Here are our upcoming seminars: 

Past talks:

1. August 26: Tye Lidman (North Carolina State University)

Title: Cosmetic surgeries and Chern-Simons invariants

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 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.

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

2. September 4: Zhenhua Liu (Princeton)

Title: General behavior of area-minimizing subvarieties

Abstract: We will review some recent progress on the general geometric behavior of homologically area-minimizing subvarieties, namely, objects that minimize area with respect to homologous competitors. They are prevalent in geometry, for instance, as holomorphic subvarieties of a Kahler manifold, or as special Lagrangians on a Calabi-Yau, etc. A fine understanding of the geometric structure of homological area-minimizers can give far-reaching consequences for related problems.

Location: Zoom, link TBA

Time: 11am to 12 pm

3. September 11: Hunter Stufflebeam (Penn)

Title: Stability and Scalar Curvature Lower Bounds

Abstract: In recent years much attention has turned to rigidity, and almost-rigidity, problems involving lower scalar curvature bounds. In this talk, I'll discuss some contributions to this area, including some new stability theorems for spheres. Some of this is joint work with Davi Maximo, and some is joint with Paul Sweeney Jr.

Location: NCSU, SAS 4201

Time: 11am to 12pm

4. September 13: Isabella Khan (Princeton)

Title: Koszul duality for partial Heegaard diagrams

Abstract: By slicing a Heegaard diagram for a knot K in S3, it is possible to retrieve the knot Floer homology of K as a tensor product of bimodules over an \A∞ algebra corresponding to the slice. The first step in this process is to assign an ∞ algebra to this slice, which can also be written as a planar graph. In this talk, we construct a pair of Koszul dual ∞ algebras corresponding to planar graphs arising as slices of these Heegaard diagrams, and discuss how to verify the Koszul duality relation.

Location: NCSU, SAS 4201

Time: 3:00 pm to 4:00 pm

5. September 13: (University of Texas)

Title: New constructions and invariants of exotic 4-manifolds

Abstract: Dimension four is the lowest dimension where smooth and topological manifolds can differ; any difference between these categories is known as exotica. In particular, a smooth 4-manifold is exotic if there is another smooth 4-manifold which is homeomorphic but not diffeomorphic to it. There is a wealth of literature, mostly written between 1983 and 2008, on producing exotic manifolds, but the techniques pioneered in this era can be hard to use in practice. I will discuss joint work with triangle locals Adam Levine and Tye Lidman in which we give some new techniques for producing exotic 4-manifolds. 

Location: NCSU, SAS 4201

Time: 4:15 pm to 5:15 pm

6. September 16: Sergey Cherkis (University of Arizona)

Title: Gravitational Instantons: the Tesseron Landscape

Abstract: Since their introduction in Euclidean quantum gravity in mid-70’s, hyperkaehler Gravitational Instantons (aka tesserons) found their use in string theory and in supersymmetric quantum field theory.  Their classification was recently completed and now their parameter space is being explored.  We propose a systematic program of realizing each of these spaces as a moduli space of monopoles: the monopolization program.    

Monopolization reveals the combinatorial and geometric structure of the parameter space of all these spaces, equips each space with various natural structures (tautological bundles, Dirac-type operators, etc), and connects different types of integrable systems associated to these gravitational instantons.  

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

7. September 23: Calvin McPhail-Snyder (Duke University)

Title: Towards quantum complex Chern-Simons theory

Abstract: I will discuss recent joint work (with N. Reshetikhin) defining invariants 𝒥 of knot (and link and tangle) exteriors with flat 𝔰𝔩₂ connections. The construction is via a geometric version of the Reshetikhin-Turaev construction: it is algebraic and relies on the representation theory of quantum groups. In this talk I will instead focus on the properties of these invariants and explain why I think they are a good candidate for quantum Chern-Simons theory with noncompact gauge group SL₂(ℂ). I will also discuss a connection with (and a generalization of) the Volume Conjecture. 

