## 2020 Cascade Topology Seminar (Online)

- Start Date: 09/26/2020
- End Date: 09/27/2020

Online

The Cascade Topology Seminar is a semi-annual conference hosted in the Pacific Northwest and Western Canada. It features topology of all kinds. This year's event will be hosted online.

**Confirmed Speakers:**

**Hannah Alpert (UBC)**

Title: Representation stability and configurations of disks in a strip

Abstract: Representation stability, formalized in 2012 by Church, Ellenberg, and Farb, is a property exhibited by the homology of the configuration space of points in the plane: even as the number of points goes to infinity, the jth homology is generated by cycles in which at most 2j of the points move. What about the configuration space of disks of width 1 in an infinite strip of width w? This disks in a strip space behaves more like the no-k-equal configuration space of the line, where k-1 but not k points may be collocated; we show that the homology of this no-k-equal space exhibits generalized representation stability as defined by Sam–Snowden and Ramos. The method is to compute homology combinatorially using discrete Morse theory. Unlike other examples of homology with generalized representation stability, here the asymptotic behavior depends on the degree of homology.

**Ahmad Issa (UBC)**

Title: Symmetric knots and the equivariant 4-ball genus

Abstract: Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g., K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss ongoing work with Keegan Boyle trying to understand the equivariant 4-genus.

**Kyle Ormsby (Reed College)**

Title: Weak factorization and transfer systems

Abstract: Transfer systems are discrete objects that encode the homotopy theory of N∞ operads, i.e., the operads whose algebras are homotopy commutative monoids with a class of equivariant transfer (or norm) maps. They have a rich combinatorial structure defined in terms of the subgroup lattice of the group of equivariance, G. Indeed, if G is a cyclic p-group, there are Catalan-many transfer systems that assemble into the Tamari lattice (i.e., associahedron). In this talk, I will show that when G is finite Abelian, transfer systems are in natural bijection with weak factorization systems on the poset category of subgroups of G. This leads to a novel involution on the lattice of transfer systems, generalizing an observation of Balchin–Bearup–Pech–Roitzheim for cyclic groups of squarefree order. I will conclude with an enumeration of saturated transfer systems and comments on the Rubin and Blumberg–Hill saturation conjecture.

This is joint work with Angélica Osorno and a team of Reed College undergraduates: Evan Franchere, Usman Hafeez, Peter Marcus, Weihang Qin, and Riley Waugh (the Electronic Collaborative Mathematics Research Group, or eCMRG).

**Sabrina Pauli (University of Oslo)**

Title: Enumerative geometry via the A^1-degree

Abstract: Morel's A^1 -degree in A^1 -homotopy theory is the analog of the Brouwer degree in classical topology. It takes values in the Grothendieck-Witt ring GW(k) of a field k, that is the group completion of isometry classes of non-degenerate symmetric bilinear forms. We can use the A^1 -degree to count algebro-geometric objects in GW(k), giving an A^1 -enumerative geometry over non-algebraically closed fields. Taking the rank and the signature recovers classical counts over the complex and the real numbers, respectively. For example, the count of lines on a smooth cubic surface enriched in GW(k) has rank 27 and signature 3.

**Andrew Lobb (Durham University)**

Title: The smooth rectangular peg problem

Abstract: For any smooth Jordan curve and rectangle in the plane, we show that there exist four points on the Jordan curve forming the vertices of a rectangle similar to the given one. Joint work with Josh Greene.

**Maggie Miller (MIT****) **

Title: Characterizing handle-ribbon knots

Abstract: Kauffman conjectured that a knot K is slice if and only if it bounds a genus-g Seifert surface containing a g-component slice link as a cut system. It’s very easy to show that a knot is ribbon if and only if it bounds a genus-g Seifert surface containing a g-component unlink as a cut system. Alex Zupan and I proved something in the middle of these statements: a knot is handle-ribbon (aka strongly homotopy-ribbon, aka something I will define in the talk) if and only if it bounds a genus-g Seifert surface containing a g-component R link L as a cut system—meaning that zero-surgery on L yields #_ g S^1 × S^2 . This gives a 3-dimensional definition of a 4-dimensional property. I’ll talk about these 3.5D knot properties and maybe how we use these techniques to extend a statement of Casson and Gordon. (The work in this talk is joint with Alexander Zupan from the University of Nebraska–Lincoln.)

**Session Times:**

The conference runs from 9:00AM - 1:00PM Pacific Daylight Time

**Registration**:

This conference is free, though pre-registration is required so that a personalized link to the event can be sent to you. ** Please register here.** Registrations will close at 4:00pm on Wednesday Sept 23rd, and the Zoom details sent to particpants there after.

Please contact the organizers here if you would like to join and did not complete the registration form.

**Saturday, ****26th**** ****September****:**

9:00AM Andrew Lobb

10:30AM Sabrina Pauli

12:00PM Ahmad Issa

**Sunday, ****27th**** ****September****:**

9:00AM Maggie Miller

10:30AM Hannah Alpert

12:00PM Kyle Ormsby

Ben Williams, UBC

Liam Watson, UBC

Jeffrey Meier, Western Washington University

The 2020 Cascade Topology Seminar is suported by the PIMS CRG in Novel Techniques in Low Dimension.