Monday, April 28th, 2025
Workshop: Jen Hom
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: New tools for studying homology cobordism
Speaker: Jen Hom
Abstract: We will discuss a new family of homology cobordism invariants coming from Pin(2)-equivariant Floer homology. Our construction relies on Koszul duality and constructions inspired by the concordance invariants epsilon and upsilon. This is joint work in progress with I. Dai, M. Stoffregen, and L. Truong.
Title: New tools for studying homology cobordism
Speaker: Jen Hom
Abstract: We will discuss a new family of homology cobordism invariants coming from Pin(2)-equivariant Floer homology. Our construction relies on Koszul duality and constructions inspired by the concordance invariants epsilon and upsilon. This is joint work in progress with I. Dai, M. Stoffregen, and L. Truong.
Workshop: Zhenkun Li
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: 2-torsion in instanton Floer homology
Speaker: Zhenkun Li
Abstract: Instanton Floer homology, introduced by Floer in the 1980s, has become a power tool in the study of 3-dimensional topology. Its application has led to significant achievements, such as the proof of the Property P conjecture. While instanton Floer homology with complex coefficients is widely studied and conjectured to be isomorphic to the hat version of Heegaard Floer homology, its counterpart with integral coefficients is less understood. In this talk, we will explore the abundance of 2-torsion in instanton Floer homology with integral coefficients and demonstrate how this 2-torsion encodes intriguing topological information about relevant 3-manifolds and knots. This is a joint work with Fan Ye.
Title: 2-torsion in instanton Floer homology
Speaker: Zhenkun Li
Abstract: Instanton Floer homology, introduced by Floer in the 1980s, has become a power tool in the study of 3-dimensional topology. Its application has led to significant achievements, such as the proof of the Property P conjecture. While instanton Floer homology with complex coefficients is widely studied and conjectured to be isomorphic to the hat version of Heegaard Floer homology, its counterpart with integral coefficients is less understood. In this talk, we will explore the abundance of 2-torsion in instanton Floer homology with integral coefficients and demonstrate how this 2-torsion encodes intriguing topological information about relevant 3-manifolds and knots. This is a joint work with Fan Ye.
Workshop: Mike Miller Eismeier
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: Naturality in HF^oo
Speaker: Mike Miler Eismeier
Abstract: There is an algebraic approximation to the group HF^oo(Y; Z) called the "cup homology" of Y. They are known to have the same dimension over F_2, they can be shown to agree (up to extension problems) for b_1(Y) <= 8, and one gets the same result over Z in millions of computer calculations on examples with 9 <= b_1(Y) <= 14. One is led to guess there should be a good reason for this; perhaps the functors are naturally isomorphic. I will prove they are not by comparing them as MCG(Y)-modules for Y = Sigma_4 x S^1. To study these examples, we develop a version of the Lefschetz decomposition over Z which may be of independent interest.
Title: Naturality in HF^oo
Speaker: Mike Miler Eismeier
Abstract: There is an algebraic approximation to the group HF^oo(Y; Z) called the "cup homology" of Y. They are known to have the same dimension over F_2, they can be shown to agree (up to extension problems) for b_1(Y) <= 8, and one gets the same result over Z in millions of computer calculations on examples with 9 <= b_1(Y) <= 14. One is led to guess there should be a good reason for this; perhaps the functors are naturally isomorphic. I will prove they are not by comparing them as MCG(Y)-modules for Y = Sigma_4 x S^1. To study these examples, we develop a version of the Lefschetz decomposition over Z which may be of independent interest.
Workshop: Deeparaj Bhat
Time: 2:30 PM - 3:30 PM
Location: SCGP 102
Title: Surgery Exact Triangles in Instanton Theory
Speaker: Deeparaj Bhat
Abstract: We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil. In particular, we show the Poincaré homology sphere is not an instanton L-space (with Z/2 coefficients), in contrast with Heegaard Floer and monopole Floer theories. Finally, we sketch the proof of the triangle inspired by the Atiyah-Floer conjecture and results from symplectic geometry.
