For more information about the Seminar, to suggest speakers, or to volunteer to speak, contact Bahar Acu, Dave Bachman, Sam Nelson, Vin de Silva, or Helen Wong
| Date | Speaker | Title and Abstract |
| Tuesday
September 3rd |
Organizational Meeting | |
| Tuesday
September 10th |
Sam Nelson Claremont McKenna College
|
Title: Biquandle Module Quiver Representations
Abstract: Biquandle module enhancements are invariants of knots and links generalizing the classical Alexander module invariant. A quiver categorification of these invariants was introduced in 2020. In this work-in-progress (joint with Yewon Joung from Hanyang University in Seoul) we take the next step by defining invariant quiver representations. As an application we obtain new polynomial knot invariants ae decategorifications. |
| Tuesday
September 17th |
Migiwa Sakurai Shibaura Institute of Technology
|
Title: Clasp pass moves and arrow polynomials of virtual knots
Abstract: For classical knots, clasp pass moves are closely related to Vassiliev invariants of degree 3. Tsukamoto showed that the values of the Vassiliev invariant of degree 3 induced from the Jones polynomial for two knots differ by 0 or +36/-36, if they are related by a single clasp pass move. For virtual knots, the arrow polynomial is a generalization of the Jones polynomial and induces a Vassiliev invariant of degree 3. We show that the values of the Vassiliev invariant of degree 3 induced from the arrow polynomial of two virtual knots differ by 0 or +2304/-2304, if they are related by a single clasp pass move. We also obtain a lower bound of the distance between virtual knots by clasp pass moves. |
| Tuesday
September 24th |
NO SEMINAR |
|
| Tuesday
October 1st |
Reginald Anderson Claremont McKenna College
|
Title: Presentations of derived categories
Abstract: A modification of the cellular resolution of the diagonal given by Bayer-Popescu-Sturmfels gives a virtual resolution of the diagonal for smooth projective toric varieties and toric Deligne-Mumford stacks which are a global quotient of a smooth projective variety by a finite abelian group. In the past year, Hanlon-Hicks-Lazarev gave a minimal resolution of the diagonal for toric subvarieties of smooth projective toric varieties. We give implications for exceptional collections on smooth projective toric Fano varieties in dimensions 1-4. This is joint work with CMC undergrads Justin Son, Hill Zhang, and Jumari Querimit-Ramirez. |
| Tuesday
October 8th |
NO SEMINAR |
|
| Tuesday
October 15th |
No Meeting | Fall Break |
| Tuesday
October 22nd |
Will Hoffer UC Riverside
|
Title: Tube Formulae for Fractal Snowflakes
Abstract: Fractals like the von Koch snowflake have rough boundaries, often having nowhere defined tangent lines/spaces. However, there is a tool useful for probing the edges of such fractals: tubular neighborhoods. In this talk, we'll introduce the theory of fractal tube formulae which describe the volumes of such tubular neighborhoods, illustrating through our recent work on generalized fractal snowflakes. In the process, we'll touch on the theory of complex dimensions and tubular zeta functions that capture the (multiplicative) oscillations appearing in the geometry of fractals. |
| Tuesday
October 29th |
Orsola Capovilla-Searle UC Davis
|
Title: Exact Lagrangian fillings of Legendrian links
Abstract: An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links arise as wavefronts in optics, and can sometimes be used to distinguish contact structures. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball which are called fillings of the link. In the last seven years, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. I will talk about recent work establishing connections between fillings and Newton polytopes, as well as applications to higher dimensional Legendrian submanifolds and non-orientable fillings. |
| Tuesday
November 5th |
Vijay Higgins UCLA
|
Title: Webs and skein algebras
Abstract: The Jones polynomial of a link can be computed diagrammatically by using skein relations which encode the representation theory of SL(2). By considering the vector space spanned by links drawn on a surface and imposing these skein relations, we obtain an algebra known as the Kauffman bracket skein algebra of the surface. These algebras have been studied by many authors including F. Bonahon and H. Wong, and much is known about their structure. Replacing SL(2) by SL(3) or any other higher rank Lie group gives rise to a new skein algebra involving not only links but also certain graphs called webs. In this talk, we will discuss some of the complications involved with studying skein algebras built from webs on surfaces and then present ways of getting around them. Some of this work is joint with F. Bonahon. |
| Tuesday
November 12th |
Claudio Gomez-Gonzales Carleton College - UC Irvine
|
Title: How hard could it be? A tour of resolvent degree
Abstract: Solving algebraic equations are among the oldest problems in mathematics. In this talk, we offer a concrete, visual, and historical introduction to resolvent degree (RD), an invariant that aspires to quantify just how hard these problems are. The lineage of this theory includes the origins of topology, Klein's "hypergalois" program, and centuries-old exploits in reducing numbers of coefficients, which dare us to push beyond the solvable/unsolvable dichotomy. We will build towards the notion of versality central to Klein's vision, with a nod to our general framework implemented in joint work with Alexander Sutherland and Jesse Wolfson, that permits us to address resolvent questions via classical invariant theory. We will conclude by reflecting on the past and future of resolvent problems, along with what we do and don't know about RD. This talk is designed to be accessible for undergraduates-let's do some math! |
| Tuesday
November 19th |
Heather Lee | Title: Some examples of homological mirror symmetry
Abstract: Mirror symmetry is a duality phenomenon between symplectic geometry and complex geometry. The homological mirror symmetry (HMS) conjecture was originally formulated by M. Kontsevich in 1994 to fully capture this phenomenon for mirror pairs of compact Calabi-Yau manifolds. Since then, it has been extended to cover a much wider range of manifolds. For example, in 2 real dimensions, among the compact Riemann surfaces, the torus is Calabi-Yau, the sphere is Fano, and all others are of general type; in addition, there are punctured Riemann surfaces which are not compact. In this talk, I will present a few illustrative examples of HMS, including ones worked out by others and from my own research. |
| Tuesday
November 26th |
No Meeting | Thanksgiving Week |
| Tuesday
December 3rd |
Rhea Bakshi University of California Santa Barbara
|
Title: TBA
Abstract: TBA |