are false. 

Abstract:  A slope p/q is characterising for a knot K if the oriented homeomorphism type of the 3-manifold obtained by performing Dehn surgery of slope p/q on K uniquely determines the knot K. For any knot K, there exists a bound C(K) such that any slope p/q with |q|≥C(K) is characterising for K. This bound has previously been constructed for certain classes of knots, including torus knots, hyperbolic knots and composite knots. In this talk, I will give an overview of joint work with Patricia Sorya in which we complete this realisation problem for all remaining knots.  

Abstract:  In joint preliminary work with Daryl Cooper and Priyam Patel, we show that each of Thurston's eight model geometries is "locally combinatorially defined" or LCD. Further, we show this is equivalent to the existence of a compact branched 3-manifold, W(G), for each geometry, G, with the following property: a closed 3-manifold, M, has a G-structure if and only if there is an immersion of M into W(G). In this talk, I will give some background on branched n-manifolds, construct the branched 3-manifold in detail for spherical geometry, and discuss some of the results in the other geometries.

Abstract:  A pattern, or knot in a solid torus, induces a map on the set of knots modulo smooth concordance. In 2016, Hedden conjectured that essentially none of these maps are group homomorphisms--more precisely, that the only homomorphisms induced by satelliting are the identity map, the reversal map, and the zero map. In particular, this would imply that patterns with winding number not in the set {-1,0,1} cannot induce homomorphisms. I will discuss work with Randall Johanningsmeier and Hillary Kim in which we prove that if P is a pattern with winding number that is even but not divisible by eight, then P cannot induce a homomorphism on the smooth concordance group. This relies heavily on previous joint work with Tye Lidman and Juanita Pinzon-Caicedo, and is the first result that obstructs all patterns of a fixed winding number from inducing homomorphisms.