Topology Seminar Archives

Fall 2017
 
Speaker: David Penneys (Ohio State University)
Time: Monday, November 13, 2017, 12:00-1:00

Title: Monoidal categories enriched in braided monoidal categories

Place: Rome 771

Abstract: While the symmetries of classical mathematical objects form groups, the symmetries of ‘quantum’ mathematical objects such as subfactors and quantum groups form more general objects which are best axiomatized as monoidal categories. Early in the study of monoidal categories, Eilenberg and Kelly defined the notion of a category enriched in a monoidal category. Recently, there has been a lot of interest in super monoidal categories, which are enriched in super vector spaces. These enriched categories are examples of monoidal categories enriched in symmetric monoidal categories. In a recent article with Morrison (arXiv:1701.00567), we introduce the notion of a monoidal category enriched in a braided monoidal category V, which is not assumed to be symmetric. We then classify V-monoidal categories in terms of strictly unital oplax braided monoidal functors from V to the centers of ordinary monoidal categories

Logic-Topology Seminar

Title:   Mathathon II: Search for Interesting Torsion in Khovanov Homology

Speaker:    Jozef Przytycki, GWU

http://home.gwu.edu/~przytyck/
Date and Time: Thursday, October 5, 2017, 2:30-3:30pm

Place: Rome Hall (801 22nd Street), Room 771

Abstract:  We will describe the work of our Mathathon group (Sujoy Mukherjee, Marithania Silvero, Xiao Wang, Seung Yeop Yang), Dec. 2016–Jan. 2017, on torsion in Khovanov homology different from Z_2. Khovanov homology, one of the most important constructions at the end of XX century, has been computed for many links. However, computation is NP-hard and we are limited to generic knots of up to 35 crossings with only some families with larger number of crossings. The experimental data suggest that there is abundance of Z_2-torsion but other torsion seems to be rather rear phenomenon. The first Z_4 torsion appears in 15 crossing torus knot T(4,5), and the first Z_3 and Z_5 torsion in the torus knot T(5,6). Generally, calculations by Bar-Nathan, Shumakovitch, and Lewark suggest Z_p^k torsion in the torus knot T(p^k,p^k+1), p^k >3, but this has not yet been proven. We show, with Mathathoners, the existence of Z_n-torsion, n>3, for some infinite family of knots.  The simplest of them is obtained by deforming the torus knot T(5,7) by a t_2k-moves. We also prove the existence of knots with other torsion, the largest being Z_2^23, so the cyclic group of over 8 millions elements. We combine computer calculations (and struggle with NP hardness) with homological algebra technique.

The talk will be elementary and all needed notions will be defined.


Date: Wednesday, September 6, 2017
Time:2:30PM – 3:30PM
Place: Rome 771
Speaker: Xiao Wang, GWU
Title:  Introduction to Khovanov homology and Hochschild homology

 
Fall 2016
 
 
 
 
 
 
Title: Stability in Knot Polynomials 

Speaker:  Katherine Walsh (University of Arizona) 

Date and Time:  November 10, 2016Thursday6:15 -7:15 pm

Place:   Phillips 736

Abstract: The colored Jones polynomial assigns to each knot a sequence of Laurent polynomials.  This talk will start with a discussion on the patterns and stability in the coefficients of these polynomials.  While much work has been done looking at the leading sequence of coefficients, we will discuss work moving towards understanding the middle coefficients.  This will include small steps like looking at the second N coefficients of the Nth colored Jones polynomial and larger steps like looking at the growth rate of thecoefficients and looking the the patterns present in the coefficients under various re-normalizations. We will also touch on new work looking at related stability in different sequences.


Title: New potential counterexamples to the Generalized Property R Conjecture

Speaker:  Alexander Zupan (University of Nebraska–Lincoln)

Date and Time:  September 13, 2016Tuesday6:15 -7:15 pm

Place:   Rome 771

Abstract:  Kirby Problem 1.82 conjectures a characterization of n-component links in the 3-sphere which have a Dehn surgery to the connected sum of n copies of S^2 X S^1. This conjecture generalizes Property R, proved by Gabai in the late 1980s.  In 2010, Gompf, Scharlemann, and Thompson offered an infinite family of 2-component links which are potential counterexamples to the Generalized Property R Conjecture.  For each n, we give an infinite family of n-component possible counterexamples to the conjecture.  Notably, these links are connected to the famous Slice-Ribbon Conjecture and can be used to produce slice knots which do not appear to be ribbon. This talk is based on joint work with Jeff Meier.


