IMSA Topology Seminar
Hans Boden, McMaster University
Virtual knots and (algebraic) concordance
Wednesday, April 26, 2023, 10:30am
Ungar, Room 528B
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Abstract: The motivation for this talk is a desire to better understand the algebraic structure of the concordance group of virtual knots. This group is known to contain, as a proper subgroup, the concordance group of classical knots. It is also known, by results of Chrisman, to be nonabelian. Apart from that, it is quite mysterious. We will explain how to use the Gordon-Litherland pairing to, given a virtual knot with a spanning surface, associate a square integral matrix. For classical knots, the matrix is always symmetric but for virtual knots, that is no longer true. Using these so-called mock Seifert matrices, we introduce a new set of knot invariants (signatures, LT signatures, and Alexander polynomials) and explore their behavior under virtual concordance. Using the mock Seifert matrices, we introduce a new algebraic concordance group defined in terms of non-orientable spanning surfaces. Morally speaking, this group is a linear approximation to the mystery group, with linearization given by a virtual analogue of the Levine homomorphism. The group can be seen to be abelian and infinite rank. It also contains lots of 2- and 4-torsion. We will present some results and a few open problems. Everything is joint with Homayun Karimi.
IMSA Topology Seminar
Isacco Nonino, University of Glasgow
Tight contact structures on hyperbolic 3-manifolds constructed using Dehn Surgery on GOF-knots in lens spaces, Part 2
Wednesday, April 12, 2023, 10:30am
Ungar, Room 528B
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Abstract: Continuing last week, I will discuss how to obtain the desired upper bound on the number of tight contact structures on certain hyperbolic 3-manifolds constructed by Dehn surgery on some GOF-knots in lens spaces.
IMSA Topology Seminar
Isacco Nonino, University of Glasgow
Tight contact structures on hyperbolic 3-manifolds constructed using Dehn Surgery on GOF-knots in lens spaces
Wednesday, April 5, 2023, 10:30am
Ungar, Room 528B
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Abstract: The aim of my current research project is to classify and better understand tight contact structures on a large class of hyperbolic 3-manifolds. More precisely, these manifolds are constructed using Dehn surgery on genus one fibered knots in lens spaces. In this talk, I will introduce a first classification result, which determines the number of (non-isotopic) tight contact structures on a subclass of these manifolds. Moreover, I will also show how to immediately derive Stein fillability for some of these structures. The strategy of the proof will be outlined, together with some key facts that are helpful in this specific setting and some hopes for future developments and generalizations.
FIU & IMSA Joint Seminar
Anna Fino, Florida international University
Nilmanifolds and their cohomologies
Tuesday, April 4, 2023, 6:00pm
Ungar, Room 528B
Abstract: Nilmanifolds constitute a well-known class of compact manifolds providing interesting explicit examples of geometric structures with special properties. A nilmanifold is a compact quotient of a connected and simply connected nilpotent Lie group G by a lattice of maximal rank in G. Hence, any left-invariant geometric structure on G descends to the nilmanifold. We will refer to such structures as invariant. In the talk we will review cohomologies of nilmanifolds endowed with an invariant (generalized) complex structure.
IMSA Seminar
Charles Doran, University of Alberta
Motivic Geometry of Two-Loop Feynman Integrals
Tuesday, April 4, 2023, 5:00pm
Ungar, Room 528B
Abstract: We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into a mixed Tate piece and a variation of Hodge structure from families of hyperelliptic curves, elliptic curves, or rational curves depending on the space-time dimension. We give more precise results for two-loop graphs with a small number of edges. In particular, we recover a result of Spencer Bloch that in the well-known double box example there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the “non-planar” two-loop tardigrade graph is that of a family of K3 surfaces of generic Picard number 11. Lastly, we show that generic members of the multi-scoop ice cream cone family of graph hypersurfaces correspond to pairs of multi-loop sunset Calabi-Yau varieties. Our geometric realization of these motives permits us in many cases to derive in full the homogeneous differential operators for the corresponding Feynman integrals. This is joint work with Andrew Harder and Pierre Vanhove.