Title: Surgery Exact Triangles in Instanton Theory
Speaker: Deeparaj Bhat
Abstract: We prove an exact triangle relating the knot instanton homology to the instanton homology of surgeries along the knot. As the knot instanton homology is computable in many instances, this sheds some light on the instanton homology of closed 3-manifolds. We illustrate this with computations in the case of some surgeries on the trefoil. In particular, we show the Poincaré homology sphere is not an instanton L-space (with Z/2 coefficients), in contrast with Heegaard Floer and monopole Floer theories. Finally, we sketch the proof of the triangle inspired by the Atiyah-Floer conjecture and results from symplectic geometry.
Workshop: Anubhav Mukherjee
Time: 4:00 PM - 5:00 PM
Location: SCGP 102
Title: Complete Riemannian 4-manifolds with uniformly positive scalar curvature metric
Speaker: Anubhav Mukherjee
Abstract: In three dimensions, geometry plays a crucial role in classifying the topology of manifolds. Inspired by this, we set out to explore the intricate world of smooth 4-manifolds through the lens of geometry. Specifically, we aim to understand under what conditions a contractible 4-manifold admits a uniform positive scalar curvature metric. In collaboration with Otis Chodosh and Davi Maximo, we demonstrated that in certain cases, the existence of such a metric can provide insight into the topology of 4-manifolds. Moreover, by utilizing Floer theory, we identified obstructions to the existence of such metrics in 4-manifolds.
Title: Complete Riemannian 4-manifolds with uniformly positive scalar curvature metric
Speaker: Anubhav Mukherjee
Abstract: In three dimensions, geometry plays a crucial role in classifying the topology of manifolds. Inspired by this, we set out to explore the intricate world of smooth 4-manifolds through the lens of geometry. Specifically, we aim to understand under what conditions a contractible 4-manifold admits a uniform positive scalar curvature metric. In collaboration with Otis Chodosh and Davi Maximo, we demonstrated that in certain cases, the existence of such a metric can provide insight into the topology of 4-manifolds. Moreover, by utilizing Floer theory, we identified obstructions to the existence of such metrics in 4-manifolds.
Tuesday, April 29th, 2025
Workshop: Lenny Ng
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: Braid varieties from several perspectives
Speaker: Lenny Ng
Abstract: Braid varieties, a family of algebraic varieties associated to any positive braid, have recently emerged in several distinct areas of mathematics. They've appeared in symplectic topology through Floer theory and Legendrian contact homology, but also in algebraic geometry through flags and constructible sheaves, and in algebraic combinatorics through cluster theory. I'll discuss the surprising interrelations between these areas, and especially the way that cluster theory provides some new insight into a well-studied low-dimensional symplectic problem: classifying Lagrangian fillings of Legendrian links.
Title: Braid varieties from several perspectives
Speaker: Lenny Ng
Abstract: Braid varieties, a family of algebraic varieties associated to any positive braid, have recently emerged in several distinct areas of mathematics. They've appeared in symplectic topology through Floer theory and Legendrian contact homology, but also in algebraic geometry through flags and constructible sheaves, and in algebraic combinatorics through cluster theory. I'll discuss the surprising interrelations between these areas, and especially the way that cluster theory provides some new insight into a well-studied low-dimensional symplectic problem: classifying Lagrangian fillings of Legendrian links.
Workshop: Kenji Fukaya
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: Monotone A infinity category and Atiyah-Floer functor
Speaker: Kenji Fukaya
Abstract: In this talk I will explain a work in progress with A. Daemi. We formulate the notion of Monotone A infinity category which for example can be used to study monotone (Immgersed) Lagrangian Floer theory. (It uses different kind of filtration from Floer theory of more general Lagrangian submanifold.) Then I will explain Atiyah-Floer conjecture can be formulated as a functorial equivalence between certain monotone A infinity categories.
Title: Monotone A infinity category and Atiyah-Floer functor
Speaker: Kenji Fukaya
Abstract: In this talk I will explain a work in progress with A. Daemi. We formulate the notion of Monotone A infinity category which for example can be used to study monotone (Immgersed) Lagrangian Floer theory. (It uses different kind of filtration from Floer theory of more general Lagrangian submanifold.) Then I will explain Atiyah-Floer conjecture can be formulated as a functorial equivalence between certain monotone A infinity categories.
Workshop: Chris Scaduto
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: The mapping class group action on the odd character variety is faithful
Speaker: Chris Scaduto
Abstract: The moduli space of holomorphic rank 2 bundles of odd degree and fixed determinant over a given Riemann surface is a symplectic manifold which has an interpretation as a certain PU(2) character variety. There is a homomorphism from a finite extension of the mapping class group of the surface to the symplectic mapping class group of this moduli space. When the genus is 2 or more, we prove that this homomorphism is injective. The proof uses Floer's instanton homology for 3-manifolds. This is joint work with Ali Daemi.
