Date |
Speaker |
Title |

Nov 4 | Sergey Maksimenko Corresponding member of the National Academy of Science of Ukraine, President of the Kiev Mathematical Society, Institute of mathematics of NAS of Ukraine, Kyiv, Ukraine |
Combinatorial symplectic symmetries of smooth functions on surfaces |

Oct 28 | Samuel Tripp,
Dartmouth College |
Integral Grid Homology for Links in Lens Spaces |

Oct 21 | Zachary Winkeler,
Dartmouth College |
Multipunctured Khovanov homology |

Oct 14 | Daniel Alvarez-Gavela,
MIT |
Legendrian invariants from algebraic K-theory |

Oct 7 | Misha Tyomkin,
Dartmouth College |
Various facets of torsion theory: Whitehead, Reidemeister and Milnor |

Sept 30 | Misha Tyomkin,
Dartmouth College |
Numbers on barcode of a strong Morse function |

Sept 23 | Emanuele Zappala,
University of Tartu, Estonia |
Cocycle invariants of knots and knotted surfaces |

**Abstracts**

#### November 4, 2021: Sergy Maksimenko "Combinatorial symplectic symmetries of smooth functions on surfaces"

*Abstract:*Let M be a compact orientable surface equipped with a symplectic form w, P be either the real line of the circle, and f:M→P a smooth function.
Let also S(f,w) = {h ∈ Diff(M) | foh=f and h^{*}w=w} be the group of diffeomorphisms of M which mutually preserve f and w, and p:S(f,w)→π_{0}S(f,w) be the natural projection into the group π_{0}S(f,w) of path components of S(f,w).
It is shown that for a large class of smooth maps f:M→P, which includes all Morse maps, the homomorphism p admits a section homomorphism s: π_{0}S(f,w)→S(f,w).
In other words, in each isotopy class γ of diffeomorphisms mutually preserving f and w one can choose a representative h_{γ} ∈ S(f,w) so that the collection {h_{γ}}_{γ} constitute a subgroup of S(f,w), i.e. h_{γ}h_{γ'} = h_{γ γ'}.
A variant of such result for non-orientable surfaces is also given.
As an application it is also shown that every finite group is a subgroup of S(f,w) for some M, f and w.

#### October 28, 2021: Samuel Tripp "Integral Grid Homology for Links in Lens Spaces"

*Abstract:* Baker, Grigsby and Hedden presented a combinatorial description for the knot Flor homology of links in lens spaces. We provide a combinatorial proof of invariance of this grid homology, and adapt this to prove the invariance of the generalization of this grid homology to integral coefficients.

#### October 21, 2021: Zachary Winkeler "Multipunctured Khovanov homology"

*Abstract:* Khovanov homology is a knot invariant that categorifies the Jones polynomial. For knots in a thickened annulus, we can define a filtration on the Khovanov chain complex that gives us an invariant called annular Khovanov homology. In this talk, we will discuss a generalized notion of filtration that allows us to construct an analogous invariant for knots in thickened disks with any number of punctures.

#### October 14, 2021: Daniel Alvarez-Gavela "Legendrian invariants from algebraic K-theory"

*Abstract:*The theory of generating functions allows one to build Legendrian invariants out of Morse theoretic invariants.
While such homological invariants have been extensively studied and compared to pseudo-holomorphic curve invariants, K-theoretic invariants
are somewhat more mysterious. In this talk I will discuss K-theoretic torsion invariants for Legendrians in 1-jet spaces, focusing on an
example which appeared in previous work of Igusa and Klein, where it was used to provide a picture for a generator of K_3(Z)=Z/48.
Joint work with K. Igusa.

#### October 7, 2021: Misha Tyomkin "Various facets of torsion theory: Whitehead, Reidemeister and Milnor"

*Abstract:*Torsion is a topological invariant defined algebraically by an
appropriate generalization of a notion of determinant. Some of its
incarnations are:

* Whitehead torsion is an obstruction for a famous h-cobordism theorem
to hold true in the non-simply connected case.

* Reidemeister torsion is an invariant which is able to detect
homotopy equivalent, but non-homeomorphic manifolds, such as lens
spaces.

* Milnor torsion is a low-dimensional variation of a Reidemeister one,
which in some cases equals to the Alexander polynomial of knot.

In this overview talk I will try to touch on these incarnations
without delving into proofs or technical details.

#### September 30, 2021: Misha Tyomkin "Numbers on barcode of a strong Morse function"

*Abstract:*
Morse function f on a manifold M is called strong if all its critical
values are pairwise distinct. For a given field F, Barannikov
decomposition (a.k.a. barcode) is a canonical pairing of some critical
points of such f with neighboring indices. We present a construction
which naturally associates a number (i.e. an element of F) to each
Barannikov pair (a.k.a. bar in the barcode), defined up to a sign. It
turns out that if homology of M over F is that of a sphere, then the
product of all the numbers is independent of f (up to sign). We then
proceed to consider homology with twisted coefficients (still in some
field F). It is in this setting where the Reidemeister torsion is
defined. We construct the twisted barcode and prove that the mentioned
product equals to the Reidemeister torsion (provided that the latter
is defined); in particular, it's again independent of f. Based on a
joint work with Petya Pushkar

#### September 23, 2021: Emanuele Zappala "Cocycle invariants of knots and knotted surfaces "

*Abstract:* In this talk I will introduce the notions of racks and quandles,
as well as their cohomology. I will discuss the diagrammatic interpretation
of these algebraic structures, and show how to use this to construct a
partition function that is an invariant of Reidemeister moves. Using the
same principles, I will also show that it is possible to obtain an invariant
of knotted surfaces in 4-space, defining certain partition functions based
on quandle 3-cocycles. These results were introduced in Trans. Amer.
Math. Soc. 355 (2003) 3947-3989 by Carter, Jelsovsky, Kamada, Langford
and Saito. Then, I will discuss how to generalize the previous results by means
of ternary quandle cohomology to get invariants of framed links and compact
oriented surfaces with boundary (embedded in 3-space). I will present
applications of all these invariants throughout the talk, including a generalization
of the cocycle invariants to quantum invariants.