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

8. September 25: Hannah Kim (UNC)

Title: Upper bounds of second eigenvalues on the sphere and the real projective space

Abstract: In this talk, we discuss sharp upper bounds for the second nonzero eigenvalue of the Laplacian on higher dimensional spheres and real projective spaces. These isoperimetric inequalities are applications of classical isoperimetric problem and extending this problem to optimizing a certain physical quantity, which is the eigenvalue. Our method consists of constructing trial functions based on recent developments in the hyperbolic center of mass and on topological degree theory used to verify that the trial functions are valid. We lastly talk about one-step method of proving that the trial functions satisfy orthogonality conditions.  

Location: UNC-Chapel Hill, Phillips 385

Time: 3.30 to 4.30 pm

9. September 30: Zhenyi Chen (Northwestern University)

Title: A-infinity Sabloff Duality via the LSFT Algebra

Abstract: The Chekanov-Eliashberg dga is a powerful invariant for Legendrian links. Using augmentations of this dga, one can truncate its differential to produce linearized contact homology. About two decades ago, Sabloff established a duality in this setting, closely linked to the Poincaré duality of Lagrangian fillings. This truncation has since been generalized into a unital A-infinity category, Aug_+. In this talk, I will present new results that extend Sabloff duality from the level of cochain complexes to A-infinity bimodules over Aug_+. The key tool in this extension is Ng's LSFT algebra, which enlarges the Chekanov-Eliashberg dga. If time permits, I will also discuss how the LSFT algebra encodes additional homotopy coherent data, providing further insights into Sabloff duality. 

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

10. October 7: Luya Wang (IAS)

Title: Deformation inequivalent symplectic structures and Donaldson's four-six question

Abstract: Studying symplectic structures up to deformation equivalences is a fundamental question in symplectic geometry. Donaldson asked: given two homeomorphic closed symplectic four-manifolds, are they diffeomorphic if and only if their stabilized symplectic six-manifolds, obtained by taking products with CP^1 with the standard symplectic form, are deformation equivalent? I will discuss joint work with Amanda Hirschi on showing how deformation inequivalent symplectic forms remain deformation inequivalent when stabilized, under certain algebraic conditions. This gives the first counterexamples to one direction of Donaldson’s “four-six” question and the related Stabilizing Conjecture by Ruan. In the other direction, I will also discuss more supporting evidence via Gromov-Witten invariants.

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

11. October 11: Mohammed Abouzaid (Stanford)

Title: Bordism of derived orbifolds 

Abstract: Among the first significant results of algebraic topology is the computation, by Thom, Milnor, Novikov, and Wall among others, of the bordism groups of stably complex and oriented manifolds. After reviewing these results, I will discuss the notion of derived orbifolds, and briefly indicate how the bordism groups of these objects appear as universal recipients of invariants arising in Gromov-Witten theory and symplectic topology. Finally, I will state what is known about them, as well as some conjectures about the structure of these groups. 

Location: Duke, Physics 119

Time: 2.30 to 3.30 pm

12. October 21: Da Rong (Daren) Cheng  (University of Miami)

Title: Existence of capillary constant mean curvature disks in R^3

Abstract: Given a surface S in R^3 diffeomorphic to the sphere, a classical question is to prove the existence of disk-type surfaces with constant mean curvature whose boundary meets S at a constant angle, the value of the mean curvature and that of the contact angle both being prescribed. Such boundary value problems originated from the study of capillary phenomenon and have a simple variational description in terms of areas and volumes. In this talk, I will report on recent progress on the existence question where we refine Struwe's foundational work (1988) on this subject, and partially generalize it to contact angles other than \pi/2. 

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

13. October 23: Stefan Steinerberger (UW)

Title: Conformally Rigid Graphs

Abstract: Given a graph G=(V,E), the smallest (nonzero) eigenvalue of its Laplacian matrix is a fantastic measure for how connected the graph is (also known as the `algebraic connectivity’); for example: adding edges increases it. What if we instead shift the weights of the edges around, can we make that eigenvalue larger? (A toy example would be: given a computer network, can we move some of the cables around to get a higher efficiency network without having to buy any new cables?). It depends on the graph but for some graphs the answer is No. These graphs are already maximizing their algebraic connectivity, shifting weights around won’t help. These are quite interesting graphs which we call “conformally rigid” (and the name will be explained). The family of conformally rigid graphs is combinatorially surprisingly rich, we don’t understand it fully, and we will survey what we know (and don’t know!). Joint work with Rekha Thomas.  

