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## Columbia Geometric Topology SeminarSpring 2024 |
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Organizers: Francesco Lin, Siddhi Krishna, Ross Akhmechet

The GT seminar typically meets on Fridays at 2:00pm Eastern time in Room 307, Mathematics Department, Columbia University.

Other area seminars. Our e-mail list. Archive of previous semesters

Date |
Time (Eastern) |
Speaker |
Title |
---|---|---|---|

January 26 |
3:30pm Eastern (note unusual time!) |
Bojun Zhao |
Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy |

February 2 |
3:30pm Eastern (note unusual time!) | Sebastian Hurtado-Salazar |
Length functions on Lie groups and lattices |

February 9 |
2pm Eastern |
Mike Willis |
Spectral Jones-Wenzl projectors and differences between Khovanov homology and stable homotopy |

February 16 |
3:30pm Eastern (note unusual time!) |
Alex Zupan |
The square knot bounds infinitely many non-isotopic ribbon disks |

February 23 |
2pm Eastern |
Sally Collins |
Homology cobordism & smooth knot concordance |

March 1 |
2pm Eastern |
Eugen Rogozinnikov |
Parametrizing spaces of positive representations |

March 7 (Thursday) |
3pm Eastern in Room 407 (note unusual day, time, and room!) |
Francesco Costantino |
Stated skein modules of 3-manifolds and TQFTs |

March 8 |
2pm Eastern |
Onkar Gujral |
Supverised learning methods applied to low dimensional topology |

March 15 |
2pm Eastern |
no seminar |
Spring Break |

March 22 |
2pm Eastern |
Emmanuel Wagner |
From representation theory to topology: there and back again |

March 29 |
2pm Eastern |
no seminar |
Simons Annual Meeting |

April 5 |
2pm Eastern |
Emily Stark |
Graphically discrete groups and rigidity |

April 12 |
2pm Eastern |
Ian Zemke |
L-space links and formality |

April 19 |
2pm Eastern | Henry Segerman |
Avoiding inessential edges |

April 26 |
2pm Eastern | Patrick Orson |
Unknotting nonorientable surfaces topologically |

April 30 (Tuesday) |
11am (room TBA) | Bruno Martelli |
Negative curvature and fibrations |

**Bojun Zhao**

**Title**: Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy

**Abstract: **Let f be a pseudo-Anosov homeomorphism on a compact orientable surface with nonempty boundary, such that f has a co-orientable stable foliation and reverses the co-orientation on it. Let M denote the mapping torus of f. In this talk, I will discuss some constructions of co-orientable taut foliations in those Dehn fillings of M with filling multislopes constrained by a bound from the degeneracy loci on the boundary components of M. In certain cases where M has connected boundary and is Floer simple, we can construct co-orientable taut foliations in all non-L-space Dehn fillings of M.

**Sebastian Hurtado-Salazar**

**Title**: Length functions on Lie groups and lattices

**Abstract: **We will discuss the notion of a length function on a group, focusing on lattices in Lie groups, such as SL_n(Z), and discuss how techniques from dynamics can help to understand some important questions about these groups.

**Mike Willis**

**Title**: Spectral Jones-Wenzl projectors and differences between Khovanov homology and stable homotopy

**Abstract: **Categorified Jones-Wenzl projectors P_n are widely studied, with endomorphisms related to the Khovanov homology of torus links. We will discuss the lifts of such projectors to the category of spectra, focusing on certain properties of P_n that fail to lift. These are some of the first structural differences found between Khovanov homology and Khovanov stable homotopy, and include the surprising negative answer to a question of Lawson-Lipshitz-Sarkar asking whether topological Hochschild homology for tangle spectra can be used to define a spectral invariant for links in S1xS2. This work is joint with Matt Stoffregen.

**Alex Zupan**

**Title**: The square knot bounds infinitely many non-isotopic ribbon disks

**Abstract: **A knot K in S^3 is (smoothly) slice if K is the boundary of a properly embedded disk D in B^4, and K is ribbon if this disk can be realized without any local maxima with respect to the radial Morse function on B^4. In dimension three, a knot K with nice topology – that is, a fibered knot – bounds a unique fiber surface up to isotopy. Thus, it is natural to wonder whether this sort of simplicity could extend to the set of ribbon disks for K, arguably the simplest class of surfaces bounded by a knot in B^4. Surprisingly, we demonstrate that the square knot, one of the two non-trivial ribbon knots with the lowest crossing number, bounds infinitely many distinct ribbon disks up to isotopy. This is joint work with Jeffrey Meier.

**Sally Collins**

**Title**: Homology cobordism & smooth knot concordance

**Abstract: **The 0-surgeries of two knots K1 and K2 are homology cobordant rel meridians if there exists an integer homology cobordism X between them such that the two positive knot meridians are in the same homology class of X. It is a natural question to ask: if two knots have 0-surgeries related in this sense, must they be smoothly concordant? We give a pair of rationally slice knots as counterexample, and along the way expand upon an involutive knot Floer homology technique for obstructing torsion in the smooth concordance group first introduced by Hom, Kang, Park, and Stoffregen. No previous knowledge of Heegaard Floer theory will be assumed.