Title: Playing with homotopy: Does Khovanov lead to bouqets of spheres in polynomial time?

Speaker:  Jozef Przytycki, GWU

Date and Time:  September 7, 2016Tuesday5:30 -6:30 pm

Place:   Rome 771

Abstract:  The motivation for my talk and related research comes from the confluence of two fascinating recent developments: Khovanov homology in knot theory
and homological algebra and pseudo-knots in RNA folding theory in computational biology.
 

Spring 2016
 
Speaker: Leonid Chekhov (Steklov Math Inst, Moscow, and Niels Bohr Inst., Copenhagen)

Title: Quantum cluster algebras and geometry of string worldsheet
Date andTime: Tuesday, May 10, 2016, 5:00-6:00 pm
Place: Phillips 730

Abstract: 

We describe the Teichmuller space $T_{g,s,n}$ of Riemann surfaces of genus g, s holes and n bordered cusps on boundaries of holes in the Poincare uniformization of constant negative curvature. The description uses structures on fat graphs, observables are geodesic functions and we derive Poisson and quantum structures on sets of observables relating them to quantum cluster algebras of Berenstein and Zelevinsky. We obtain MCG-invariant quantum Ptolemy transformation and quantum skein relations. We propose to associate coordinates of  $T_{g,s,n}$ (the Thurston shear coordinates and the Penner lambda-lengths) with the coordinates of a string worldsheet in which bordered cusps naturally correspond to open string asymptotic states. (joint with M.Mazzocco, Loughborough Univ., UK)


Speaker: Adam S. Sikora (SUNY at Buffalo)

Title: Skein algebras of surfaces
Date and Time: Tuesday April 19, 2016; 5:00-6:00
Place: Phillips 108


Abstract: 

We show that the Kauffman bracket skein algebra of any oriented surface F has no zero-divisors and that its center is generated by knots parallel to the boundary of F.
Furthermore, we generalize the notion of skein algebras to skein algebras of marked surfaces and we prove analogous results them.
Our proofs rely on certain filtrations of skein algebras induced by pants decompositions of surfaces and by the associated Dehn-Thurston intersection numbers.

It is a joint work with Jozef Przytyck


Speaker: Jim Hoste, Pitzer College

Title: Links with finite n-quandles
Date and Time: Wednesday, April 13, 6:15-7:15pm
Place: Phillips 704

Abstract: Associated to every knot is its fundamental quandle Q(K), which Joyce proved is a complete knot invariant. A somewhat more tractable, but less sensitive invariant is the n-quandle, a quotient of Q(K) defined for every natural number n. I will describe these quandles and show that the n-quandle of a knot is isomorphic to the set of cosets of the peripheral subgroup of a certain quotient of the fundamental group of the knot. This characterization proves a conjecture of Przytycki: The n-quandle of a knot is finite if and only if the fundamental group of the n-fold cyclic cover of S^3 branched over the knot is finite. I will outline a program to catalog all finite quandles that appear as n-quandles of some knot or link. Some of this is joint work with Pat Shanahan.


Speaker: Zhiyun Cheng (Beijing Normal University, and GWU)

Title: When is region crossing change an unknotting operation?
Date and Time: Wednesday, March 23, 4:45- 5:45pm
Place: Phil 109

Abstract: In recent years, more and more android/ios puzzle games are designed based on mathematics. For example, Region Select, a puzzle game which was developed by Akio Kawauchi, Ayaka Shimizu and Kengo Kishimoto, is based on the study of a local operation (region crossing change) on link diagrams. In this talk I will discuss the mathematical background behind this game.


Speaker: Marithania Silvero Casanova (Universidad de Sevilla, Spain)

Title: On extreme Khovanov homology and its Geometric Realization
Date and Time: Wednesday, March 96:00-7:00pm
Place: Phillips Hall #704

Abstract: In this talk we present a new approach to extreme Khovanov homology in terms of a specific graph constructed from the link diagram. With this point of view, we review a well-known conjecture related to the existence of torsion in extreme Khovanov homology. We discuss in more detail the more general conjecture that  Independence simplicial complex of (bipartite) circle graph is a wedge of spheres. We show that conjecture holds in many special cases, e.g. for trees, permutation graphs or graphs which are wedge sums of polygons. 