Location: UNC-Chapel Hill, Phillips 385

Time: 3.30 to 4.30 pm

14. October 28: Saman Habibi Esfahani (Duke)

Title: Non-linear Dirac operators and multi-valued harmonic forms 

Abstract: This talk is based on joint work with Yang Li. I will discuss non-linear Dirac operators and related regularity questions, which arise in various problems in gauge theory, Floer theory, DT theory, and minimal submanifolds. These operators are used to define generalized Seiberg-Witten equations on 3- and 4-manifolds. Taubes proposed that counting harmonic spinors with respect to these operators on 3-manifolds could lead to new 3-manifold invariants, while Donaldson and Segal suggested counting spinors over special Lagrangians to define Calabi-Yau invariants. Meanwhile, Doan and Rezchikov outlined a Fukaya 2-category for hyperkähler manifolds based on such counts. The central question is whether the space of such harmonic spinors is compact. We address this question in certain cases, proving and disproving several conjectures and questions in the field, particularly one raised by Taubes in 1999. The key observation is that multivalued harmonic forms, in the sense of Almgren and De Lellis-Spadaro's Q-valued functions, play a crucial role in the problem. 

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

15. November 4: Boyu Zhang (The University of Maryland)

Title: Mapping class groups of 4-manifolds with 1-handles  

Abstract: Budney-Gabai proved in 2019 that the mapping class group of S^1xD^3 relative to the boundary is an abelian group with infinite rank.  In this talk, we will discuss a generalization of Budney-Gabai's result to 4-manifolds with 1-handles.  We will show that if a manifold M is the boundary connected sum of a compact manifold N with S^1xD^3, such that either N has a trivial pi_2 or N is of the form IxY, then the mapping class group of M has a center with infinite rank.  This talk is based on joint work with Jianfeng Lin and Yi Xie.  

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

16. November 11: Joshua Greene (Boston College)

Title: Symplectic geometry and inscription problems 

Abstract: The Square Peg Problem was posed by Otto Toeplitz in 1911. It asks whether every Jordan curve in the plane contains the vertices of a square, and it is still open to this day. I will survey the approaches to this problem and its relatives using symplectic geometry. This talk is based on joint work with Andrew Lobb. 

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

17. December 2: Jiajun Yan (Rice University)

Title: A Gauge-Theoretic Construction of 4-Dimensional Hyperkähler ALE Spaces and the McKay Correspondence

Abstract: Non-compact hyperkähler spaces arise frequently in gauge theory. The 4-dimensional hyperkähler ALE spaces are a special class of non-compact hyperkähler spaces. They are in one-to-one correspondence with the finite subgroups of SU(2) and have interesting connections with representation theory and singularity theory, captured by the McKay Correspondence. In this talk, we give a gauge-theoretic construction of these spaces, inspired by Kronheimer’s original construction via a finite-dimensional hyperkähler reduction. In the gauge-theoretic construction, we realize each ALE space as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a hyperkähler gauge group action. Time permitting, we will discuss an application of the gauge-theoretic construction to give a new proof of the McKay correspondence. Let Gamma be a finite subgroup of SU(2). The McKay correspondence states that the McKay quiver of Gamma is isomorphic to the graph of the minimal resolution of C^2/Gamma. The main approach here is Morse-theoretic, inspired by Atiyah-Bott-Kirwan theory and Chern-Simons theory, as we aim to identify the critical sets of a Morse function coming from the norm square of some moment map with certain flat connections that induce Gamma-representations as in the McKay correspondence.

