Year: 2019

Probabilistic Methods in Number Theory

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Doga Can Sertbaş, Cumhuriyet University

Date and Time: 22/03/2019, 10:00

Place: B256

In 1947, Erdos gave a lower bound for the diagonal Ramsey numbers R(k,k). His proof contains purely probabilistic arguments where the original problem is not related to the probability theory. This pioneering work of Erdos gave rise to a new proof technique which is so called the probabilistic method. According to this method, one just obtains the existence of a particular mathematical object in a non-constructive way. In this talk, we first introduce the Ramsey numbers and then explain the basics of the probability theory. After mentioning the fundamentals of the probabilistic method, we give several examples from the number theory. In particular using probabilistic inequalities, we show how one can prove some number theoretic results which seem completely unrelated to the probability theory.

Knotted Strings in the plane

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Neslihan Güğümcü, Technical University of Athens

Date and Time: 15/03/2019, 10:00

Place: B256

Abstract: Planar curves have been studied since the time of Gauss. Gauss was one of the first to notice that they can be handled combinatorially by codes (named as Guass codes) that are strings of labels encoding self-intersections. Whitney classified all immersed curves up to a topological relation called regular homotopy by using the winding number of immersion maps. In the first half of the 20th century Reidemeister showed that classical knot theory is equivalent to the study of immersed curves in the plane, whose self-intersections are endowed with a combinatorial structure, with an under/over-data. With this extra structure, regular homotopy needs to transforms into a richer equivalence relation generated by Reidemeister moves. Since then knot theory is a classical subject of topology, bringing us many interesting questions relating to combinatorial topology.

In this talk, we will talk about knotoids (introduced by Turaev) that provide us a new diagrammatic theory that is an extension of classical knot theory. Problem of classifying knotoids lies at the center of the theory of knotoids. We will construct a Laurent polynomial with integer coefficients for knotoids called the affine index polynomial and we will show how it contributes to the classification problem.

Counting from the Bottom to the Top

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Noyan Er, Dokuz Eylül University.
Date: 13th of March, 2019, Wednesday. Time: 09:30-10:30
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: After finishing up a leftover result from my last talk, we will discuss, as a natural follow-up, a characterization of Artinian principal ideal rings due to Eisenbud and Griffith the proof of which involves counting vertically.

An Invitation to Algebraic Number Theory

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Hikmet Burak Özcan, Dokuz Eylül University.
Date: 6th of March, 2019, Wednesday. Time: 09:30 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: This will be the first of a series of talks in algebraic number theory. During the series of talks, we will cover several chapters of the book “Algebraic Theory of Numbers” by Pierre Samuel. In the first talk, we will mention some theorems and their corollaries from Chapter 1.
References
[1] Pierre Samuel. Algebraic Theory of Numbers, Hermann, 1970.

Counting from Left to Right

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Noyan Er, Dokuz Eylül University.
Date: 27th of February, 2019, Wednesday. Time: 09:30-10:30
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: This will the the beginning of a series of sporadic talks related to some classical methods bridging Ring Theory and Representation Theory, which will feature some well known results due to Eisenbud and Griffith.

Topological Equivalences of E∞ DGAs

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Haldun Özgür Bayındır, The University of Haifa.
Date: 6th of February, 2019, Thursday. Time: 11:00 – 12:00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.
Abstract: In algebraic topology we often encounter chain complexes with extra multiplicative structure. For example, the cochain complex of a topological space has what is called the E∞-algebra structure which comes from the cup product. In this talk I present an idea for studying such chain complexes, E∞ differential graded algebras (E∞ DGAs), using stable homotopy theory. Namely, I discuss new equivalences between E∞ DGAS that are defined using commutative ring spectra.We say E∞ DGAs are E∞ topologically equivalent when the corresponding commutative ring spectra are equivalent. Quasi-isomorphic E∞ DGAs are E∞ topologically equivalent. However, the examples I am going to present show that the opposite is not true; there are E∞ DGAs that are E∞ topologically equivalent but not quasi-isomorphic. This says that between E∞ DGAs, we have more equivalences than just the quasi-isomorphisms. I also discuss interaction of E∞ topological equivalences with the Dyer-Lashof operations and cases where E∞ topological equivalences and quasi-isomorphisms agree.