Fall 2015

Title:Real enumeration problems.

Speaker:  Prof. Oleg Viro, Stony Brook University
Date and Time: Tuesday, September 22,  5:30--6:30pm
Place: Monroe 252

Abstract:

We will consider problems of mixed setup, in which the initial object belongs to the elementary differential geometry (say, a smoothly immersed generic planar or spherical curve) and we are counting with certain weights the simplest algebraic curves in special position to the original curve. Say, bitangent lines or tritangent circles. The resulting quantity happens to be a topological invariant of the curve, which can be calculated combinatorially.


Title:Applications of Spectral Sequence arising out of a Bicomplex


Speaker: Harpreet Bedi - GWU
Date and Time: Thursday, September 24,  5:30--6:30pm
Place: Monroe 350

Abstract: 

We will carefully construct the bicomplex lying in the first quadrant and then state general results arising from application of spectral sequence to this bicomplex.
After that specific examples would be constructed e.g Universal Coeff. Theorem, 5-Lemma, Balancing Tor etc. (time permitting)

Title:Torsion in rack homology

Speaker: Seung Yeop Yang - GWU
Date and Time: Thursday, September 29,  5:30--6:30pm
Place: Monroe 252

Abstract: 

  It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order (see Journal of Pure and Applied Algebra, 2015). But it does not hold for connected quandles in general. In this paper, we define an m-almost quasigroup quandle which is a generalization of a quasigroup quandle and study annihilation of its rack and quandle homology groups. We then construct some families of connected quandles for which torsion subgroups of rack and quandle homology are annihilated by the orders of their quandle inner automorphism groups.

Title: Homology Spectral Sequence from Filtered Complex.


Speaker: Harpreet Bedi, George Washington University
Date and Time: Tuesday, October 13, 5:30pm-6:30pm
Place: Monroe 252

Abstract: In this talk we introduce filtrations and then prove construction and convergence of homology spectral sequence.


Title: Homology Spectral Sequence
 
Speaker: Harpreet Bedi, George Washington University
Date and Time : Tuesday, October 27,  2:20PM -3:20 PM
Place: Monroe 267
 
Abstract:
 
In this talk we show the construction of the homology spectral sequence and prove its existence and convergence.
 

Title:Roping more by ringing less in topology
http://insaindia.org/detail.php?id=N99-1262

Speaker: Ajit Iqbal Singh, Indian National Science Academy
Date and Time: Wednesday, November 4, 5:30-6:30pm
Place: Monroe 267
 
Abstract:

We shall present Kakeya’s interval-filling sequences, Liapunov’s convexity theorem for finite-dimensional vector-valued measures and Hobby-Rice theorem for integrals of finitely many continuous functions on sub-intervals together with applications to fair division. The concept of division in rings of continuous functions will be introduced via module homomorphisms.


Title: Khovanov homology, independence complex and H-thick knots.

Speaker:  Marithania Silvero   (Universidad de Sevilla, Spain and GWU)
Date and TimeThursday December 3, 5:30 -- 6:30pm
Place:   Corcoran Hall 101

Abstracthttp://at.yorku.ca/cgi-bin/abstract/cblw-16


Title:Homotopy Theory of Link Homology via the Dold-Kan Theorem 

Speaker: Louis H Kauffman,  UIC
Date and Time: Monday, December 7, 11:00-1:00pm
Place: Monroe 267

Abstract:

The Dold-Kan construction in simplicial homotopy theory can be applied to convert link homology theories into homotopy theories. We construct a mapping  F: L -----> S taking link diagrams L to a category of simplicial spaces S such that up to looping or delooping, link diagrams related by Reidemeister moves will give rise to homotopy equivalent  simplicial objects, and the homotopy groups of these objects will be equal to the link homolgy groups of the original link homology theory. The construction is independent of the particular link homology theory, applying equally well to Knovanov Homology and to Knot Floer Homology and other theories of these types. The construction is of particular interest for Khovanov Homology where there is a natural pre-simplicial structure already present in the definition of the Khovanov category. This allows us to define an embedding of the cube category into a simplicial category so that the map to Frobenius algebras determined by a knot or link produces a simplicial module. The homology of this simplicial module is Khovanov homology. The homotopy type of this simplicial module is the homotopy type to which we refer above.
 