IMSA Topology Seminar
Lev Tovstopyat-Nelip, UGA
Quasipositive surfaces and decomposable Lagrangians
Wednesday, March 29, 2023, 10:30am
Ungar, Room 528B
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Abstract: I'll explain how an invariant of Legendrian links in knot Floer homology can be used to obstruct the existence of decomposable Lagrangian link cobordisms in a very general setting. The argument involves braiding the ends of the link cobordism about open books and appealing to an algebraic property of the Legendrian invariant called comultiplication. Much of the talk will be spent describing the topological and contact geometric ingredients.
IMSA Topology Seminar
Sinem Onaran
Contact surgeries on Contact 3-manifolds
Wednesday, March 8, 2023, 10:30am
Ungar, Room 528B
Abstract: In this talk, I will discuss contact surgeries. I will discuss the behavior of contact structures under contact (+1/n) and contact (+n)-surgeries along Legendrian knots. Then, I will focus on a question: which tight contact structures on a given lens space can be obtained by a single contact (-1)-surgery along a Legendrian knot in S^3 with some contact structure? (joint with Geiges) Further, I will discuss various versions of contact surgery numbers, the minimal number of components of a surgery link L describing a contact 3-manifold under consideration. (joint with Etnyre and Kegel)
IMSA Topology Seminar
Feride Ceren Kose, University of Georgia
Symmetric Unions and Ribbon Knots
Wednesday, March 1, 2023, 10:30am
Ungar, Room 528B
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Abstract: A symmetric union of a knot is a classical construction in knot theory introduced in the 1950s by Kinoshita and Terasaka. Because of the flexibility in their construction and the fact that they are ribbon, hence smoothly slice, symmetric unions appear quite frequently in the literature. It is still unknown whether there exists a ribbon knot which cannot be presented as a symmetric union. Thus, similar to the Slice-Ribbon conjecture, one may ask whether every ribbon knot is a symmetric union. In this talk, I will classify the simplest type of symmetric unions that are composite, two-bridge, Montesinos, or amphichiral, and in doing so, give infinite families of ribbon knots that cannot admit the simplest type of symmetric union diagrams.
IMSA Seminar
Gueo Grantcharov, Florida International University
Generalized Calabi-Yau Problem
Thursday, February 23, 2023, 6:00pm
Ungar, Room 528B
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Abstract: This is a continuation of the previous talk on generalized complex geometry. We'll review the definitions of holomorphically trivial generalized canonical bundle, Calabi-Yau structure and Calabi-Yau problem. Then we'll report on its current status.
IMSA Seminar
Christopher Scaduto, University of Miami
Skein Exact Triangles in Equivariant Singular Instanton Theory
Wednesday, February 22, 2023, 10:30am
Ungar, Room 528B
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Abstract: Given a knot or link in the 3-sphere, its Murasugi signature is an integer-valued invariant which can easily be computed from a diagram. Work of Herald and Lin gives an alternative description of knot signatures, as signed counts of SU(2)-representations of the knot group which are traceless around meridians. There is a version of singular instanton homology for links which categorifies the Murasugi signature. We construct unoriented skein exact triangles for these Floer groups, categorifying the behavior of the Murasugi signature under unoriented skein relations. More generally, we construct skein exact triangles in the setting of equivariant singular instanton theory. This is joint work with Ali Daemi.
IMSA Seminar
Ya Deng, Université de Lorraine
A More Comprehensible Proof of the Reductive Shafarevich Conjecture
Thursday, February 16, 2023, 5:00pm & Friday, February 17, 2023, 5:00pm
Ungar, Room 528B
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Abstract: The Shafarevich conjecture is one of the most beautiful mathematical problems in complex geometry. It connects many different subjects, especially non-abelian Hodge theories. It was first proved by Katzarkov-Ramachandran in 1998 for surfaces with reductive linear fundamental groups. In 2004 Eyssidieux proved the reductive Shafarevich conjecture for projective varieties, and this result was an important building block in the later proof of the linear case by Eyssidieux-Katzarkov-Pantev-Ramachandran. In the two lectures I will explain the recent work on the new construction of Shafarevich morphisms, and the more comprehensible proof of the reductive Shafarevich conjecture. This is a joint work with Katsutoshi Yamanoi.
