# Founded 1989

The Claremont Topology Seminar meets on Tuesday's from 3:00-4:00 pm in Millikan 2099, Pomona College. Millikan is located on the NE corner of College Avenue and 6th Street. Following the talk we go for coffee/tea in downtown Claremont. Sometimes we meet a little later and then go to dinner. (Look for special times on the calendar.) Click here for a map of the Pomona Campus.

To reach Pomona College from the 10 freeway, exit at Indian Hill, go north, turn right (east) on 6th street and then turn left (north) on College. To reach Pomona College from the 210 freeway, if traveling East, exit at Towne Avenue, turn right (south) on Towne, turn left (east) on Foothill Blvd, and turn right (south) onto College Avenue. If traveling West on the 210 freeway, exit at Baseline/Padua, turn right (west) onto Baseline, turn left (south) onto Padua at the first light, turn right (west) onto Foothill Blvd at the third light, turn left (south) onto College Avenue.

Parking on College Avenue is free.

For more information about the Seminar, or to suggest speakers, contact Jim Hoste , Dave Bachman , Sam Nelson , Erica Flapan or Vin de Silva.

# Fall, 2017

special days, times or locations are in red

 Date Speaker Title and Abstract Tuesday Sept 5, 2017 3:00 pm Organizational Meeting Meet at Some Crust Bakery for organizational meeting. Tuesday Sept 12, 2017 3:00 pm Title: Abstract: Tuesday Sept 19, 2017 3:00 pm Allison K HenrichSeattle University Title: An Intransitive Relation on Knots Abstract: In this talk, we will introduce and explore the following relation on knots. A knot D is said to be a descendant of another knot P if there is a minimal crossing diagram of P on which some subset of crossings can be changed to produce a diagram of D. In this case, P is said to be a parent of D. The descendant-parent relation has many interesting properties--for instance, intransitivity--and it enjoys useful connections with other previously studied relations on knots. We explore several such properties and connections, and we provide a variety of computational results. This is joint work with Jason Cantarella, Elsa Magness, Oliver O’Keefe, Kayla Perez, Eric Rawdon, and Briana Zimmer. Tuesday Sept 26, 2017 3:00 pm Sam Nelson Claremont McKenna College Title: Psyquandles, Singular Knots and Pseudoknots Abstract: Singular knots are 4-valent spatial graphs considered up to rigid vertex isotopy. Pseudoknots are knots with some precrossings, classical crossings where we don't know which strand is on top. Psyquandles are a new algebraic structure which defines invariants of both singular and pseudoknots. In particular we will define the Jablan Polynomial, a generalization of the Alexander polynomial for singular/pseudoknots arising from psyquandles. This is joint work with Natsumi Oyamaguchi (Shumei University) and Radmila Sazdanovic (NCSU). Tuesday Oct 3, 2017 3:00 pm Ellie Dannenberg Pomona College Title: Circle Packings on Surfaces with Complex Projective Structures. Abstract: The classical circle packing theorem of Koebe, Andreev, and Thurston says that given a triangulation $\tau$ of a closed, orientable surface, there is a unique constant curvature Riemannian metric on the surface so that the surface with this metric admits a circle packing with dual graph $\tau$. Kojima, Mizushima, and Tan give a definition of a circle packing on surfaces with complex projective structures. Unlike in the metric case, there is a deformation space of complex projective circle packings with combinatorics given by $\tau$. They conjecture that this space is homeomorphic to Teichmuller space. I'll present progress towards this conjecture for certain classes of triangulations. Abstract: Tuesday Oct 10, 2017 3:00 pm Teddy Einstein Cornell University Title: Hierarchies of Non-Positively Curved Cube Complexes Abstract: A non-positively curved (NPC) cube complex is a combinatorial complex constructed by gluing Euclidean cubes along faces in a way that satisfies a combinatorial local non-positive curvature condition. A hierarchy is an inductive method of decomposing the fundamental group of a cube complex. Cube complexes and hierarchies of cube complexes have been studied extensively by Wise and feature prominently in Agol's proof of the Virtual Haken Conjecture for hyperbolic 3-manifolds. Abstract: Tuesday Oct 17, 2017 3:00 pm No Meeting Fall Break Monday Oct 23, 2017 3:00 pm John Stillwell University of San Francisco History of