Our Works

• We will introduce the notion of two Point Sets. Then, we will discuss some results regarding two points sets and their constructions .

• Resources used.

• Symposium presentation 

• In our presentation we will tell you step by step how Brouwer’s Fixed Point Theorem is amazingly constructed and we will see two of its interesting results.

• Resources used : 

  1. Munkres, J. R. (1974). Topology; a first course. Prentice-Hall.

• Symposium Presentation 

• We will start with defining affine varieties and the correspondence between  finitely generated k-algebras and affine varieties

• Resources used:

1. Algebraic Geometry - Andreas Gathmann (Class notes TU Kaiserslautern 2021/22)

2. Algebra: Chapter 0 - Paolo Aluffi 

3. www.math3ma.com

• Symposium Presentatiton 

• We try to explain Game theory and GANs for the mathematical motivations for the minimax game for training generative adversarial networks and how we use the minimax equation to understand the source code of the generative adversarial network implementations.

• Resources used:

1. Avinash K Dixit and Barry J Nalebuff. Thinking strategically: The compet- itive edge in business, politics, and everyday life. WW Norton & Company, 1993.

2. Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde- Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative ad- versarial nets. Advances in neural information processing systems, 27, 2014.

3. John Maynard Smith. Evolution and the theory of games. American scien- tist, 64(1):41–45, 1976.

4. John Nash. Non-cooperative games. Annals of mathematics, pages 286–295, 1951.
    

5. J von Neumann, Oskar Morgenstern, et al. Theory of games and economic behavior, 1947.  

• Symposium Presentatiton 

• Smooth Morse Theory is a field of mathematics that requires the theory of

smooth manifolds. For the first 10 weeks we tried to prepare for the upcoming topics of morse theory and consuquently we generaly only studied theory of smooth manifolds.

• Resources used:

1. Milnor, J. W. (1963). Morse theory (No. 51). Princeton university press.

2. • Hutchings, Michael (2002). Lecture notes on Morse homology.

3. • Lee, J. M., & Lee, J. M. (2012). Smooth manifolds (pp. 1-31). Springer

4. New York.

• Symposium Presentatiton 

This report studies the basics of Graph Theory focusing on Vertex coloring and Planar Graphs with an article called 5-Coloring Reconfiguration of Planar Graphs with No Short Odd Cycles.

• Resources used:

1. J.A. Bondy, U.S.R Murty, Graph Theory with Applications. London: Macmillan, 1976.

2. D.W. Cranston, R. Mahmoud, 5-Coloring Reconfiguration of Planar Graphs with No Short Odd Cycles. Richmond, 2022.

• Symposium Presentatiton 

In today’s world, uncovering meaningful patterns in complex datasets poses sig- nificant challenges, and Topological Data Analysis (TDA) is an up-and-coming approach to data analysis that focuses on understanding the shape of data, which is a convenient way to overcome these challenges. This report provides a brief introduction to the powerful data analysis technique Topological Data Analysis. Before that, to inform basics about Topological Data Analysis, there is an introduction part, a section where fundamental concepts are addressed step by step according to the persistent homology pipeline.

• Resources used:

1. Topological Data Analysis with 1-Parameter Persistent Homology - DRP METU Symposium. Michael Lesnick. Lecture notes for math840: Multiparameter persistence.  

• Symposium Presentatiton 

Throughout our program, we focused on various properties of Fibonacci cubes. Before formally defining Fibonacci cubes, we will define Hypercubes and briefly learn their role in network sciences. Examining Hypercubes not just as graphs but as an interconnection network allows us to understand the motivation behind studying Fibonacci cubes. Besides learning the different properties of a graph, we might as well consider a network system behind every property. For example, studying edge connectivity is important since we wish to know which parts of our network system can be removed without losing connectedness. This report aims to serve as a starting point for someone interested in studying Fibonacci cubes and their appealing features.

1. Azarija, J., Klavˇzar, S., Rho, Y., & Sim, S. (2017). On domination-type invariants of Fibonacci cubes and hypercubes. Ars Mathematica Contemporanea, 14(2), 387-395.

2. Castro, A., Klavˇzar, S., Mollard, M., & Rho, Y. (2011). On the domination number and the 2-packing number of Fibonacci cubes and Lucas cubes. Computers & Mathematics with Applications, 61(9), 2655-2660.

3. Cong, B., Zheng, S. Q., & Sharma, S. (1993, April). On simulations of linear arrays, rings and 2D meshes on Fibonacci cube networks. In [1993] Proceedings Seventh International Parallel Processing Symposium (pp. 748-751). IEEE.

4. Ellis-Monaghan, J. A., Pike, D. A., & Zou, Y. (2006). Decycling of Fibonacci cubes. Australasian Journal of Combinatorics, 35, 31.

5. Hertz, A. (2020). Decycling bipartite graphs. GERAD, HEC Montreal.

6. Hsu, W. J. (1993). Fibonacci cubes-a new interconnection topology. IEEE Transactions on Parallel and Distributed Systems, 4(1), 3-12.

7. Klavzar, S. (2013). Structure of Fibonacci cubes: a survey. Journal of Combinatorial Optimization, 25, 505-522.

8. Saygi, E. (2019). On the domination number and the total domination number of Fibonacci cubes. Ars Mathematica Contemporanea, 16(1).

9. Yılmaz, M. B. (2022). Fibonacci k ̈uplerinin Roman tipi baskınlık sayıları (Master’s thesis, TOBB ETU).  ̈

10. Zou, Y. (2006). Decycling and dominating cubes and grids (Doctoral dissertation, Memorial University of Newfoundland).

• Symposium Presentatiton