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Workshop on Algebraic and Applied Topology
Dokuz Eylül University, İzmir
April 19, 2019

The goal of the workshop is to bring together the researchers from the fields of Algebraic Topology and Applied Topology in Turkey to discuss their field of interests and to initiate new collabrations for future research projects.

The morning session will be held at Room B256, Department of Mathematics and the afternoon session will be held at Room B258, Department of Mathematics. For more information, send an e-mail to asli.ilhan at .deu.edu.tr.

Program

10:00-11:00 Ayşe Borat
11:00-11:20 Coffee Break
11:20-12:10 Matthew Gelvin
12:10-14:00 Lunch
14:00-14:50 Mehmet Akif Erdal
14:50-15:00 Coffee Break
15:00-15:50 Hanife Varlı
15:50-16:00 Coffee Break
16:00-16:30 Sabri Kaan Gürbüzer
16:30-16:40 Coffee Break
16:40-17:10 Derya Bayrıl Aykut

A Survey on Topological Robotics
Ayşe Borat

Topological robotics is a field initiated by Michael Farber in 2003. This new field tries to answer topological questions which are inspired by robotics and engineering. In this talk, we will give a brief survey in topological robotics mainly focusing on an important homotopy invariant called Topological Complexity which measures how far a space away from admitting a motion planning algorithm.

Euler Characteristics of Categories and Control of Homotopy Type
Matthew Gelvin

The Euler characteristic of a simplicial complex is a well-known and important combinatorial invariant. When considering small categories and their geometric realizations, one might hope that there is a similar invariant, ideally one that generalizes the classical Euler characteristic in the case of posets. Leinster defined such an object and proved some of its basic properties.

In this talk, I will outline Leinster’s notion of the Euler characteristic of a category and describe how it was used in joint work with Jesper Møller to guide our search for objects that control the homotopy type of certain categories that arise in the study of p-local finite groups.

Fibration Categories from Enrichments
Mehmet Akif Erdal

Fibration categories, as introduced by Brown [1], provide convenient models for homotopy theories as weaker alternatives to model categories. In this talk we will discuss fibration category structures that are induced by enrichments in symmetric monoidal model categories. We will also show that various categories of operator algebras, including Schocket and Uuye’s homotopy theory for $C^*$-algebras [4,5], and their equivariant versions are examples of fibration categories induced by enrichments. By using this, we recover known results that equivariant $KK$-theories and $E$-theories are triangulated categories (see [2,3]).

References
  1. Kenneth S. Brown. Abstract homotopy theory and generalized sheaf cohomology. Trans. Amer. Math. Soc., 186:419–458, 1973.
  2. Ralf Meyer and Ryszard Nest. The baum–connes conjecture via localisation of categories. Topology, 45(2):209–259, 2006.
  3. Ryszard Nest and Christian Voigt. Equivariant Poincar ́e duality for quantum group actions. Journal of Functional Analysis, 258(5):1466–1503, 2010.
  4. Claude Schochet. Topological methods for c-algebras. i. spectral sequences. Pacific Journal of Mathematics, 96(1):193–211, 1981.
  5. Otgonbayar Uuye. Homotopical algebra for $C^*$-algebras. Journal of Non- commutative Geometry, 7(4):981–1006, 2013.

Discrete (and Smooth) Morse Theory
Hanife Varlı

The primary concern of Morse theory is the relation between spaces and functions. The center of interest lies in how the critical points of a function defined on a space affect the topological shape of the space and conversely. Discrete Morse theory, developed by Robin Forman, is a discrete version of Morse theory that turned out to be also an efficient method to study of the topology of the discrete objects such as simplicial and cellular complexes.

In this talk, we will briefly mention smooth Morse theory, then talk about discrete Morse theory. In particular, we will talk about perfect discrete Morse functions, and the problem of composing and decomposing perfect discrete Morse functions on the connected sum of triangulated manifolds.

On a Decomposition of the Bicomplex of Planar Binary Trees
Sabri Kaan Gürbüzer

In this talk, we will introduce some simplicial properties of the set of planar binary trees and a decomposition of the bicomplex into vertical towers given Frabetti [1].

References
  1. Frabetti, A., Simplicial properties of the set of planar binary trees. Journal of Algebraic Combinatorics, 32, 41-65,(2001).

On the Lie Algebra of Spatial Kinematics
Derya Bayrıl Aykut

A spatial displacement is a composition of a spatial rotation followed by a spatial translation. There is an invariant line of these transformations, called screw axis. In this talk we will mention about velocity analaysis of a general spatial motion.

References
  1. TSAI, Lung-Wen (1999). Robot Analysis: The Mechanics of Serial and Parallel Ma- nipulators . A Wiley-Interscience Publication
  2. Selig, J. M. (2005). Geometric Fundamentals of Robotics. Springer(USA).