**Eugen Rogozinnikov**

**Title**:Parametrizing spaces of positive representations

**Abstract: **Higher Techmüller theory deals with spaces of representations of the fundamental group of a surface into a reductive Lie group $G$, modulo the conjugation, especially with the connected components (called higher Teichmüller spaces) that consist entirely of injective representations with discrete image.

In the last two decades in works of Fock, Goncharov, Burger, Iozzi, Guichard, Wienhard, and others researchers, it was discovered that the most interesting higher Teichmüller spaces are emerging from the groups $G$ having a positive structure, i.e. certain submonoid $G_+$ with no invertible non-unit elements. Some of these submonoids have been known since 1930’s as totally positive matrices and then generalized by Lustzig for split real Lie groups. However it left out a large class of non-split reductive Lie groups such as $SO(p,q)$. O. Guichard and A. Wienhard filled this gap in 2018 by introducing the Theta-positivity, which also includes submonoids $SO(p,q)_+$ sitting in a unipotent group of $SO(p,q)$ and $Sp(2n,R)_+$ which is the set of upper uni-triangular block 2x2-matrices with a symmetric positive definite matrix in the upper right corner.

In my talk, I introduce the Theta-positivity for Lie groups and explain how the spaces of positive representations of the fundamental group of a punctured surface into a Lie group with a positive structure can be parametrized, and how we can describe the topology of these spaces using this parametrization. This is a joint work with O. Guichard and A. Wienhard.

**Francesco Costantino **

**Title**:Stated skein modules of 3-manifolds and TQFTs

**Abstract: **After reviewing the definition of stated skein modules for surfaces and 3 manifolds, I will detail how this recent notion allows to relate topological constructions (related to cut and paste techniques) to algebraic ones (braided tensor products of algebra objects in braided categories for instance). I will explain how the stated skein algebra of some special surfaces provides a topological description for some notable algebras (e.g. the quantised functions ring $O_q(\mathfrak{sl_2})$ or its ``transmutation’’ BSL_2(q)). Then I will describe how stated skein moduli of 3-manifolds fit into a TQFT framework albeit a non completely standard one. If time permits I will also discuss some unexpected non injectivity results in dimension 3. (Joint work with Thang Le)

**Onkar Gujral**

**Title**: Supverised learning methods applied to low dimensional topology

**Abstract: **In this talk we will survey recent work applying neural networks to knot theory. We will start with the necessary background on neural networks and then describe various knot invariants and properties that people have been able to predict with them. These include quasipositivity, the slice genus, the tau-invariant and unknottedness. We will also describe some work in which authors have used invariants like the Khovanov polynomial and Jones polynomial to predict invariants like the slice genus and s-invariant.

**Emmanuel Wagner**

**Title**: From representation theory to topology: there and back again

**Abstract: **In this talk, I will explain how foam evaluation allows to see an sl(2) action on the equivariant gl(n) Khovanov-Rozansky link homologies. We will also see how to extend the previous action functorially. Joint work with You Qi, Louis-Hadrien Robert and Joshua Sussan.

**Emily Stark**

**Title**: Graphically discrete groups and rigidity

**Abstract: **Rigidity problems in geometric group theory frequently have the following form: if two finitely generated groups share a geometric structure, do they share algebraic structure? The work of Papasoglu--Whyte demonstrates that infinite-ended groups are quasi-isometrically flexible; our results show that if you assume a common geometric model, then there is often rigidity. To do this, we introduce the notion of a graphically discrete group, which imposes a discreteness criterion on the group's lattice envelopes. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds; free groups are non-examples. We will discuss new examples and rigidity phenomena for free products of graphically discrete groups.This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

**Ian Zemke**

**Title**: L-space links and formality

**Henry Segerman**

**Title**: Avoiding inessential edges

**Abstract: **Results of Matveev, Piergallini, and Amendola show that any two triangulations of a three-manifold with the same number of vertices are related to each other by a sequence of local combinatorial moves (namely, 2-3 and 3-2 moves). For some applications however, we need our triangulations to have certain properties, for example that all edges are essential. (An edge is inessential if both ends are incident to a single vertex, into which the edge can be homotoped.) We show that if the universal cover of the manifold has infinitely many boundary components, then the set of essential ideal triangulations is connected under 2-3, 3-2, 0-2, and 2-0 moves. Our results have applications in veering triangulations and in quantum invariants such as the 1-loop invariant. This is joint work with Tejas Kalelkar and Saul Schleimer.

**Patrick Orson**

**Title**: Unknotting nonorientable surfaces topologically

**Abstract: **Knot invariants are typically used to give a negative answer to the question of when two embeddings are ambiently isotopic, and rarely to give a positive answer. An exception is the celebrated result of Freedman and Quinn that if the complement of a 2-sphere embedded in the 4-sphere has cyclic fundamental group then that 2-sphere is topologically unknotted. We recently showed that the analogous result for closed nonorientable surfaces in the 4-sphere is also true (in most cases). This talk will describe this recent work and highlight some key ideas from the proof. This is joint work with Anthony Conway and Mark Powell.

**Bruno Martelli**

**Title**: Negative curvature and fibrations

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