Noyan Er, Dokuz Eylül University.

Date: 27th of March, 2019, Wednesday. Time: 10:00-12:00

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will finish up what was started last week. Counting things, however, will continue in the upcoming seminars.

# Month: March 2019

## Mathematical Physics Seminars 1- Basic Physics Concepts, Wave Equation and Time-Independent Schrödinger Equation.

Muhammed Deniz, Dokuz Eylül Univ., Department of Physics, Date: Mar 25, 2019, Monday. Time: 14:30 – 16:00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206.

## Rational Points on Curves

Hikmet Burak Özcan (DEU)

Date&Time: 29/03/2019, 10:00

Place: B256

In this talk, after defining rational points on a curve we will address the problem of finding the rational points on curves. We will give a recipe in order to generate a new rational point from already known ones. After that we will introduce the notion of elliptic curves and mention the rational points on elliptic curves. Finally we will refer to the well-known results, Mordell’s Theorem and Siegel’s Theorem.

## The Second Invitation to Algebraic Number Theory

Hikmet Burak Özcan, Dokuz Eylül University.

Date: 20th of March, 2019, Wednesday. Time: 10:00 – 12:00.

Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: This will be the second talk of the series devoted to algebraic number theory. First, we will briefly recall what we did in the first talk. Then we will state the elegant theorem, proved by Chevalley, which concerns about diophantine equations over a finite field.

## Probabilistic Methods in Number Theory

**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

**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 20^{th} 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

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

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.