undergraduate-research

Abstract: Last spring, we gave a presentation on chaos theory, where we discussed many concepts within the scope of the theory. In this presentation, we aim to explore the behavior of dynamical systems through examples drawn from everyday life. From changes in the Earth’s climate to predator–prey interactions observed in nature; from the operating principles of lasers to the extinction of languages spoken in the past, we examine these processes from a mathematical perspective. We would be pleased to have you join our presentation.

Date: December 24, 2025 | 12.00 – 13.00

Place: B256

Origami Axioms and Constructible Numbers by Origami

December 3 2025 by Engin Mermut

Atakan Özcan and Pınar Ecem Akyıldırım, Dokuz Eylül University.
Date: 4th of December, 2025, Thursday, Time: 15.00 – 16.00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will firstly mention straightedge-compass constructions and five axioms for them. By straightedge and compass constructions, we will talk about, drawing lines and circles, constructions of a perpendicular to a given line through a given point, drawing a parallel line to a given line passing through a given point and bisecting an angle. Then we will see basic properties of origami and the Huzita-Hatori seven postulates for origami. Another set of axioms for origami is obtained by just adding another axiom to the five axioms of straightedge-compass constructions. We will show the equivalence between these two sets of axioms for origami. Trisection of an angle is known to be impossible using straightedge and compass constructions, but we will see that it is possible using origami constructions.

Undergraduate Mathematics Marathon Day 2023-2024 Spring

June 13 2025 by cem çelik

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ProjeProgramDuyuru13062025

Student Seminar: To Chaos and Beyond

June 10 2025June 10 2025 by cem çelik

Title: To Chaos and Beyond

Speakers: Şüheda Dilmaç, Naciye Çağlar

Abstract: In this student-to-student presentation, we explore the fascinating world of chaos theory
together! Concepts like the butterfly effect, the Lorenz attractor, and fractals are explained in a
clear and fun way through real-life examples and interactive content. A delightful introduction for
anyone curious about chaos!

Date: June 11, 2025 | 12.00 – 13.00

Location: B256

Fundamental Theorem of Symmetric Polynomials, Newton’s Identities and Discriminants

December 23 2024December 20 2024 by Engin Mermut

Mustafa Eren Taşlı, Dokuz Eylül University. Date: 26th of December, 2024, Wednesday, Time: 15.00 – 16.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will define symmetric polynomials and the elementary symmetric polynomials in $n$ indeterminates over a field $F$ . The elementary symmetric polynomials in the indeterminates $x_1, x_2, \ldots, x_n$ are as follows:

$\begin{align*} \sigma_1=&x_1+x_2+\ldots+x_n \\ \sigma_2=&\sum_{1 \le i < j \le n} x_ix_j \\ \vdots\\ \sigma_n =&x_1 x_2 \ldots x_n \end{align*}$

The Fundamental Theorem of Symmetric Polynomials states that any symmetric polynomial can be expressed as a polynomial in the elementary symmetric polynomials, that is:

Theorem. Let $f(x_1, x_2, \ldots, x_n)$ be a symmetric polynomial in the $n$ indeterminates $x_1, x_2, \ldots, x_n$ over a field $F$ . Then, there exists a polynomial $g(y_1, y_2, \ldots, y_n)$ in the $n$ indeterminates $y_1, y_2, \ldots, y_n$ such that

$f(x_1, x_2, \ldots, x_n) = g(\sigma_1, \sigma_2, \ldots, \sigma_n),$

where $\sigma_1, \sigma_2, \ldots, \sigma_n$ are the above elementary symmetric polynomials of the $n$ indeterminates $x_1, x_2, \ldots, x_n$ . Moreover, the polynomial $g(y_1, y_2, \ldots, y_n)$ is uniquely determined.

We will prove this theorem using the graded lexicographic order for multivariable polynomials.

Using the recurrence relation from the Newton Identities, we will learn how to express the sum of powers of the indeterminates, that is, the polyomials

$s_k = x_1^k + x_2^k + \ldots + x_n^k$

for a positive integer $k$ , as polynomials in terms of the elementary symmetric polynomials. We will reinforce this understanding with examples.

The discriminant in the indeterminates $x_1, x_2, \ldots, x_n$ over the field $F$ is given by:

$\Delta = \prod_{1 \leq i < j \leq n} (x_i - x_j)^2 \in F[x_1, \dots, x_n].$

The discriminant is a symmetric polynomial, and we will express it in terms of the elementary symmetric polynomials using determinants.

This seminar, as part of my graduation project titled Symmetric Polynomials, Newton’s Identities, Discriminants, and Resultants, serves as an introduction to a method for calculating the discriminant ( $\Delta$ ) of an $n$ -th degree polynomial without finding its roots.

Mustafa-Eren-Taşlı_Fundamental-Theorem-of-Symmetric-Polynomials_26-December-2024-Thursday_time-15-00_Classroom-B254

Cardano’s Formula and Casus Irreducibilis

December 23 2024December 13 2024 by Engin Mermut

Çağdaş Çiğdemoğlu, Dokuz Eylül University. Date: 18th of December, 2024, Wednesday, Time: 15.00 – 16.00. Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Classroom B255.

Abstract: We will start with a general monic cubic equation in the form

$x^3 + bx^2 + cx + d = 0,$

and transform it into the following form

$y^3 + py + q = 0,$

using a substitution. Then, we will construct the Cardano Formulas to find the roots of the equation. The roots are expressed as follows:

$y_1 = \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

$y_2 = \omega \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega^2 \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

$y_3 = \omega^2 \sqrt[3]{\frac{-q + \sqrt{q^2 + \frac{4p^3}{27}}}{2}} + \omega \sqrt[3]{\frac{-q - \sqrt{q^2 + \frac{4p^3}{27}}}{2}},$

where $\omega = e^{i \frac{2\pi}{3}} = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ is a primitive cube root of unity and where the above cube roots are one of the three complex cube roots whose product is $-\frac{p}{3}$ and these are fixed in the above formulas.

We will learn about the discriminant $\Delta$ of the above monic cubic polynomial, understand its significance, and examine how the roots change depending on the value of the discriminant.

For the monic cubic polynomial in the form $y^3 + py + q$ , the discriminant is expressed by:

$\Delta = -27q^2 - 4p^3 = (y_1-y_2)^2(y_1-y_3)^2(y_2-y_3)^2.$

For the general monic cubic polynomial $x^3 + bx^2 + cx + d$ , whose roots are $x_1, x_2, x_3$ , the discriminant is expressed by:

$\Delta = b^2c^2 + 18bcd - 4c^3 - 4b^3d - 27d^2 = (x_1-x_2)^2(x_1-x_3)^2(x_2-x_3)^2.$

We will also discuss Casus Irreducibilis, which occurs when the coefficients $b, c, d$ (or $p, q$ ) are in a subfield of $F$ of $\mathbb{R}$ , the cubic polynomial is irreducible over $F$ (equivalently, the cubic equation has no roots in $F$ ) and when the discriminant is positive. In this case, the cubic equation has three distinct real roots that cannot be expressed using real radicals.