For more information about the Seminar, to suggest speakers, or to volunteer to speak, contact Bahar Acu, Dave Bachman, Sam Nelson, Vin de Silva, or Helen Wong.
| Date | Speaker | Title and Abstract |
| Tuesday
January 23 |
Organizational Meeting |
|
| Tuesday
January 30 |
Song Yu California Institute of Technology and Tsinghua Yau Mathematical Sciences Center
|
Title: Knot invariants, Gromov-Witten invariants, and integrality conjectures
Abstract: In this talk, we will take a peek at large N duality which is a deep correspondence between invariants of knots in 3-manifolds and enumerative geometry in symplectic 6-manifolds discovered in physics in the 1980-90s. On the numerical level, the correspondence relates Chern-Simons knot invariants to open Gromov-Witten invariants which are counts of bordered Riemann surfaces with Lagrangian boundary conditions, and has led to predictions on the integrality structures of both invariants. We will discuss recent progress on the enumerative geometry side and connections to known integrality properties in Gromov-Witten theory. |
| Tuesday
February 6 |
NO SEMINAR
|
|
| Tuesday
February 13 |
Luya Wang Stanford University
|
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. |
| Tuesday
February 20 |
Puttipong Pongtanapaisan Arizona State University
|
Title: Building Knotted Objects Efficiently Abstract: Knotted objects can be constructed by gluing together standard pieces called handles. Understanding the minimum number of handles required for construction and their sequential attachment provides valuable insights into the complexity of entanglement. Certain knots require specific types of handles to be attached first, preventing them from fitting into small lattice tubes. This is particularly relevant as polymers in confinement are modeled as knots within lattice tubes. In this talk, I will discuss methods for studying these handles and their attachment order using coloring games applied to link diagrams. |
| Tuesday
February 27 |
UC Davis
|
CANCELLED |
| Tuesday
March 5 |
Adam Yassine Pomona College
|
Title: A Structural Approach to Classical Mechanics Abstract: A structural approach to the study of classical mechanics clarifies the physical heuristics that physicists use in constructing mathematical models of classical mechanical systems. The focus of our current program is to develop a category theoretic framework that captures certain compositional features of classical mechanics. The framework is both flexible enough to support the description of a wide variety of systems and rigid enough to uniquely determine the physicists' models. |
| Tuesday
March 12 |
No Meeting | Spring Break |
| Tuesday
March 19 |
Wesleyan College |
CANCELLED |
| Tuesday
March 26 |
Qing Zhang UC Santa Barbara |
Title: Super-modular categories from near-group centers Abstract: A super-modular category is a unitary pre-modular category with Müger center equivalent to the symmetric unitary category of super-vector spaces. The modular data for a super-modular category gives a projective representation of the group: $\Gamma_\theta<\mathrm{SL}(2, \mathbb{Z})$. Adapting work of Ng-Rowell-Wang-Wen, Cho- Kim-Seo-You computed modular data from congruence representations of $\Gamma_\theta $ using the congruence subgroup theorem for super-modular categories of Bonderson-Rowell-Wang-Z and the minimal modular extension theorem of Reutter-Johnson-Freyd. They found two classes of previously unknown modular data for rank 10 super-modular categories. We show that these data are realized by modifying the Drinfeld centers of near-group fusion categories associated with the groups $\Z/6$ and $\Z/2\times \Z/4$. The methods we develop have more general applications, and we describe some of them. This talk is based on joint work with Eric Rowell and Hannah Solomon. |
| Tuesday
April 2 |
Jim Hoste Pitzer College |
Title: Variations on the Kauffman Bracket Abstract: Forty years ago, Lou Kauffman formulated his "bracket" polynomial, a function from link diagrams to Laurent polynomials in one variable. This elementary construction leads to a simple definition of the Jones Polynomial. The simplifying assumptions made by Kauffman in producing the bracket polynomial are not strictly necessary, leading to the question: Can a more general invariant of links be obtained using variations of the Kauffman bracket? In this talk I will explore this question. |
| Tuesday
April 9 |
NO SEMINAR
|
|
| Tuesday
April 16 |
Ryan Maguire Dartmouth College |
Title: Relative Strengths of Knot Invariants by Experiment Abstract: Four knot polynomials have been well studied by topologists, graph theorists, and algebraists alike: The Alexander, Jones, HOMFLY-PT, and Khovanov polynomials. It is known that the Khovanov polynomial is "stronger" than the Jones polynomial, and similarly one may state that HOMFLY-PT is stronger than both the Alexander and Jones polynomials. No comparison can be made between the Jones and Alexander polynomials since there are families of knots with identical Alexander polynomials but distinct Jones polynomials, and vice-versa, but experiment tells us the Jones polynomial is stronger, on average, at distinguishing knots. We have tabulated the Alexander, Jones, and HOMFLY-PT polynomials for all knots up to 19 crossings, and the Khovanov polynomial for up to 17 crossings. Using this, we can experiment on the relative strengths of these knot invariants and generate statistics on them. |