Location: Duke, Physics 119

Time: 3.15 pm to 4.15 pm

18. Monday, January 13, 2025, 3:15pm, Location: Duke, Physics 119
    Valentino Tosatti (NYU Courant), Immortal solutions of the Kähler-Ricci flow 

Abstract: I will discuss the problem of understanding the long-time behavior of Ricci flow on a compact Kähler manifold, assuming that a solution exists for all positive time. Inspired by an analogy with the minimal model program in algebraic geometry, Song and Tian posed several conjectures which describe this behavior. I will report on recent work (joint with Hein and Lee) which confirms these conjectures.

19. Monday, February 3, 2025, 3:15pm, Location: Duke, Physics 119
    Luca Di Cerbo (University of Florida), 

Title: Curvature, Macroscopic Dimensions, and Symmetric Products of Surfaces

Abstract: In this talk, we present a study of the curvature properties of symmetric products of surfaces. Surprisingly, these spaces turn out to be important for questions arising in the study of the macroscopic dimension of Riemannian manifolds with positive scalar curvature. Among other things, we provide supporting evidence for the well-known Gromov-Lawson and Gromov conjectures. Based on joint work with A. Dranishnikov and E. Jahuari. 

20. Monday, February 17, 2025, 3:15pm, Location: Duke, Physics 119
    Aliakbar Daemi (Washington University in St. Louis), 

Title: Higher rank Yang--Mills gauge theory and knots 

Abstract: Yang--Mills gauge theory with gauge group SU(2) has played a significant role in the study of the topology of 3- and 4-manifolds. It is natural to ask whether we obtain more topological information by working with other choices of gauge groups such as SU(n) for higher values of n. In this talk, I will discuss some conjectures and results related to SU(n) Yang--Mills gauge theory and its applications in low dimensional topology. The main part of the talk is based on a joint work with Nobuo Iida and Chris Scaduto.

21. Monday, March 17, 2025, 3:15pm, Physics 119, Geometry/topology Seminar
    Refined counts of holomorphic curves and applications
    Shaoyun Bai (MIT)
    
    Much of the development of symplectic topology is based on studying moduli spaces of pseudo-holomorphic maps. The general scheme is to extract enumeration invariants from these moduli spaces to reveal geometric information. In this talk, I will discuss a recipe for refining the usual counting, which produces integers and bordism classes rather than rational numbers. Applications to symplectic topology and Hamiltonian dynamics will also be covered.

22. Wednesday, March 19, 2025, 2:00pm, Physics 119, Geometry/topology Seminar
    Interior Hessian Estimates for Singularities of the Lagrangian Mean Curvature Flow
    Jeremy Wall (UNC)

In this talk, we discuss the history of the Lagrangian Mean Curvature equation beginning with the special Lagrangian equation of Harvey and Lawson. We consider the hypercritical case which results in a convex potential function for our Lagrangian submanifold. We will also discuss how this extends to the supercitical case where the potential function is no longer convex. From there, we will talk about the Lagrangian Mean Curvature flow and some of its singularities, namely, shrinker, expander, translator, and rotator solutions. We then extend our results to a broader class of Lagrangian Mean Curvature equations. This talk is based on joint work with Arunima Bhattacharya.

23. Monday, April 14, 2025, 3:15pm, Location: Duke, Physics 119
    Yaiza Canzani (UNC): Eigenfunction concentration and Weyl Laws via geodesic beams

Abstract: A broad spectrum of physical phenomena, including the localization of quantum particles, is governed by the behavior of Laplace eigenvalues and eigenfunctions. This behavior is intrinsically connected to that of the geodesic flow, reflecting the deep correspondence between quantum and classical dynamics. To exploit this connection, in collaboration with J. Galkowski, we have developed a framework that hinges on decomposing eigenfunctions into a superposition of geodesic beams. In this talk, I will introduce these techniques and explain how to use them to derive refined bounds on the standard estimates for the eigenfunction’s pointwise behavior and the Weyl Law for the eigenvalue counting function. A significant consequence of this method is that a quantitatively improved Weyl Law holds on most manifolds.