IMSA Seminar
Gueo Grantcharov, Florida International University
Generalized Kaehler and Bihermitian Structures
Thursday, February 16, 2023, 6:00pm
Ungar, Room 528B
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Abstract: In the talk we'll review some of the properties of generalized Kaehler structures which are similar to the Kaehloer case. It includes analog of Kodaira's result on stability under small deformations, Calabi-Yau problem and period map.
IMSA Seminar
Vladmir Baranovsky, UC Irvine
Mapping Spaces and E_2 Algebras.
Thursday, February 16, 2023, 4:00pm
Ungar, Room 528B
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Abstract: In a 1991 paper, Bendersky and Gitler have constructed a spectral sequence converging to the cohomology of a mapping space Maps(K, Y) where K is a simplicial set and Y is a space, such as a smooth compact manifold. The E_1 term of that spectral sequence involves chains on configuration spaces of K and cochains on Cartesian powers of Y. We assume that K is a graph (or rather its ribbon thickening) and explain a conjecture that expresses the differential of the E_1 term via standard “surjection operations” on the cochains of Y. One application is a theorem asserting that the cohomology of Maps(K, Y) may be approximated by the cohomology of Cartesian powers of Y with certain double diagonals removed. When cochains of Y are replaced by an E_2 algebra or a category with appropriate structure, one expects similar results involving factorization homology of the 2-dimensional ribbon thickening of K.
IMSA Seminar
Danny Ruberman, Brandeis University
Diffeomorphism groups of 4-manifolds
Wednesday, February 15, 2023, 10:30am
Ungar, Room 528B
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Abstract: A phenomenon that is unique to dimension 4 is the existence of infinite families of manifolds that are homeomorphic but not diffeomorphic. This is shown via a combination of gauge theory (Seiberg-Witten theory or Yang-Mills theory) with Freedman’s topological classification results. In a joint project with Dave Auckly, we find similar `exotic’ behavior comparing the topology of the groups of diffeomorphisms and homeomorphisms of a smooth 4-manifold. Our main theorem is that the kernel of the map on homotopy groups induced by the inclusion Diff(X) -> Homeo(X) can be infinitely generated. The same techniques yield similar results about spaces of embeddings of surfaces and 3-manifolds in 4-manifolds.
IMSA Seminar
Inkyung Ahn, Korea University, Korea
Population Models with Fokker-Planck-Type Diffusions Incorporating Perceptual Constraints of Species in a Habitat with Spatial Heterogeneity
Tuesday, February 14, 2023, 5:00 pm
Ungar, Room 528B
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Abstract: In this talk, we examine the predator-prey models in a spatially heterogeneous region where the dispersal of predators is non-uniform, a process known as prey-induced dispersal (PYID). We incorporate the perceptual constraints of species in our model and examine how PYID affects the dynamics and coexistence of the system. Our analysis is based on a Holling-type II functional response model under no-flux boundary conditions. We analyze the local stability of the semi-trivial solution in models with PID and linear dispersal in the absence of predators. Furthermore, we investigate the local and global bifurcation from the semi-trivial solution in models with two different dispersals. Our results show that if a predator's satisfaction with the prey density is higher than a certain level, it may not be beneficial in terms of their fitness. Meanwhile, if predators change their motility when they are appropriately satisfied with the amount of prey, they will obtain a survival advantage. Additionally, if time allows, we discuss competition models with starvation-driven diffusions (SDD) and perceptual constraints of species.
Topology Seminar
Kenneth L Baker
University of Miami & IMSA
Handle Numbers of Nearly Fibered Knots
Wednesday, February 1, 2023, 10:30am
Ungar 528B
Abstract: In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2.Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types. We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4. Furthermore, we show that nearly fibered knots have unique incompressible Seifert surfaces rather than just unique minimal genus Siefert surfaces. This is joint work with Fabiola Manjarrez-Gutierrez.
