[Turkmath:9341] MSGSU Seminer: M. Hakan Güntürkün / Different Approaches to Tropical Geometry and A Point-Line Incidence Structure / 21.11.13

[Mustafa Topkara]

Mimar Sinan Güzel Sanatlar Üniversitesi - Matematik Bölümü
Genel Seminer


Konuşmacı:
M. Hakan Güntürkün


Başlık:
Different Approaches to Tropical Geometry and A Point-Line Incidence
Structure

Özet:
This will be an introductory talk on Tropical Algebraic Geometry. We will
introduce tropical algebra that has two operations, max and plus. The set
of extended real numbers with these operations forms a semifield which is
called "Tropical Semifield".  We will show how to sketch the tropical lines
on the tropical semifield. Then we will introduce a central concept in
tropical algebraic geometry, "Amoeba". This was first mentioned in 1994 by
Gelfand, Kapranov and Zelevinsky. This might be seemed as a bridge between
classical algebraic geometry and tropical geometry. Sometimes it is helpful
to use the tropical counterparts of the algebraic varieties since we can
use combinatorics extensively on these simpler objects.  We will sketch the
tropical line by using amoebas. Then we will pass to a field theoretic
notion, Puiseux Series to explain the tropical varieties. We will give the
essential definitions and sketch the tropical line by using Puiseux series.
Then if time permits, we will try to explain higher dimensional tropical
linear varieties. We will also give the relation with the tropical
varieties and the analytification of a variety in sense of Berkovich.

Then we will give a point-line incidence structure called k-nets. We will
give some open problems related to these Sylvester-Gallai type structures.
 We will explain how to prove the nonexistence of one case by using
tropical algebraic geometry. We will tropicalize the lines and the points
and get a contradiction to the existence of the structure.

Yer:
MSGSÜ Bomonti Kampüsü (Harita <http://math.msgsu.edu.tr/iletisim.html>),
Matematik Bölümü Seminer Odası.

Zaman:
21 Kasım 2013 Perşembe, 16:00
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