Spring 2015

Title: Polygonal gluings: classic problems and computational biology applications

Speaker: Nikita Alexeev (GWU)
Time:
 Tuesday, April 7, 4:45-6:00pm
Place: Philips Hall 348

Abstract:

In the first part of my talk I will address the following enumerative problem: How many ways are there to glue a 2k-gon into a genus g surface? It is relatively easy to enumerate these ways for a small k, for example, if k=2, there are two ways to obtain a sphere and one way to obtain a torus. It turns out, that this problem is related to Theory of Random Matrices: evaluating corresponding moments of eigenvalue distribution of Gaussian random matrices gives a solution for our problem. In the talk I am going to elaborate on this relation and give a short sketch of the proof. There are several variations of the polygon gluing problem with some restrictions, and in the second part of my talk I will consider one of
such variation, which is applicable in Computational Biology. Namely, a glued graph, embedded into a surface, is a dual (in some specific sense) to the so-called Breakpoint graph, which is used in Bioinformatics to evaluate "evolutionary" distances between genomes.

I am going to explain, what an evolutionary distance is and how a breakpoint graph can help evaluate it. I am also going to show the application of a topological recursion to compute evolutionary distance distribution by utilizing properties of the breakpoint graph.


Title: Introduction to Lie Groups, Lie algebras and their representations
Speaker: Prasad Senesi (The Catholic University of America)
Time: Thursday, April 9, 4:45-6:00pm
Place: Philips Hall 348

Abstract: 
I will give an introductory talk concerning Lie groups and algebras, and conclude with some some comments about my own work.


Topology and History of Science Seminar

Title: A Knot's Tale: Three great men, two smoking boxes, one brilliant wrong idea.
Speaker: Julia Collins (Edinburgh)
Time: April 13, 2015, 11:00-12:00
Place: Monroe Hall 251

Abstract:
I will tell the story of three best friends in 19th century Scotland and their attempt to develop an atomic theory based on knots and links. Tait, Kelvin and Maxwell were inspired by a fantastic experiment involving smoke rings, and their theories, whilst being completely wrong, inspired a new field of mathematical study which is once again becoming important in physics, chemistry and biology.
https://www.youtube.com/watch?v=SwG_uRuYkhg


Title:    Invariants of finite type for curves on surfaces revisited.
Speaker:  Oleg Viro, Stony Brook University
Time: Tuesday, May 12, 4:00-5:00pm
Place:Monroe 250

Abstract:

 Curves on surfaces and curves in the 3-space form stratified spaces which
 have many similar properties and are closely related to each other, but
 the relations are not yet well understood. Curves on surfaces are more
 complicated objects than curves in the space. In particular, the natural
 stratifications of the spaces of curves and finite type invariants are
 more complicated in the case of curves on surfaces. We will consider these
 phenomena in the simplest non-trivial situations.


 

Speaker: Carl Hammarsten (GWU)
Time: 3:45 - 5:00 Friday, February 27, 2015
Place: Monroe 250
Title: Combinatorial Heegaard Floer Homology and Decorated Heegaard Diagrams.

Abstract: Heegaard Floer homology is a collection of invariants for closed oriented three-manifolds, introduced by Ozsvath and Szabo in 2004. The simplest version is defined as the homology of a chain complex coming from a Heegaard diagram of the three manifold. In the original definition, the differentials count the number of points in certain moduli spaces of holomorphic disks, which are hard to compute in general.

More recently, Sarkar and Wang (2008) and Ozsvath, Stipsicz, and Szabo (2010) have determined combinatorial methods for computing this homology with Z_2 coefficients. Both methods rely on the construction of very specific Heegaard diagrams for the manifold, which are generally very complicated.

We introduce the idea of a decorated Heegaard diagram. That is, a Heegaard diagram together with a collection of embedded paths satisfying certain criteria. Using this decorated Heegaard diagram, we present a combinatorial definition of a chain complex which is homotopically equivalent to the Heegaard Floer one, yet significantly smaller.