Title: The mapping class group action on the odd character variety is faithful
Speaker: Chris Scaduto
Abstract: The moduli space of holomorphic rank 2 bundles of odd degree and fixed determinant over a given Riemann surface is a symplectic manifold which has an interpretation as a certain PU(2) character variety. There is a homomorphism from a finite extension of the mapping class group of the surface to the symplectic mapping class group of this moduli space. When the genus is 2 or more, we prove that this homomorphism is injective. The proof uses Floer's instanton homology for 3-manifolds. This is joint work with Ali Daemi.
Workshop: Akram Alishahi
Time: 2:30 PM - 3:30 PM
Location: SCGP 102
Title: Compressing surface diffeomorphisms via bordered Floer homology
Speaker: Akram Alishahi
Abstract: Let F be a closed surface, and \psi be a diffeomorphism of F. An interesting question with nice topological implications for detecting homotopy ribbon fibered knots is whether \psi extends over some handlebody with boundary F. In 1985, Casson-Long gave an algorithm for answering this question. In this talk, first we will discuss how bordered Floer homology can be used to detect whether \psi extends over a specific handlebody. Then, we will outline how to adapt ideas of Casson-Long to use bordered Floer homology to detect whether \psi extends over any compression body. This is a joint work with Robert Lipshitz.
Title: Compressing surface diffeomorphisms via bordered Floer homology
Speaker: Akram Alishahi
Abstract: Let F be a closed surface, and \psi be a diffeomorphism of F. An interesting question with nice topological implications for detecting homotopy ribbon fibered knots is whether \psi extends over some handlebody with boundary F. In 1985, Casson-Long gave an algorithm for answering this question. In this talk, first we will discuss how bordered Floer homology can be used to detect whether \psi extends over a specific handlebody. Then, we will outline how to adapt ideas of Casson-Long to use bordered Floer homology to detect whether \psi extends over any compression body. This is a joint work with Robert Lipshitz.
Math Event: Geometry/Topology Seminar: Gioacchino Antonelli - Isoperimetric Sets in Nonnegatively Curved Manifolds
Time: 4:00 PM - 5:15 PM
Location: P-131
Title: Isoperimetric Sets in Nonnegatively Curved Manifolds
Speaker: Gioacchino Antonelli [NYU Courant]
Abstract: Abstract: In this talk, I will explore how the isoperimetric structure of a complete Riemannian manifold is influenced by nonnegative curvature. I will begin by presenting some positive and negative results concerning the existence of isoperimetric sets. Specifically, on a manifold with nonnegative sectional curvature and Euclidean volume growth, isoperimetric sets always exist for large volumes, but may fail to exist for small volumes. Next, I will examine the question of uniqueness for large volumes. I will show that on a manifold with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay, there exists a set G of positive real numbers with density one at infinity such that, for all volumes in G, isoperimetric sets are unique. View Details
Title: Isoperimetric Sets in Nonnegatively Curved Manifolds
Speaker: Gioacchino Antonelli [NYU Courant]
Abstract: Abstract: In this talk, I will explore how the isoperimetric structure of a complete Riemannian manifold is influenced by nonnegative curvature. I will begin by presenting some positive and negative results concerning the existence of isoperimetric sets. Specifically, on a manifold with nonnegative sectional curvature and Euclidean volume growth, isoperimetric sets always exist for large volumes, but may fail to exist for small volumes. Next, I will examine the question of uniqueness for large volumes. I will show that on a manifold with nonnegative Ricci curvature, Euclidean volume growth, and quadratic curvature decay, there exists a set G of positive real numbers with density one at infinity such that, for all volumes in G, isoperimetric sets are unique. View Details
Wednesday, April 30th, 2025
Workshop: Jiakai Li
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: Monopoles and webs
Speaker: Jiakai Li
Abstract: Webs are embedded trivalent graphs in the 3-sphere and foams are singular cobordisms between webs. I will talk about a construction of monopole Floer homology for webs that is functorial under foam cobordisms. Our approach is based on Kronheimer and Mrowka’s monopole Floer homology; the ingredients include both real and orbifold Seiberg-Witten theory. There is an interesting analogy between this monopole web homology and the instanton web homology J^♯, introduced by Kronheimer and Mrowka.
