**Meeting time and place:** Fridays from 4pm-5pm in Herman Brown 227 and on Zoom (email me for a link)

**Instructor:** Shelly Harvey

### Current Participants (listed in alphabetic order):

Faculty:

- Shelly Harvey

Chris Leininger

Alan Reid

Postdocs:

- Jason Joseph

Allison Miller

Jacob Russell

Chandrika Sadanard (UIUC)

Eric Samperton (UIUC)

Morgan Weiler

Will Worden

Graduate Students:

- Tam Cheetham-West

Leo Digiosia

Christopher Loa (UIUC)

Alex Manchester

Marissa Miller (UIUC)

Sara Edelman-Munoz

Alex Nolte

Kate O'Connor

Nick Rouse

Connor Sell

Emily Shinkle (UIUC)

Will Stagner

Shawn Williams

Undergraduate student:

- Gabe Gress

### Fall 2020

Date | Speaker | Topic |
---|---|---|

8-28 | Shelly Harvey | Massey products of a DGA, A_infinity algebras, and their relationships to Milnor's invariants for link complements |

9-4 | Shelly Harvey | Massey products of a DGA, A_infinity algebras, and their relationships to Milnor's invariants for link complements (part II) |

9-11 | Chris Leininger | Geodesic currents for surfaces |

9-18 | Tam Cheetham-West | Galois rigidity and finite quotients |

9-25 | Emily Shinkle | The Flip Graph |

10-2 | Shawn Williams | Strong S-Equivalence of Ordered Links |

10-9 | Connor Sell | Cusps of hyperbolic 4-manifolds |

10-16 | Will Stagner | Hyperbolic knots in compact 3-manifolds |

10-23 | Sara Edelman-Munoz | The Congruence Subgroup Property in $SL_3(\mathbb(Z))$ |

10-30 | Alex Nolte | Minimal surfaces in three-manifolds |

11-6 | Nicholas Rouse | Introduction to Burger-Mozes groups |

11-13 | Alex Manchester | (n)-solvable filtration and satellite operators |

11-20 | Kate O'Connor | Milnor's Invariants |

**
Abstracts: **
**Shelly Harvey (Massey products and their relation to Milnor's invariants of links):** Massey products were defined by William Massey in the late
50s as a higher order cohomology operation generalizing the cup
product. For a differential graded algebra, one has an A_infinity
structure and it is known that the Massey products can be interpreted
as the higher order multiplications. For a link complement, Turaev
and Porter show that the Massey products could be interpreted as
Milnor's invariants of the link, answering a question of John
Stallings. In this talk, we will define Massey products on the
cohomology of a DGA (differential graded algebra) and show how this
gives the Massey product for a topological space. If there is time,
we will discuss the relationship between Milnor's invariants of a link
and Massey products of the link complement.
**Chris Leininger (Geodesic currents for surface):**I'll talk about Bonahon's construction of the space of
geodesic currents associated for a surface with negative Euler
characteristic. The goal is to say what a geodesic current is,
describe different ways to think about them, give important examples,
and talk about some key tools for using them.
**Tam Cheetham-West (Galois rigidity and finite quotients):** I'll talk about the property of Galois rigidity defined by
Bridson, McReynolds, Reid, and Spitler and how they use this property
to show that certain Fuchsian and Kleinian groups are completely
determined (up to isomorphism) by their finite quotients.
**Emily Shinkle (The Flip Graph):** I will introduce the flip graph, a graph measuring
"distance" between different ways of triangulating a surface. We'll
discuss basic properties and fundamental results about the flip graph,
and explore ways the flip graph has been used as a tool for studying
other objects, such as the mapping class group and cluster algebras.
**Shawn Williams (Strong S-Equivalence of Ordered Links):** S-equivalence is an important notion in the study of knots
and links. Any two S-equivalent links will have the same Alexander
module, Blanchfield form, and other abelian invariants. In 1999, Naik
and Stanford related this algebraic notion to a geometric one--showing
that two knots are S-equivalent if and only if you can relate them by
a sequence of certain moves on their knot diagrams. In this talk we
will explore the work of Gee in her 2004 paper of the same title,
presenting a stronger version of this result to ordered links.
**Connor Sell (Cusps of hyperbolic 4-manifolds):**
It's easy to take for granted how well we understand the
cusp cross-sections of hyperbolic 2- and 3-manifolds. In this talk,
I'll survey what is known and what is unknown about the cusps of
cusped hyperbolic manifolds in 4 or more dimensions.
**Will Stagner (Hyperbolic knots in compact 3-manifold)s:**Thurston’s geometrization program demonstrated the
importance of understanding hyperbolic 3-manifolds as fundamental
building blocks in 3-dimensional topology. In this talk, we will
explore a result by Myers that further exemplifies the ubiquity of
hyperbolic structures — that every compact orientable 3-manifold
contains a knot whose complement is hyperbolic. This result can be
used to show that such a 3-manifold is completely determined by its
set of knot groups.
**Sara Edelman-Munoz (The Congruence Subgroup Property in $SL_3(\mathbb(Z))$):**
A group has the congruence subgroup property if all of its finite index subgroups are virtually congruence subgroups. For an arithmetic group, it is interesting to ask whether or not the group has this property. In 1964 Bass, Lazard and Serre showed that $SL_n(\mathbb(Z))$ has the congruence subgroup property when n $\geq$ 3. This was also shown by Mennick in 1965. In this talk we will use Mennicke symbols to show that $SL_3(\mathbb(Z))$ has the congruence subgroup property and discuss the generalization of this proof to $SL_n(\mathbb(Z))$ when $n>3$.
**Alex Nolte (Minimal surfaces in three-manifolds):**In the 1950s, Papakyriakopoulos proved three foundational results in 3-manifold topology on the existence of embedded disks and spheres. Around 1980, Meeks and Yau showed that these results could be strengthened by realizing their conclusions with area-minimizing embedded minimal surfaces. These results have found fruitful application in studying group actions on 3-manifolds. In this talk, we will introduce minimal surfaces, discuss how Meeks and Yau use them to strengthen Dehn's lemma and the loop and sphere theorems, and sketch some applications to topology.
**Nicholas Rouse (Introduction to Burger-Mozes groups):**In this expository talk, we discuss the basic properties and
construction of lattices in the product of the automorphism groups of
two trees. These groups were studied by Burger, Mozes, and others
beginning in the 1990s. Using these techniques, one can construct
groups that are finitely presented, torsion-free, simple, and split as
a free product of free groups amalgamated along a subgroup of finite
index. We also present some examples of these groups and analogies
with Lie theory.
**Alex Manchester: (n)-solvable filtration and satellite operators**This will unintentionally be a sequel to Shawn's CMS talk on Thursday. We will discuss some of the motivation for the definition of the (n)-solvable filtration, what is known about it, and its relationships with satellite operators. We will also at least sketch the proof that every genus 1 algebraically slice knot is (1)-solvable (proved by Davis, Martin, Otto, and Park).
**Kate O'Connor (Milnor's Invariants):**In the 1950s, John Milnor introduced a family of algebraic link invariants known as Milnor’s invariants, which generalize linking number. These roughly measure how deep the longitudes of each link component lie in the lower central series of the link group. In this talk, we will first discuss information we can extract from the lower central series of a link group. Then we will define Milnor’s invariants and discuss some of their properties. Finally, we will give geometric ways of defining certain Milnor’s invariants.