 


Speaker: Jing Wang (GWU)
Title: Homology of Small Categories and Its Applications to Quiver Cohomology: I
Time: Friday January 30, 3:45pm -5:00pm

Place:  Monroe Hall 267

Abstract:

Motivated from knot theory, we introduce a homology theory for small categories with functor coefficients. Under this general framework, different familiar homology theories such as group homology, chromatic homology, poset homology and Khovanov homology can be realized as homology of small categories whose coe fficients are specifed functors. For

the category of an abstract simplicial complex, we dene chain groups via two di fferent approaches and prove that these two definitions are equivalent in the sense that homology groups under these two defnitions are isomorphic via an interpretation of barycentric subdivision. As an application, we develop cohomology theory for quivers (directed graphs).
We introduce quiver cohomology for non-commutative algebras motivated by Wagner and Turner's work. We analyze and speculate on properties of the quiver cohomology groups via some calculations.
 

Fall 2014
 
Speaker: Evgeny Fominykh (Chelyabinsk State University)
Title: On complexity of 3-manifolds
Time: Tuesday 5:00-6:00Nov. 4, 2014
Place: Monroe HallSeminar Room 267

AbstractThe most useful approach to a classication of 3-manifolds is the complexity theory founded by S. Matveev. Unfortunately, exact values of complexity are known for few infinite series of 3-manifold only. We present the results (joint with Andrei Vesnin) on complexity for two infinite series of hyperbolic 3-manifolds with boundary.
 

Speaker: Carl Hammarsten (GWU)
Title: Combinatorial Heegaard Floer Homology and Decorated Heegaard Diagrams.
Time: 1:00-2:00pm, Oct. 31
Abstract: We introduce the idea of a decorated Heegaard diagram. That is, a Heegaard diagram together with a collection of embedded paths satisfying certain criteria. Using this decorated Heegaard diagram, we present a combinatorial definition of a chain complex which is homotopically equivalent to the Heegaard Floer one, yet significantly smaller. Finally, we use branched spines to show that every manifold admits a decorated Heegaard diagram. Moreover, these diagrams exhibit some unexpected useful extra properties.
 

Speaker: Jing Wang(GWU)
Title:Homology of a small Category and its application to Quiver homology

Time: 2:30-3:30pm, Oct 31
Abstract: Motivated from homology of an abstract simplicial complex, we introduce homology of a small category with functor coefficients. Several homology theories (e.g group homology, Khovanov homology, chromatic homology, etc) can be realized under this framework. If time allows, I will describe its application to quiver homology.


Speaker: Harpreet Bedi (GWU)
Title: Introduction to spectral sequences
Time: Friday, September 12, 2014, 1:00 - 2:00pm
Place: Monroe Hall, Seminar Room 267
 


Logic -Topology Seminar

Speaker: Przytycki, Jozef (GWU)
Title: Simplicial modules, quantum plane and q-polynomial of rooted trees
Time: Friday, September 12, 2014, 2:30 - 3:30pm
Place: Monroe Hall, Seminar Room 267


Abstract: For the 30th anniversary of the Homflypt polynomial of links, I propose a new polynomial invariant of rooted trees. I will relate this to the Kauffman bracket (version of the Jones polynomial) and to (pre)simplicial categories. 


Speaker: Maciej Borodzik (Warsaw University)
Title: Gordian distance of torus knots
Time: Monday December 1, 5:15-6:15pm

Place: Probably seminar room, Monroe 267

Abstract: We study various bounds for the Gordian distance of torus knots (more generally, for algebraic knots or L--space knots) coming from Heegaard--Floer theory. We compare the bounds obtained independently by several authors, explain similarities and differences.


Summer 2014

Title: HOMFLY-PT Complexes in the Knot Floer Cube of Resolutions
Speaker: Nathan Dowlin, Princeton University*
Time: Thursday, June 26, 2014, 2:15 - 3:15pm
Place: Monroe Hall, Seminar Room 267

* the speaker is a graduate student visiting GW.
Abstract:
Using the untwisted cube of resolutions for knot Floer homology, we define a spectral sequence from a direct sum of HOMFLY-PT homologies to knot Floer homology, where the direct sum is taken over oriented multi-cycles satisfying certain properties. We relate this spectral sequence to the conjectured spectral sequence from HOMFLY-PT homology to knot Floer homology on the level of Poincare polynomials, and use this relationship to define the sl(n) polynomial in terms of a sum of sl(n-1) polynomials.