| Tuesday
April 23 |
Joe Breen University of Iowa |
Title: Open books in all dimensions Abstract: I will discuss recent work (joint with K. Honda and Y. Huang) on establishing a relationship, first discovered by Giroux, between "contact structures" and "open books". This relationship has been widely used in 3-dimensional contact topology, and mathematicians are beginning to investigate the consequences in higher-dimensional contact topology. No background knowledge of contact topology or open book decompositions will be assumed. I will even motivate why higher-dimensional contact topology could be useful for questions in low-dimensional topology. |
| Tuesday
April 30 |
Elena Wang Michigan State University |
Title: A Distance for Geometric Graphs via the Labeled Merge Tree Interleaving Distance Abstract: Geometric graphs appear in many real-world data sets, such as road networks, sensor networks, and molecules. We investigate the notion of distance between graphs and present a metric to measure the distance between two geometric graphs via merge trees. In order to preserve as much useful information as possible from the original data, we introduce a way of rotating the sublevel set to obtain the merge trees via the idea of the directional transform. We represent the merge trees using a surjective multi-labeling scheme and then compute the distance between two representative matrices. Our distance not only has theoretically desirable qualities but can also be approximated in polynomial time. We illustrate its utility by implementation on a Passiflora leaf data set. |
| Date | Speaker | Title and Abstract |
| Tuesday
Sept 5 |
Organizational Meeting |
|
| Tuesday
Sept 12 |
Robert Bowden Harvey Mudd College |
Title: Chebyshev Threadings in Skein Algebras for Punctured Surfaces Abstract: Skein algebras are algebras of links in a surface quotiented by diagram-based equivalence relations based on the Kauffman bracket. In the case of surfaces with punctures, the skein algebra is generated by links as well as arcs between the punctures, and there are additional skein relations for the arcs. We examine the algebraic structure of the punctured case, finding a description of the central elements at certain roots of unity. Our construction is closely related to the one for the usual skein algebra, where central elements come from threading links by Chebyshev polynomials. |
| Tuesday
Sept 19 |
Reginald Anderson Claremont McKenna College |
Title: Cellular resolutions of the diagonal and exceptional collections for toric Deligne-Mumford stacks Abstract: Beilinson gave a resolution of the diagonal for complex projective space which yields a strong, full exceptional collection of line bundles. Bayer-Popescu-Sturmfels generalized Beilinson's result to a cellular resolution of the diagonal for what they called "unimodular" toric varieties (a more restrictive condition than being smooth), which can also be extended to smooth toric varieties and global quotient toric DM stacks of a smooth toric variety by a finite abelian group, if we allow our resolution to have cokernel which is supported only along the vanishing of the irrelevant ideal. Here we show implications for exceptional collections of line bundles and a positive example for the modified King's conjecture by giving a strong, full exceptional collection of line bundles on a smooth, non-unimodular nef-Fano complete toric surface. |
| Tuesday
Sept 26 |
Reginald Anderson Claremont McKenna College |
Title: Cellular resolutions of the diagonal and exceptional collections for toric Deligne-Mumford stacks - Continued Abstract: Beilinson gave a resolution of the diagonal for complex projective space which yields a strong, full exceptional collection of line bundles. Bayer-Popescu-Sturmfels generalized Beilinson's result to a cellular resolution of the diagonal for what they called "unimodular" toric varieties (a more restrictive condition than being smooth), which can also be extended to smooth toric varieties and global quotient toric DM stacks of a smooth toric variety by a finite abelian group, if we allow our resolution to have cokernel which is supported only along the vanishing of the irrelevant ideal. Here we show implications for exceptional collections of line bundles and a positive example for the modified King's conjecture by giving a strong, full exceptional collection of line bundles on a smooth, non-unimodular nef-Fano complete toric surface. |
| Tuesday
Oct 3 |
Julian Chaidez USC |
