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Title: Classifying Lie Quandles Speaker: Bryce Virgin Time: 2:00pm–3:00pm Place: CMC 118
Lie quandles are smooth manifolds with one parameter families of compatible smooth quandle operations, which generalize the dynamical structure of classical mechanics in the Hamiltonian formalism and quantum mechanics in the Heisenberg formalism. In this presentation, we will introduce this connection between dynamics and Lie quandles, and describe a conjectured form of Noether's theorem in this new framework. We will conclude with the discusion of a simple family of Lie quandles called the Alexander-Lie quandles associated to a given Lie Group, together with a classification of the Alexander-Lie quandles on all Euclidean spaces.
Title: Handle numbers of knots, Seifert surfaces, and sutured manifolds Speaker: Kenneth Baker, University of Miami Time: 2:00pm–3:00pm Place: CMC 118
A sutured manifold without toroidal sutures admits a Heegaard splitting into compression bodies. Generalizing Heegaard genus, its handle number is the minimum number of handles needed to assemble such a splitting. The handle number of a Seifert surface is that of its complementary sutured manifold, and the handle number (or Morse-Novikov number) of a knot is the minimum handle number among its Seifert surfaces. We will overview properties and curiosities of handle numbers of knots and sutured manifolds as we build towards a construction of knots whose handle number is not realized by the handle number of any of its minimal genus Seifert surfaces. Much of this is joint work with Fabiola Manjarrez-Gutiérrez.
Title: Quiver Categorification of Homset Invariants Speaker: Sam Nelson, Claremont McKenna College, CA Time: 2:00pm–3:00pm Place: Joint Zoom Meeting
The quandle coloring quiver and similar constructions provide a way of turning homset- based invariants into categories. In this talk we will examine this phenomenon with several examples.
Title: Word problem for groups as equivalence relations Speaker: Meng-Che "Turbo" Ho Time: 2:00pm–3:00pm Place: CMC 118
In 1911, Dehn proposed three decision problems for finitely presented groups: the word problem, the conjugacy problem, and the isomorphism problem. These problems have been central to both group theory and logic, and were each proven to be undecidable in the 50s. There is much current research studying the decidability of these problems in certain classes of groups.
Classically, when a decision problem is undecidable, its complexity is measured using Turing reductions. However, Dehn's problems can also be naturally thought of as equivalence relations. We take this point of view and measure their complexity using computable reductions, which is a finer measure than Turing reductions. This yields behaviors different from the classical context: for instance, every decision problem has the same complexity as a word problem under Turing reductions, but not every equivalence relation has the same complexity as a word problem under computable reductions. This leads us to study the structure of complexity classes (of equivalence relations) containing a word problem and other related questions.
This is a joint work with Uri Andrews, Matthew Harrison-Trainor, and Luca San Mauro.
Title: Exotic disks and involutions via instanton Floer theory Speaker: Irving Dai Time: 2:00pm–3:00pm Place: CMC 118
We develop a suite of instanton-theoretic invariants associated to a strongly invertible knot. We give some applications to constructing exotic disks for Whitehead doubles, stabilization questions for slice surface, and equivariant \(Z\)-sliceness. This is joint work with Abhishek Mallick and Masaki Taniguchi.
Title: Category Theory for Classifying Complex Geometric Objects of High Symmetry Speaker: Greg McColm Time: 2:00pm–3:00pm Place: CMC 118
Automated systems are generating millions of highly symmetric finite and infinite geometric figures, making automated classification, cataloguing and indexing systems necessary for making sense of the collections, not to mention finding figures with desired properties. This includes the classification of crystals, which include upwards of a million “crystal nets” of known crystals and millions of hypothetical ones. Category theory may provide approaches to the classification problem, and we consider some category theoretic constructions that may be useful in classifying figures. This talk presumes no prior knowledge of category theory.
Title: Foams and link homology Speaker: Mikhail Khovanov, Johns Hopkins University Time: 2:00pm–3:00pm Place: CMC 118
Some of the best-behaved link homology theories can be constructed using the category of foam cobordisms between planar webs as an intermediate step. We'll review this theory and its uses.
Title: \(K^3\) and Horikawa surfaces and exotic 4-manifolds, Part II Speaker: Sümeyra Sakallı Time: 2:00pm–3:00pm Place: CMC 118
Title: The Yang-Baxter equation Speaker: Mohamed Elhamdadi Time: 2:00pm–3:00pm Place: CMC 118
The Yang-Baxter equation (YBE) originated in theoretical physics and statistical mechanics in the work of C. N. Yang (1967) and R. Baxter (1972). The YBE appears in many areas of mathematics such as Hopf algebras, non-associative algebras and low dimensional topology to name a few. A solution to YBE gives a representation of the braid group and thus allows the construction of link invariants such as the Jones polynomial.
We will discuss set theoretical solutions to YBE, give some examples and consider the structure group of YB solutions. We will also discuss a homology theory for YBE.
Title: \(K^3\) and Horikawa surfaces and exotic 4-manifolds, Part I Speaker: Sümeyra Sakallı Time: 2:00pm–3:00pm Place: CMC 118
Complex surfaces are fundamental tools in 4-manifold theory. In this talk I will give a detailed description of two special complex surfaces; \(K^3\) and Horikawa surfaces, and discuss my ongoing research on them related to exotic 4-manifolds.
Title: Lens Space Surgeries on Simple Spaces, Part II Speaker: Shiyu Liang Time: 2:00pm–3:00pm Place: CMC 118
Title: Lens Space Surgeries on Simple Spaces, Part I Speaker: Shiyu Liang Time: 2:00pm–3:00pm Place: CMC 118
The Berge Conjecture asserts that any knot in \(S^3\) admitting a lens space surgery has a surgery dual that is a simple knot in a lens space. Berge described twelve infinite families of such knots, and work of Greene and Berge independently established that this list is complete among simple knots (that admit \(S^3\) surgeries).
This seminar will be presented in two parts. In the first talk, we will review the necessary background, including standard definitions and results in knot theory, 3-manifold topology, and Dehn surgery, and conclude with a discussion of the original Berge Conjecture. In the second talk, we will explore analogues of the Berge Conjecture in other simple spaces, such as \(S^1\times S^2\) and the Poincaré homology sphere.