Spring 2014

Title: Legendrian links, monotonic simplification, and Jones' conjecture on braids
Speaker: Ivan Dynnikov, University of Moscow
Time: Tuesday, April 8, 2014, 5:30-6:30pm
Place: Monroe Hall, Seminar Room 267
Abstract
I will talk about our recent work with Maxim Prasolov in which we unexpectedly proved Jones' conjecture which states that the algebraic crossing number of a braid whose closure is the given link is an invariant of the link provided that the braid index is minimal.
This work started from discovering a criteria when a rectangular diagram of a link admits a destabilization after applying finitely many elementary moves preserving the number of edges. It turns out that the two Legendrian links naturally associated to each rectangular diagram are responsible for this property.


Title: Invariants and Homology for Yang-Baxter Operators

Speaker: Jing Wang (GWU)
Time: Tuesday, March 18, 2014, 5:30-6:30
Place: Seminar room 267

Abstract
This talk will be divided into two parts: In the first part, I will describe Jones-Turaev's discovery about invariants of links from the Yang-Baxter equation. In particular, Jones Polynomial can be recovered using Y-B operators. Part two of my talk will be devoted to homology theory. I will discuss the "Yang-Baxter Homology" introduced by Przytycki and Lebed. Some familiar homologies such as distributive homology can be thought of as special case.

Title: Traces, cross ratios and two generator subgroups of SU(3,1)
Speaker: Krishnendu Gondgopadhyay ( IISER Mohali. India)
Time:Wednesday March 19, 2014, 5:30pm-6:30pm
Place: Seminar room 267

Abstract
In this talk, I shall discuss how traces and cross ratios of isometries of the complex hyperbolic 3-space
may be used to obtain a classification of "generic" representations of the free group F_2=<x, y> into the isometry group SU(3,1).


Speaker: Seung Yeop Yang (GWU)
Title: Introduction to twist spinning of knots; II
Time: Tuesday, February 25, 2014, 1:00pm - 2:00pm
Place:Monroe Hall, Seminar Room 267

Abstract (for part I and II)
We start by defining basic spinning from a knot to a knotted sphere, defined by E.Artin in 1925. We describe a work of Zeeman and Epstein, 1960, and construct a twist spinning of Zeeman 1965. In the more recent developments we follow the survey paper by G.Friedman, 2004, and describe our recent results on twisted spinning of 3-valent graphs.


Time: Tuesday January 14; 1:00pm
Location: Monroe 267 (Seminar Room)
Speaker: Robin Koytcheff (University of Victoria)

Abstract: Budney recently constructed an operad which encodes splicing of knots and gave a decomposition of the space of knots over this splicing operad. Infection of links by string links is a generalization of splicing from knots to links and is useful in studying knot concordance. We construct a colored operad that encodes infection. This colored operad captures all the relations in the 2-string link monoid. We also show that a certain subspace of 2-string links is freely generated over a suboperad of our infection colored operad by its subspace of prime links.

 


Speaker: Victoria Lebed (Advanced Mathematical Institute, Osaka City University,Japan)
Title: Towards topological applications of Laver tables

LOGIC--TOPOLOGY SEMINAR

Time: January 21 (Tuesday), 2014, 5:00pm
Location: Monroe 267 (Seminar Room)

(The talk will be introduced by Valentina Harizanov description of
Richard Laver (1942-2012) work in logic.)

Abstract:
(Joint work with Patrick Dehornoy) Laver tables are certain finite shelves (i.e., sets endowed with a binary operation which is distributive with respect to itself). They originate from set theory and have a profound combinatorial structure. In this talk I will discuss our dreams regarding potential braid and knot invariant constructions using Laver tables, and also present some real results in this direction, such as a detailed description of 2- and 3-cocycles for Laver tables. The rich structure of the latter promises interesting topological applications.

 


Fall 2012

Topology Seminar (and Specialty Exam talk)
Time: 1pm-2pm Dec. 1, 2011
Title: Homology of a Small Category with Functor Coefficients and Barycentric Subdivision.
Speaker: Jing Wang (GWU)
Location: Monroe Hall room 267
Abstract: We will introduce the definition of homology of a small category with functor coefficients and then compare two different homologies when we take the category of a simplical complex. We analyze the classical result that barycentric subdivision preserves homology and generalize it to the context of a small category with coefficients in a functor to R-modules.


