Dartmouth Topology Seminar
Fall 2020–Spring 2021
Thursdays 10-11am EDT
201 Kemeny Hall
Note: Special meeting times are marked in red.
Date Speaker Title
Oct 28 Samuel Tripp,
Dartmouth College
Oct 21 Zachary Winkeler,
Dartmouth College
Oct 14 Daniel Alvarez-Gavela,
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
Sept 16 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

October 28, 2021: Samuel Tripp "TBA"


October 21, 2021: Zachary Winkeler "TBA"


October 14, 2021: Daniel Alvarez-Gavela "TBA"


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


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


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.

September 16, 2021: Sergey 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)→π0S(f,w) be the natural projection into the group π0S(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: π0S(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 hγ ∈ S(f,w) so that the collection {hγ}γ constitute a subgroup of S(f,w), i.e. hγhγ' = hγ 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.