Title: Quantum 4-Manifold Invariants Via Trisections Abstract: I will describe a new family of potentially non-semisimple invariants for compact a 4-manifold whose boundary is equipped with an open book. The invariant is computed using a trisection, along with some additional combing data, and a piece of algebraic data called a Hopf triple. The relationship with other recent works on non-semisimple 4-manifold invariants, like the work of Costantino-Geer-Patureau-Mirand-Virelizier, is not yet clear. This talk is based on joint work with Shawn Cui (Purdue) and Jordan Cotler (Harvard). |
| Tuesday
Oct 10 |
Christopher Perez Loyola University New Orleans |
Title: Towers and elementary embeddings in total relatively hyperbolic groups Abstract: In a remarkable series of papers, Zlil Sela classified the first-order theories of free groups and torsion-free hyperbolic groups using geometric structures he called towers. It was later proved by Chloe Perin that if H is an elementarily embedded subgroup (or elementary submodel) of a torsion-free hyperbolic group G, then G is a tower over H. We prove a generalization of Perin's result to toral relatively hyperbolic groups using JSJ and shortening techniques. |
| Tuesday
Oct 17 |
No Meeting | Fall Break |
| Tuesday
Oct 24 |
Wenyuan Li USC |
Title: Generating families on Lagrangian cobordisms Abstract: An important question in contact topology is to understand Legendrian knots and their relations given by Lagrangian cobordisms. In the contact manifold T*M x R, an important tool to study Legendrian knots and their Lagrangian cobordisms is called generating families or generating functions, which are generalizations of the defining functions f of graphical Legendrians of the form {(x, df(x), f(x))}. When there exists a generating family with good control at infinity, interesting Legendrian invariants can be extracted. We try to understand the following basic question: when can a generating function on the Legendrian knot be extended to the Lagrangian cobordism? We will give a necessary and sufficient condition to the problem for generating families with good control at infinity. In particular, we show that such an extension always exists in the case of Lagrangian concordances. |
| Tuesday
Oct 31 |
Konstantinos Varvarezos UCLA |
Title: Cosmetic Surgeries on Knots and Heegaard Floer Homology Abstract: A common method of constructing 3-manifolds is via Dehn surgery on knots. A pair of surgeries on a knot is called purely cosmetic if the resulting 3-manifolds are homeomorphic as oriented manifolds, whereas it is said to be chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. An outstanding conjecture predicts that no nontrivial knots admit any purely cosmetic surgeries. We apply certain obstructions from Heegaard Floer homology to show that (nontrivial) knots which arise as the closure of a 3-stranded braid do not admit any purely cosmetic surgeries. Furthermore, we find new obstructions to the existence of chirally cosmetic surgeries coming from Heegaard Floer homology; in particular, we make use of immersed curve formulations of knot Floer homology and the corresponding surgery formula. Combining these with other obstructions involving finite type invariants, we completely classify chirally cosmetic surgeries on odd alternating pretzel knots. Moreover, we rule out cosmetic surgeries for L-space knots along slopes with opposite signs. |
| Tuesday
Nov 7 |
Hyunki Min UCLA |
Title: Contact structures and the mapping class group of lens spaces Abstract: One important problem in contact topology is to classify contact structures on a given manifold. Around 20 years ago, Giroux and Honda classified contact structures on lens spaces. A natural question to ask after that is how the transformations on lens spaces interact with the contact structures. In this talk, we study contactomorphisms on lens spaces, which are diffeomorphisms preserving the contact structure. We show that the contact mapping class group of a standard contact lens space is a subgroup of the mapping class group of the lens space. |
| Tuesday
Nov 14 |
Claremont Colleges Course Previews for Spring 2024 | Title: Geometry Topology Course Previews for Spring 2024 Abstract: Geometry and Topology Seminar invites students and faculty to a course preview session devoted to a discussion and presentations about upcoming Spring 2024 courses in geometry, topology and/or with applications in geometry and topology to help students make their enrollment choices. |
| Tuesday
Nov 21 |
No Meeting | Thanksgiving Week |
| Tuesday
Nov 28 |
Melody Molander UC Santa Barbara |
Title: Skein Theory of Affine ADE Subfactor Planar Algebras Abstract: Subfactor planar algebras first were constructed by Vaughan Jones as a diagrammatic axiomatization of the standard invariant of a subfactor. These planar algebras also encode two other invariants of the subfactors: the index and the principal graph. The Kuperberg Program asks to find all diagrammatic presentations of subfactor planar algebras. This program has been completed for index less than 4. In this talk, I will introduce subfactor planar algebras and give some presentations of subfactor planar algebras of index 4 which have affine ADE Dynkin diagrams as their principal graphs. |