Time: November 22 (Tuesday); 11:10-12:10, Monroe 252
Speaker: Adam Sikoa, University at Buffalo.
Title: Character Varieties of surfaces as completely integrable systems
Abstract: It is known that the trace functions of a maximal set of disjoint simple closed curves on a closed surface make its SU(2)-character variety into an (almost) completely integrable dynamical system. We prove an analogous statement for all rank 2 Lie groups. We will discuss the possible generalizations of this result to higher ranks and, if time permits, its applications to quantization of character varieties.

 


Time: Wednesday October 26; 6:30-7:30 pm
Speaker: Mark Kidwell (U.S. Naval Academy and GWU)
Place: Monroe 267 (seminar room)
Title: The Bonahon Metric and Topology
Abstract: In his book “Low-Dimensional Geometry: From Euclidean Spaces to Hyperbolic Knots”, Francis Bonahon considers no structure more abstract than a metric space. He then needs to define a metric on a quotient space, such as the torus obtained by identifying opposite sides of a rectangle. We explore
some quirky consequences of Bonahon’s definition of a (pseudo)-metric on a quotient space. We then answer the question: does the topology defined by the Bohahon metric on a quotient space coincide with the quotient topology?


Special Topology Seminar
Friday October 7, 2011;
5:00 - 6:00pm
Speaker: Takefumi Nosaka, RIMS at Kyoto University
Place: Monroe 451
Title: Mochizuki's quandle 3-cocycle invariant of links
S^3 is one of the Dijkgraaf-Witten invariants

Abstract: Let p be an odd prime, and \phi the Mochizuki 3-cocycle of "the dihedral quandle" of order p. Using the 3-cocycle, Carter-Kamada-Saito
combinatorially defined a shadow quandle cocycle invariant of links in S^3. Let M_L be the double covering branched along a link L. Our main
result is that the cocycle invariant of L equals the Dijkgraaf-Witten invariant of M_L with respect to the group Z/pZ. We further compute
Dijkgraaf-Witten invariants of some 3-manifolds. In this talk, I introduce a simple proof of the equality. This is a joint work with Eri Hatakenaka.


Time: Tuesday, September 27, 11:10am – 12:10pm.
Speaker: Noel Brady, NSF and University of Oklahoma.
Place: Monroe Hall (2115 G Street), Room 267
Title: Modified right-angled Artin groups.
Abstract: The family of right-angled Artin groups (RAAGs) interpolates between the family of finitely generated free groups on one hand and the family of finitely generated free abelian groups on the other. RAAGs are easy to define (their definition can be encoded in a finite graph) and have very nice geometric and topological properties (they have non-positively curved cubical classifying spaces). There is a standard map from a RAAG to the integers, and the topological properties of the kernel is reflected in the topology of the clique complex associated to the defining finite graph. We introduce a new class of groups called modified RAAGs. Like classical RAAGs these can be encoded using finite graphs (with some extra decoration), and admit non-positively curved cubical classifying spaces. There are standard maps from modified RAAGs to the integers, and the kernels exhibit a wide range of geometric and topological properties. We will sketch the ideas involved in the construction of modified RAAGs, and will give some applications.


Logic-Topology Seminar
Time: Tuesday, September 20, 2011, 11:10a.m.–12:10p.m.
Speaker: Jozef Przytycki, GWUhttp://home.gwu.edu/~przytyck/
Place: Monroe Hall (2115 G Street), Room 267
Title: Homology of Distributive Lattices
Abstract: While homology theory of associative structures, such as groups and rings, was extensively studied in the past, beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, has been neglected until recently. Distributive structures have been studied for a long time. In 1880, C.S. ~Peirce emphasized the importance of (right-) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term and 2-term homology, and then by discussing 4-term homology for Boolean algebras and distributive lattices. We will start with a gentle introduction to distributive lattices and Boolean algebras (and their generalizations) for topologists, and with homology theory of distributive structures for logicians. We will end by outlining potential relations to Khovanov homology, via the Yang-Baxter operator.


Special Topology Seminar
Speaker: Krzysztof Putyra (Columbia University)
Title: A connection between odd and even Khovanov homology
Time: 5:40pm, Monday July 25, 2011
Place: Seminar room (Monroe Hall (2115 G Street), Room 267