Talk_id | Date | Speaker | Title |
31527
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Tuesday 1/10 2:30 PM
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G&T Seminar Organizational Meeting
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- G&T Seminar Organizational Meeting
- 01/10/2023
- 2:30 PM - 3:00 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
No abstract available.
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31562
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Tuesday 2/7 3:30 PM
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Amitesh Datta, Princeton University
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Does the Jones polynomial of a knot detect the unknot? A novel approach via braid group representations and class numbers of number fields.
- Amitesh Datta, Princeton University
- Does the Jones polynomial of a knot detect the unknot? A novel approach via braid group representations and class numbers of number fields.
- 02/07/2023
- 3:30 PM - 4:30 PM
- Online (virtual meeting)
(Virtual Meeting Link)
- Peter Kilgore Johnson (john8251@msu.edu)
How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation.
In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s and is equivalent to whether B_4 is a group of 3 x 3 matrices). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones.
I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.
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31504
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Tuesday 2/14 2:00 PM
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Renaud Detcherry, Institut de Mathématiques de Bourgogne
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Title: On the kernel of Witten-Reshetikhin-Turaev quantum representations
- Renaud Detcherry, Institut de Mathématiques de Bourgogne
- Title: On the kernel of Witten-Reshetikhin-Turaev quantum representations
- 02/14/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Efstratia Kalfagianni (kalfagia@msu.edu)
Abstract: Witten-Reshetikhin-Turaev SO(3) quantum representations are a family of representations of mapping class groups of surfaces. The family is asymptotically faithful, but each representation has kernel: indeed, r-th powers of Dehn twists are in the kernel of the level r quantum representation.
An open question is whether the kernel is generated by r-th powers of Dehn twists; we will present partial results on this question, by relating the so-called "h-adic expansion" of quantum representations to Johnson homomorphisms.
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31556
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Tuesday 2/21 2:00 PM
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Ian Montague , Brandeis University
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Seiberg-Witten Floer K-Theory and Cyclic Group Actions
- Ian Montague , Brandeis University
- Seiberg-Witten Floer K-Theory and Cyclic Group Actions
- 02/21/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
Given a spin rational homology sphere equipped with a cyclic group action, I will introduce equivariant refinements of Manolescu's kappa invariant, derived from the equivariant K-theory of the Seiberg--Witten Floer spectrum. These invariants give rise to equivariant relative 10/8-ths type inequalities for equivariant spin cobordisms between rational homology spheres. I will explain how these inequalities provide applications to knot concordance, obstruct cyclic group actions on spin fillings, and give genus bounds for knots in punctured 4-manifolds. If time permits I will explain how these invariants are related to equivariant eta-invariants of the Dirac operator, and describe work-in-progress which provides explicit formulas for the $S^1$-equivariant eta-invariants on Seifert-fibered spaces.
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31547
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Tuesday 2/28 2:00 PM
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Ryan Stees, Indiana University
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Milnor's invariants for knots and links in closed orientable 3-manifolds
- Ryan Stees, Indiana University
- Milnor's invariants for knots and links in closed orientable 3-manifolds
- 02/28/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
Early in his career, John Milnor defined his seminal link invariants, now called Milnor's $\overline{\mu}$-invariants. They are topological concordance invariants of links in $S^3$, and much is known about them. However, until recently, few results have extended Milnor's work to links in other closed orientable 3-manifolds, and such extensions have done so for special classes of 3-manifolds or specific types of links. In this talk, I will discuss an extension of these invariants to concordance invariants of knots and links in any closed orientable 3-manifold, discuss some theorems that justify calling them ``Milnor's invariants", and study their properties.
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31573
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Tuesday 3/14 11:30 AM
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Shunyu Wan , University of Virginia
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Naturality of Legendrian LOSS invariant under positive contact surgery and application
- Shunyu Wan , University of Virginia
- Naturality of Legendrian LOSS invariant under positive contact surgery and application
- 03/14/2023
- 11:30 AM - 12:30 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
Given a Legendrian Knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L which lives in the knot Floer homology group. We prove that the LOSS invariant is natural under the positive contact surgery. In this talk I will review some background and definition, get the idea of the proof and try to focus on the application which is about new examples of non-simple knots.