Title: Monopoles and webs
Speaker: Jiakai Li
Abstract: Webs are embedded trivalent graphs in the 3-sphere and foams are singular cobordisms between webs. I will talk about a construction of monopole Floer homology for webs that is functorial under foam cobordisms. Our approach is based on Kronheimer and Mrowka’s monopole Floer homology; the ingredients include both real and orbifold Seiberg-Witten theory. There is an interesting analogy between this monopole web homology and the instanton web homology J^♯, introduced by Kronheimer and Mrowka.
Workshop: Tom Mrowka
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: SU(3) instanton homology for knots and webs
Speaker: Tom Mrowka
Abstract: I describe some work with Peter Kronheimer on a few variants of SU(3)-instanton homology for knots and webs and potential connections to the corresponding SO(3) groups.
Title: SU(3) instanton homology for knots and webs
Speaker: Tom Mrowka
Abstract: I describe some work with Peter Kronheimer on a few variants of SU(3)-instanton homology for knots and webs and potential connections to the corresponding SO(3) groups.
Workshop: Allison Moore
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: Root puzzles and plumbed 3-manifolds
Speaker: Allison Moore
Abstract: Given a plumbing tree and a spin-c structure, I will discuss how to construct a plumbed 3-manifold invariant in the form of a Laurent series twisted by a root lattice. Such a series is invariant under the Neumann moves on plumbing trees and the action of the Weyl group. For irreducible root lattices of rank at least 2, there are infinitely many such series, each depending on a combinatorial puzzle defined on the root lattice. These families of invariants generalize the Z-hat series of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park and Ri. They are motivated by the study of the WRT invariants, and work of Akhmechet-Johnson-Krushkal makes connections of a related series to lattice cohomology. Time permitting, I will also discuss a multivariable generalization of the root lattice-twisted series for knot complements and gluing formulas. This is joint work with N. Tarasca.
Title: Root puzzles and plumbed 3-manifolds
Speaker: Allison Moore
Abstract: Given a plumbing tree and a spin-c structure, I will discuss how to construct a plumbed 3-manifold invariant in the form of a Laurent series twisted by a root lattice. Such a series is invariant under the Neumann moves on plumbing trees and the action of the Weyl group. For irreducible root lattices of rank at least 2, there are infinitely many such series, each depending on a combinatorial puzzle defined on the root lattice. These families of invariants generalize the Z-hat series of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park and Ri. They are motivated by the study of the WRT invariants, and work of Akhmechet-Johnson-Krushkal makes connections of a related series to lattice cohomology. Time permitting, I will also discuss a multivariable generalization of the root lattice-twisted series for knot complements and gluing formulas. This is joint work with N. Tarasca.
Physics Seminar: Umang Mehta
Time: 2:00 PM - 3:00 PM
Location: 313
Title: Anyon scattering
Abstract: I'll present a general formalism for how topological data is encoded in scattering amplitudes of gapped theories with topological ground states. Starting from a natural basis for scattering amplitudes expressed in the language of Unitary Modular Tensor Categories (UMTCs) which describe the line operators in the low energy TQFT, I'll derive modified crossing relations for these amplitudes, as well as a consistency condition for a conjecture for forward scattering. Finally, I'll present an explicit realization of these results in Chern-Simons matter theories where these basis elements can be expressed as conformal blocks in the boundary chiral Wess-Zumino-Witten theory, and comment on potential applications of this formalism to S-matrix bootstrap as well as quantum Hall experiments that aim to measure anyon data.
Title: Anyon scattering
Abstract: I'll present a general formalism for how topological data is encoded in scattering amplitudes of gapped theories with topological ground states. Starting from a natural basis for scattering amplitudes expressed in the language of Unitary Modular Tensor Categories (UMTCs) which describe the line operators in the low energy TQFT, I'll derive modified crossing relations for these amplitudes, as well as a consistency condition for a conjecture for forward scattering. Finally, I'll present an explicit realization of these results in Chern-Simons matter theories where these basis elements can be expressed as conformal blocks in the boundary chiral Wess-Zumino-Witten theory, and comment on potential applications of this formalism to S-matrix bootstrap as well as quantum Hall experiments that aim to measure anyon data.