Mathematics Seminar Tuesday Oct 24, 2017 3:00 pm Aaron Mazel-Gee USC Title:K-theory and traces Abstract: A vector bundle over a topological space X is a family of vector spaces parametrized by X. The study of spaces through their vector bundles is called topological K-theory. If the space has a more algebraic flavor, then we can also study its algebraic K-theory; for example, on a complex manifold we can restrict to holomorphic vector bundles. As K-theory can be very difficult to compute, we are quite lucky to have an assortment of trace maps'' (in the sense of traces of matrices) that compare it to more computable invariants. In particular, the cyclotomic trace is responsible for the vast majority of computations of algebraic K-theory. However, despite its importance, there has been no geometric basis for the cyclotomic trace: it has just served as an extremely useful but conceptually unmotivated tool. In this talk, I will give an overview of K-theory and traces, with the end goal of explaining forthcoming joint work with David Ayala and Nick Rozenblyum that provides a precise geometric explanation of the cyclotomic trace. Tuesday Oct 31, 2017 3:00 pm Iris Yoon Title : Persistence of Sheaf Cohomology: Studying Data Evolving Over Time Abstract: Persistent homology is a widely used tool in Topological Data Analysis (TDA) that infers topological features from data. I will discuss the extension of persistence to cellular sheaf cohomology. Cellular sheaves are useful tools for extracting global structure from local data and relations. By virtue of being cellular, one can take advantage of sheaf theory in a computable manner. Persistent cellular sheaf cohomology then allows one to examine global changes that result from local changes of time-varying data. While there are many applications of persistent sheaf cohomology to be explored, I will focus on its application to distributed computation of persistent homology. It turns out, such distributed computation can be useful in studying multi-density data by recovering information that gets lost by persistent homology. Tuesday Nov 7, 2017 3:00 pm Radmila Sazdanovic North Carolina State University Title:Patterns in Khovanov link and chromatic graph homology Abstract: Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. In this talk, we discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams. Tuesday Nov 14, 2017 3:00 pm Dmitri Gekhtman Cal Tech Title: Holomorphic retractions onto Teichmuller disks. Abstract: A complex manifold is a space built by gluing patches of complex coordinate coordinate space using holomorphic transition maps. There is a finite-dimensional space of ways to equip a closed surface with the structure of a complex manifold. This space of complex structures is called the Teichmuller space of the surface. It turns out that the Teichmüller space is itself a complex manifold. In this talk, we will see how the topology and geometry of a surface is reflected in the complex-analytic structure of its Teichmuller space. We will see how polygonal decompositions of the surface give rise to holomorphically embedded copies of the unit disk inside of Teichmuller space. These embedded disks are called Teichmuller disks. We will describe some recent results which give partial answers to the following question: Which Teichmuller disks are holomorphic retracts of Teichmuller space? Tuesday Nov 21, 2017 3:00 pm Yen Duong Title: Random groups and cubulations Abstract: Special cube complexes have been a hot topic for a few years now for their versatility and usefulness. Following a construction of Sageev we can cubulate all sorts of groups, and specifically I'll explain Gromov's density model of random groups, the square model of random groups, and examples of Sageev's cubulation with random groups, and why we would want to do all this. Tuesday Nov 28, 2017 3:00 pm Title: Abstract: Tuesday Dec 5, 2017 3:00 pm Dave Bachman Pitzer College Title: Applied Mathematics problems in art and design Abstract: We typically don't think of art and design as settings for applied mathematics. However, advanced mathematics can be a powerful tool for the artist who is willing to use it. In this talk, we'll see examples where knowledge from advanced areas such as differential geometry, linear algebra, and differential equations were crucial to the design process. We'll also see cases of the converse, where design problems led to interesting mathematical questions.