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32601
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Tuesday 3/14 2:00 PM
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John Baldwin, Boston College
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Small Dehn surgery and SU(2)-representations
- John Baldwin, Boston College
- Small Dehn surgery and SU(2)-representations
- 03/14/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Matthew Edward Hedden (heddenma@msu.edu)
In their celebrated proof of the Property P Conjecture and its sequel, Kronheimer and Mrowka proved that the fundamental group of r-surgery on a nontrivial knot in the 3-sphere admits an irreducible SU(2)-representation whenever r is at most 2 in absolute value (which implies in particular that surgery on a nontrivial knot is never a homotopy 3-sphere). They asked whether the same is true for other small values of r -- in particular, for r = 3 and 4 -- noting that it's false for r = 5 since 5-surgery on the right-handed trefoil is a lens space. I'll describe recent work which answers their question in the affirmative. Our proof involves Floer homology and also the dynamics of surface homeomorphisms. All of this work is joint with Steven Sivek, and significant parts are also joint with Zhenkun Li and Fan Ye.
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31548
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Monday 3/20 2:00 PM
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Siddhi Krishna, Columbia University
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RTG Seminar: Fibered knots: what, why and how
- Siddhi Krishna, Columbia University
- RTG Seminar: Fibered knots: what, why and how
- 03/20/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
Fibered knots show up all over low-dimensional topology, as they provide a robust way to investigate interactions between phenomena of different dimensions. In this talk, I'll survey what they are, why you should care, and how to identify them. Then, as time permits, I'll also sketch a proof that positive braid knots are fibered. I will assume very little background for this talk -- all are welcome!
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31549
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Tuesday 3/21 11:30 AM
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Siddhi Krishna, Columbia University
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Twist positivity, L-space knots, and concordance
- Siddhi Krishna, Columbia University
- Twist positivity, L-space knots, and concordance
- 03/21/2023
- 11:30 AM - 12:30 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
In this talk, I’ll describe a braid word theoretic property, called “twist positivity”, which often puts strong restrictions on quantitative and geometric properties of a braid. I’ll describe some old and new results about twist positivity, as well as some new applications towards knot concordance. In particular, I’ll describe how using a suite of numerical knot invariants (including the braid index) in tandem allows one to prove that there is an infinite family of L-space knots (containing all positive torus knots and also an infinite family of hyperbolic knots) where every knot represents a distinct smooth concordance class. This confirms a prediction of the slice-ribbon conjecture. Everything I’ll discuss is joint work with Hugh Morton. I will assume little background about knot invariants for this talk – all are welcome!
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31538
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Tuesday 3/21 2:00 PM
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Anup Poudel, Ohio State
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A comparison between $SL_n$ spider categories.
- Anup Poudel, Ohio State
- A comparison between $SL_n$ spider categories.
- 03/21/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Vijay B Higgins (higgi231@msu.edu)
In this talk, we will explore and make comparisons between various models that exist for spherical tensor categories associated to the category of representations of the quantum group $U_q(sl_n).$ In particular, we will discuss the combinatorial model of Murakami-Ohtsuki-Yamada (MOY), the n-valent ribbon model of Sikora and the trivalent spider category of Cautis-Kamnitzer-Morrison (CKM). We conclude by showing that the full subcategory of the spider category from CKM, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This proves a conjecture of Le and Sikora and also answers a question from Morrison's Ph.D. thesis.
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31554
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Tuesday 4/4 2:00 PM
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David Chan, Vanderbilt University
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TBA
- David Chan, Vanderbilt University
- TBA
- 04/04/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
No abstract available.
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31543
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Monday 4/10 2:00 PM
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Sam Gunningham, Montana State
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RTG Seminar: TBA
- Sam Gunningham, Montana State
- RTG Seminar: TBA
- 04/10/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Vijay B Higgins (higgi231@msu.edu)
TBA
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31542
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Tuesday 4/11 2:00 PM
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Sam Gunningham, Montana State
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TBA
- Sam Gunningham, Montana State
- TBA
- 04/11/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Vijay B Higgins (higgi231@msu.edu)
TBA
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32612
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Tuesday 4/18 2:00 PM
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Pierre-Louis Blayac, University of Michigan
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TBA
- Pierre-Louis Blayac, University of Michigan
- TBA
- 04/18/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Vijay B Higgins (higgi231@msu.edu)
TBA
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31537
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Monday 4/24 2:00 PM
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Sarah Petersen, University of Colorado, Boulder
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RTG Seminar: TBA
- Sarah Petersen, University of Colorado, Boulder
- RTG Seminar: TBA
- 04/24/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
No abstract available.
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31536
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Tuesday 4/25 2:00 PM
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Sarah Petersen, University of Colorado, Boulder
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TBA
- Sarah Petersen, University of Colorado, Boulder
- TBA
- 04/25/2023
- 2:00 PM - 3:00 PM
- C304 Wells Hall
- Peter Kilgore Johnson (john8251@msu.edu)
No abstract available.
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