Math Event: Colloquium: SPRING RECITALS - SPRING RECITALS
Time: 2:15 PM - 3:30 PM
Location:
Title: SPRING RECITALS
Abstract: Instead of the usual colloquium, spring recitals will take place on April 30 and May 1. Todays
Title: SPRING RECITALS
Abstract: Instead of the usual colloquium, spring recitals will take place on April 30 and May 1. Todays
Workshop: Greg Moore
Time: 2:30 PM - 3:30 PM
Location: SCGP 102
Title: Field Theory and Four-Manifolds: Topological Twisting, K-Theoretic Invariants, and Family Invariants
Speaker: Greg Moore
Abstract: We will review some recent progress in the application of quantum field theory to four-manifold invariants. The talk will begin with a review of some aspects of 2411.14396, written with V. Saxena and R. Singh, on topological twisting of arbitrary d=4, N=2 quantum field theories. We then give an update on a project with J. Manschot, H. Kim, R. Tao, and X. Zhang on evaluation of ``K-theoretic Donaldson Invariants'' using 5d supersymmetric Yang-Mills path integrals to reproduce and generalize results of N. Nekrasov and of Gottsche-Nakajima-Yoshioka. Finally, we comment on 2311.08394, written with J. Cushing, M. Rocek, and V. Saxena where coupling d=4 N=2 field theories to background conformal N=2 supergravity gives a generalization of Donaldson invariants of a 4-manifold X to invariants valued in the cohomology of BDiff(X).
Title: Field Theory and Four-Manifolds: Topological Twisting, K-Theoretic Invariants, and Family Invariants
Speaker: Greg Moore
Abstract: We will review some recent progress in the application of quantum field theory to four-manifold invariants. The talk will begin with a review of some aspects of 2411.14396, written with V. Saxena and R. Singh, on topological twisting of arbitrary d=4, N=2 quantum field theories. We then give an update on a project with J. Manschot, H. Kim, R. Tao, and X. Zhang on evaluation of ``K-theoretic Donaldson Invariants'' using 5d supersymmetric Yang-Mills path integrals to reproduce and generalize results of N. Nekrasov and of Gottsche-Nakajima-Yoshioka. Finally, we comment on 2311.08394, written with J. Cushing, M. Rocek, and V. Saxena where coupling d=4 N=2 field theories to background conformal N=2 supergravity gives a generalization of Donaldson invariants of a 4-manifold X to invariants valued in the cohomology of BDiff(X).
Workshop: John Baldwin
Time: 4:00 PM - 5:00 PM
Location: SCGP 102
Title: Instanton Floer homology from Heegaard diagrams
Speaker: John Baldwin
Abstract: Heegaard Floer homology and monopole Floer homology are known to be isomorphic thanks to the monumental work of Taubes et al. But is there a simpler, more axiomatic explanation? And how is instanton Floer homology related to these other theories? I'll talk about work in progress with Zhenkun Li, Steven Sivek, and Fan Ye motivated by these questions. In particular, I'll sketch the construction of a chain complex that computes sutured instanton homology, which is isomorphic as a vector space to the Heegaard Floer chain complex of the sutured manifold. We are currently trying to prove that the differentials on the two sides agree.
Title: Instanton Floer homology from Heegaard diagrams
Speaker: John Baldwin
Abstract: Heegaard Floer homology and monopole Floer homology are known to be isomorphic thanks to the monumental work of Taubes et al. But is there a simpler, more axiomatic explanation? And how is instanton Floer homology related to these other theories? I'll talk about work in progress with Zhenkun Li, Steven Sivek, and Fan Ye motivated by these questions. In particular, I'll sketch the construction of a chain complex that computes sutured instanton homology, which is isomorphic as a vector space to the Heegaard Floer chain complex of the sutured manifold. We are currently trying to prove that the differentials on the two sides agree.