# Spring 2018 Schedule

special days, times or locations are in red

 Date Speaker Title and Abstract Tuesday Jan 16, 2018 3:00 pm Edgar BeringTemple University Title: Uniform twisting in mapping class groups and outer automorphisms of free groups Abstract: There is a long running analogy between the mapping class group of a surface and the outer automorphism group of a free group. In both settings there is a notion of Dehn twist. It is well-known that in a mapping class group, two Dehn twists about intersecting curves have powers that generate a free group. Hamidi-Tehrani showed that fourth powers suffice to guarantee a free group; Leininger and Margalit conjecture that taking squares should suffice. In this talk I will present Hamidi-Tehrani's result, and then my analogous theorem for Dehn twists of a free group. The situation for free groups is complicated by a lack of geometry; this is compensated for with combinatorics. To prevent the talk from being overly technical, all of the results will be presented only in the case of simple twists, though they hold in greater generality. Tuesday Jan 23, 2018 3:00 pm Title: Abstract: Tuesday Jan 30, 2018 3:00 pm Title: Abstract: Tuesday Feb 6, 2018 3:00 pm Sam Nelson Claremont McKenna College Title: Twisted Virtual Bikeigebras Abstract: Twisted virtual handlebody-links are regular neighborhoods of trivalent graphs embedded in manifolds of the form $\Sigma times [0,1]$ where $\Sigma$ is a compact surface which may or may not be orientable. In this talk we will introduce combinatorial moves for representing these knotted objects and describe an algebraic structure called twisted virtual bikeigebras for distinguishing them by counting colorings. This is joint work with former CMC student Yuqi Zhao. Tuesday Feb 13, 2018 3:00 pm Title: Abstract: Tuesday Feb 20, 2018 3:00 pm Title: Abstract: Tuesday Feb 27, 2018 3:00 pm Daniel Douglas USC Title: Quantum Traces for Fock-Goncharov Coordinates Abstract: We describe work-in-progress generalizing the SL_2 quantum trace map of Bonahon and Wong (2010) to the case of SL_n. The SL_2 quantum trace is a homomorphism from the Kauffman bracket skein algebra of a punctured surface to a certain noncommutative algebra which can be thought of as a quantum Teichmuller space. The construction is modeled on the classical trace of monodromies of hyperbolic structures on surfaces. Our current work focuses on SL_3, where convex projective structures play the central role, as developed by Fock and Goncharov. Another distinction is the appearance of the HOMFLY-PT skein algebra in place of the Kauffman bracket skein algebra. Tuesday Mar 6, 2018 3:00 pm Neslihan Gugumcu National Technical University of Athens. Title: On knotoids, braidoids and topology of proteins Abstract: In this talk, we first remember the basics of knotoid theory and overview the previous results on the height of knotoids. We then discuss on the applications of knotoids to the study of proteins and see that knotoids indeed provide a finer topological analysis of proteins. Lastly, we introduce braidoids that are geometric objects analogous to classical braids, forming a "braided" counterpart theory to the theory of knotoids. *The talk contains three main parts which are joint works with Kauffman, Kauffman and Lambropoulou and Stasiak, and Lambropoulou, respectively. Tuesday Mar 13, 2018 3:00 pm No Meeting Spring Break Tuesday Mar 20, 2018 3:00 pm Eleni Panagiotou UCSB Title:Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and VIscoelasticity Abstract: We draw on mathematical results from topology to develop quantitative methods for polymeric materials to characterize the relationship between polymer chain entanglement and bulk viscoelastic responses. We generalize the mathematical notion of the linking number and writhe to be applicable to open (linear) chains. We show how our results can be used in practice by performing fully three-dimensional computational simulations of polymeric chains entangled in weaves of a few distinct topologies and with varying levels of chain densities. We investigate relationships between our topological characteristics for chain entanglement and viscoelastic responses by performing Lees-Edwards simulations of the rheology over a broad range of frequencies. Our topological measures of entanglement indicate that the global topology is the dominant factor in characterizing mechanical properties. We find an almost linear relation between the mean absolute writhe and the loss tangent and an almost inverse linear relation between the mean absolute periodic linking number and the loss tangent. These results indicate the potential for our topological methods in providing a characterization of the level of chain entanglement useful in understanding the origins of mechanical responses in polymeric materials. Tuesday Mar 27, 2018 3:00 pm Title: Abstract: Tuesday Apr 3, 2018 3:00 pm Title: Abstract: Tuesday Apr 10, 2018 3:00 pm Tuesday Apr 17, 2018 3:00 pm Title: Abstract: Tuesday Apr 24, 2018 3:00 pm Nancy Scherich UCSB Title: An Application of Salem Numbers to Braid Group Representations Abstract: Many well known braid group representations have a parameter. I will show how to carefully choose evaluations of the parameter to force the representation to land in a lattice. It is surprising and exciting to see how careful algebraic constructions can lead to geometric results. Tuesday May 1, 2018 3:00 pm Christine LeeUT Austin Title:Title: A knot with no tail Abstract: In this talk, we will discuss the stability behavior of the U_q(sl(2))-colored Jones polynomial, a quantum link invariant that assigns to a link K in S^3 a sequence of Laurent polynomials {J_K^n(q)} from n=2 to infinity. The colored Jones polynomial is said to have a tail if there is a power series whose coefficients encode the asymptotic behavior of the coefficients of J_K^n(q) for large n. Since Armond and Garoufalidis-Le proved the existence of a tail for the colored Jones polynomial of an adequate knot, first conjectured by Dasbach-Lin, it has been conjectured that multiple tails exist for all knots. Moreover, the stable coefficients of the tail have been shown to relate to the topology and the geometry of the alternating link complement, prompting the Coarse Volume Conjecture by Futer-Kalfagianni-Purcell. I will talk about an unexpected example of a knot, recently discovered in joint work with Roland van der Veen, where the colored Jones polynomial does not admit a tail, and discuss potential ways to view this example in the context of the categorification of the polynomial, the aforementioned Coarse Volume Conjecture, and a general conjecture made by Garoufalidis-Vuong concerning the stability of the colored Jones polynomial colored by irreducible representations of Lie algebras different from U_q(sl(2)).

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