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: The skein module of the connected sum of two copies of L(0,1) Abstract: Skein modules were introduced by Jozef H. Przytycki, and independently by Vladmimor Turaev, as generalisations of the Jones, Kauffman bracket, and HOMFLYPT polynomial link invariants in the 3-sphere to arbitrary 3-manifolds. The Kauffman bracket skein module (KBSM) is the most extensively studied of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially over the ring of Laurent polynomials. Marche conjectured that over the ring of Laurent polynomials, the KBSM of closed oriented 3-manifolds splits into the sum of free and torsion modules. The counterexample to this conjecture is given by the connected sum of two copies of the real projective space. With the goal of finding a definite structure of the KBSM over this ring, we compute the skein module of S^1 x S^2 # H_1 and S^1 x S^2 # S^1 x S^2. We show that it is isomorphic to the KBSM of a genus two handlebody modulo some specific handle sliding relations. Moreover, these handle sliding relations can be written in terms of Chebyshev polynomials. We also discuss whether the KBSM of these manifolds splits into the sums of free and torsion modules. This is joint work with Seongjeong Kim, Shangjun Shi, and Xiao Wang.

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