Math Event: Algebraic Geometry Seminar: Sung Gi Park - Hodge symmetries of singular varieties
Time: 4:00 PM - 5:00 PM
Location:
Title: Hodge symmetries of singular varieties
Speaker: Sung Gi Park [Princeton University]
Abstract: The Hodge diamond of a smooth projective complex variety exhibits fundamental symmetries, arising from Poincaré duality and the purity of Hodge structures. In the case of a singular projective variety, the complexity of the singularities is closely related to the symmetries of the analogous Hodge-Du Bois diamond. For example, the failure of the first nontrivial Poincaré duality is reflected in the defect of factoriality. Based on joint work with Mihnea Popa, I will discuss how local and global conditions on singularities influence the topology of algebraic varieties. View Details
Title: Hodge symmetries of singular varieties
Speaker: Sung Gi Park [Princeton University]
Abstract: The Hodge diamond of a smooth projective complex variety exhibits fundamental symmetries, arising from Poincaré duality and the purity of Hodge structures. In the case of a singular projective variety, the complexity of the singularities is closely related to the symmetries of the analogous Hodge-Du Bois diamond. For example, the failure of the first nontrivial Poincaré duality is reflected in the defect of factoriality. Based on joint work with Mihnea Popa, I will discuss how local and global conditions on singularities influence the topology of algebraic varieties. View Details
Thursday, May 1st, 2025
Workshop: Abhishek Mallick
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: Families-version of Donaldson's diagonalization theorem and its applications
Speaker: Abhishek Mallick
Abstract: In this talk, I will discuss various families-version of Donaldson's diagonalization theorem and explore their applications to studying exotic 4-manifolds, surfaces, and diffeomorphisms. Some of these applications answer open questions in the aforementioned fields. This is joint work with Hokuto Konno and Masaki Taniguchi.
Title: Families-version of Donaldson's diagonalization theorem and its applications
Speaker: Abhishek Mallick
Abstract: In this talk, I will discuss various families-version of Donaldson's diagonalization theorem and explore their applications to studying exotic 4-manifolds, surfaces, and diffeomorphisms. Some of these applications answer open questions in the aforementioned fields. This is joint work with Hokuto Konno and Masaki Taniguchi.
Workshop: Hokuto Konno
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: Symplectic structures and diffeomorphisms of 4-manifolds
Speaker: Hokuto Konno
Abstract: Using Kronheimer's computation of his invariant for families of symplectic forms, we construct new families of 4-manifolds over high-dimensional spheres with nonzero families Seiberg-Witten invariants. From this construction, we deduce some infinite generation result for the homotopy groups of diffeomorphism groups of 4-manifolds. This is joint work with Jun Li and Weiwei Wu.
Title: Symplectic structures and diffeomorphisms of 4-manifolds
Speaker: Hokuto Konno
Abstract: Using Kronheimer's computation of his invariant for families of symplectic forms, we construct new families of 4-manifolds over high-dimensional spheres with nonzero families Seiberg-Witten invariants. From this construction, we deduce some infinite generation result for the homotopy groups of diffeomorphism groups of 4-manifolds. This is joint work with Jun Li and Weiwei Wu.
Workshop: Haochen Qiu
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: The Dehn twist on a connected sum of two homology tori
Speaker: Haochen Qiu
Abstract: Kronheimer and Mrowka showed that the Dehn twist along a 3-sphere in the neck of the connected sum of two K3 surfaces is not smoothly isotopic to the identity. The tool they used is the nonequivariant family Bauer-Furuta invariant, and their result requires that the manifolds are simply connected and the signature of one of them is 16 mod 32. We generalize the S^1-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that if X_1, X_2 are two homology tori whose determinants r_1, r_2 are odd, then the Dehn twist along a 3-sphere in the neck of X_1 # X_2 is not smoothly isotopic to the identity.
Title: The Dehn twist on a connected sum of two homology tori
Speaker: Haochen Qiu
Abstract: Kronheimer and Mrowka showed that the Dehn twist along a 3-sphere in the neck of the connected sum of two K3 surfaces is not smoothly isotopic to the identity. The tool they used is the nonequivariant family Bauer-Furuta invariant, and their result requires that the manifolds are simply connected and the signature of one of them is 16 mod 32. We generalize the S^1-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that if X_1, X_2 are two homology tori whose determinants r_1, r_2 are odd, then the Dehn twist along a 3-sphere in the neck of X_1 # X_2 is not smoothly isotopic to the identity.
Math Event: Colloquium: SPRING RECITALS - SPRING RECITALS
Time: 2:15 PM - 3:30 PM
Location:
Title: SPRING RECITALS
Abstract: Instead of the usual colloquium, spring recitals will take place on April 30 and May 1. Todays
Title: SPRING RECITALS
Abstract: Instead of the usual colloquium, spring recitals will take place on April 30 and May 1. Todays
Workshop: Roberto Ladu
Time: 2:30 PM - 3:30 PM
Location: SCGP 102
Title: On the complexity of non-inertial h-cobordisms
Speaker: Roberto Ladu
Abstract: Morgan-Szabó in '99 defined a notion of complexity for h-cobordisms between 1-connected 4-manifolds and constructed a family of $h$-cobordisms of diverging complexity.Their examples are all inertial, i.e. the cobordisms have the same ends.I will show how to construct h-cobordisms of diverging complexity between exotic 4-manifolds. Furthermore, I will refine the result of Morgan-Szabó obtaining a lower-bound on the number of blow-ups needed to find h-cobordisms of non-minimal complexity.This talk is based on https://arxiv.org/abs/2501.08750
Title: On the complexity of non-inertial h-cobordisms
Speaker: Roberto Ladu
Abstract: Morgan-Szabó in '99 defined a notion of complexity for h-cobordisms between 1-connected 4-manifolds and constructed a family of $h$-cobordisms of diverging complexity.Their examples are all inertial, i.e. the cobordisms have the same ends.I will show how to construct h-cobordisms of diverging complexity between exotic 4-manifolds. Furthermore, I will refine the result of Morgan-Szabó obtaining a lower-bound on the number of blow-ups needed to find h-cobordisms of non-minimal complexity.This talk is based on https://arxiv.org/abs/2501.08750
Workshop: Steven Sivek
Time: 4:00 PM - 5:00 PM
Location: SCGP 102
Title: L-spaces and knot traces
Speaker: Steven Sivek
Abstract: Given a knot K in S^3, Gabai proved that the zero-surgery S^3_0(K) determines the Seifert genus of K and whether or not K is fibered. One might hope that in many cases the zero-surgery would uniquely determine the knot K, but this is not known to be true for even a single knot of genus greater than 1. I’ll talk about an attempt to address this for higher-genus knots, using a relationship between the knot Floer homology of a fibered knot and the dynamics of its monodromy, in which we understand just enough about zero-surgeries on L-space knots to show that they can all be detected by their zero-traces. This is joint work with John Baldwin.
Title: L-spaces and knot traces
Speaker: Steven Sivek
Abstract: Given a knot K in S^3, Gabai proved that the zero-surgery S^3_0(K) determines the Seifert genus of K and whether or not K is fibered. One might hope that in many cases the zero-surgery would uniquely determine the knot K, but this is not known to be true for even a single knot of genus greater than 1. I’ll talk about an attempt to address this for higher-genus knots, using a relationship between the knot Floer homology of a fibered knot and the dynamics of its monodromy, in which we understand just enough about zero-surgeries on L-space knots to show that they can all be detected by their zero-traces. This is joint work with John Baldwin.
Friday, May 2nd, 2025
Workshop: Tye Lidman
Time: 9:30 AM - 10:30 AM
Location: SCGP 102
Title: Instanton Floer homology and cosmetic surgeries
Speaker: Tye Lidman
Abstract: The Cosmetic Surgery Conjecture predicts two different Dehn surgeries on the same knot always gives different three-manifolds. We discuss how the Chern-Simons filtration on instanton Floer homology can help approach this problem. This allows us to reduce the conjecture to essentially one case. This is joint work with Ali Daemi and Mike Miller Eismeier.
Title: Instanton Floer homology and cosmetic surgeries
Speaker: Tye Lidman
Abstract: The Cosmetic Surgery Conjecture predicts two different Dehn surgeries on the same knot always gives different three-manifolds. We discuss how the Chern-Simons filtration on instanton Floer homology can help approach this problem. This allows us to reduce the conjecture to essentially one case. This is joint work with Ali Daemi and Mike Miller Eismeier.
Workshop: Kristen Hendricks
Time: 11:00 AM - 12:00 PM
Location: SCGP 102
Title: Equivariant surgery formulas in Heegaard Floer theory
Speaker: Kristen Hendricks
Abstract: A core computational feature of Ozsvath and Szabo's Heegaard Floer theory is its possession of a surgery formula; that is, given a knot K in S^3, the Heegaard Floer three-manifold invariants of Dehn surgery on K can be computed from the knot Floer homology of K, and similarly (albeit more complicatedly) for surgeries on links. In this talk we prove an equivariant version of the Heegaard Floer knot surgery formula for symmetric knots in S^3; that is, given a symmetry on a knot, we compute the induced action on the surgery complex. The proof goes by way of showing naturality of certain bimodules constructed by I. Zemke to encode the data of the link surgery formula. As an application, we show that the kernel of the forgetful map from the equivariant homology cobordism group to the homology cobordism group contains a Z^infty summand. This is joint work with A. Mallick, M. Stoffregen, and I. Zemke.
Title: Equivariant surgery formulas in Heegaard Floer theory
Speaker: Kristen Hendricks
Abstract: A core computational feature of Ozsvath and Szabo's Heegaard Floer theory is its possession of a surgery formula; that is, given a knot K in S^3, the Heegaard Floer three-manifold invariants of Dehn surgery on K can be computed from the knot Floer homology of K, and similarly (albeit more complicatedly) for surgeries on links. In this talk we prove an equivariant version of the Heegaard Floer knot surgery formula for symmetric knots in S^3; that is, given a symmetry on a knot, we compute the induced action on the surgery complex. The proof goes by way of showing naturality of certain bimodules constructed by I. Zemke to encode the data of the link surgery formula. As an application, we show that the kernel of the forgetful map from the equivariant homology cobordism group to the homology cobordism group contains a Z^infty summand. This is joint work with A. Mallick, M. Stoffregen, and I. Zemke.
Workshop: Juan Muñoz-Echaniz
Time: 1:00 PM - 2:00 PM
Location: SCGP 102
Title: Counting SL(2,C) connections on Seifert-fibered spaces
Speaker: Juan Muñoz-Echaniz
Abstract: In the recent years, there has been an surge in interest for developing new invariants of 3- and 4-manifolds based on SL(2,C) generalizations of the anti-self-duality equation: in dimension 3, Abouzaid and Manolescu have introduced a sheaf-theoretic SL(2,C) Floer homology; in dimension 4, Tanaka and Thomas have defined Vafa--Witten invariants for complex projective surfaces. These are defined using algebraic techniques (derived critical loci, perverse sheaves, virtual localisation), and it remains unknown how to interpret these invariants from the viewpoint of differential geometry. I will describe an approach to counting flat SL(2,C) connections on Seifert-fibered 3-manifolds using techniques from gauge theory. Namely, I will describe a perturbation of the SL(2,C) Chern--Simons functional which has the effect of `localising' its critical points around a compact set. As an application, we obtain new formulae for the Euler characteristic and Poincaré polynomial of the irreducible locus in the SL(2,C) character variety of a Seifert-fibered homology 3-sphere: in particular, we show that the Euler characteristic equals the Milnor number (divided by four) of any weighted-homogeneous isolated complete intersection singularity whose link is the given 3-manifold--which suggests an SL(2,C) analogue of the Casson Invariant Conjecture of Neumann and Wahl.
Title: Counting SL(2,C) connections on Seifert-fibered spaces
Speaker: Juan Muñoz-Echaniz
Abstract: In the recent years, there has been an surge in interest for developing new invariants of 3- and 4-manifolds based on SL(2,C) generalizations of the anti-self-duality equation: in dimension 3, Abouzaid and Manolescu have introduced a sheaf-theoretic SL(2,C) Floer homology; in dimension 4, Tanaka and Thomas have defined Vafa--Witten invariants for complex projective surfaces. These are defined using algebraic techniques (derived critical loci, perverse sheaves, virtual localisation), and it remains unknown how to interpret these invariants from the viewpoint of differential geometry. I will describe an approach to counting flat SL(2,C) connections on Seifert-fibered 3-manifolds using techniques from gauge theory. Namely, I will describe a perturbation of the SL(2,C) Chern--Simons functional which has the effect of `localising' its critical points around a compact set. As an application, we obtain new formulae for the Euler characteristic and Poincaré polynomial of the irreducible locus in the SL(2,C) character variety of a Seifert-fibered homology 3-sphere: in particular, we show that the Euler characteristic equals the Milnor number (divided by four) of any weighted-homogeneous isolated complete intersection singularity whose link is the given 3-manifold--which suggests an SL(2,C) analogue of the Casson Invariant Conjecture of Neumann and Wahl.