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My webpage at www.vadimkulikov.org

My name is Vadim Kulikov. This web page is mainly related to my studies and work at
the department of mathematics and statistics at the University of Helsinki.
My e-mail is vadim.kulikov@helsinki.fi,
my phone number is
+358(0)503444045 and
I have a mathematical blog.
I am interested in all of mathematics and things that can be understood using the understanding of mathematics. I participate in the research of the Helsinki logic group. Recently I finished my Ph.D. thesis Playing Games on Sets and Models The defence is taking place 19. November 2011 at the University Museum Arppeanum, Snellmanninkatu 3, Helsinki. My opponent is Professor Philip Welch from University of Bristol. Here is my CV and here CV in Finnish. Here is my web page at the Department. ## Teaching (except exercise classes):2011: Topology I. 2010: I supervised bachelor theses. spring 2009: Knot Theory (in Finnish) and a short course on LaTeX. fall 2008: TVT-ajokortti, Computer Driving Licence spring 2007: a short course on LaTeX ## Other stuffBelow you can find some of my work. For instance my master's and licentiate's theses. |

Ph.D. thesis Playing Games on Sets and Models |

Generalized
Descriptive Set Theory and Classification Theory
An article joint work with Tapani Hyttinen and Sy-David Friedman. Abstract:
The field of descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncount- able. We explore properties of generalized Baire and Cantor spaces and study equivalence relations and their Borel reducibility on these spaces. The study shows that the descriptive set theory looks very dierent in this generalized setting compared to the classical, countable case. We draw the connection between the stability theoretic complexity of rst-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations. |

Here is my licentiate's thesis, which is nearly the same thing as the article Generalized Descriptive Set Theory and Classification Theory listed above. |

Here is a poster about a connection of knot theory with biology, which I presented on the EWM summer school in Turku in June 2009. I had problems reducing the size of the file, so it is big for now. |

Lectures in Combinatorial Knot Theory spring 2009. Lectures in Topological Knot Theory spring 2009. |

Master's thesis: Weak Ehrenfeucht-Fraïssé Gam,
supervisor Tapani Hyttinen.
An article (in collaboration with T. Hyttinen, to appear in TAMS)
on the same subject (more results, different introduction, more prerequisites required compared to the master's thesis):
Article: Weak Ehrenfeucht-Fraïssé Games
A weak version of Ehrenfeucht-Fraïssé games is defined. This gives rise for two new relations between structures
for each ordinal alpha:
1.Player 2 wins the weak game on A and B (equivalence relation) 2. Player 1 does not win the weak game on A and B (not necessary an equivalence) we study how these relate to each other (are they same for given alpha?). Also we compare it to the ordinary Ehrenfeucht-Fraïssé game |

Analytic sets A proof that analytic sets (continuous images of Borel sets) are measurable and that there are analytic non-Borel sets. Based on a presentation I gave spring 2008 on a course on geometric measure theory by Pertti Mattila. |

Axiom of Determinacy (in finnish) Uncountable axiom of choice can consistently be replaced by the axiom of determinacy. In this presentation I gave a proof that this axiom implies Lebesgue-measurability of all subsets of real numbers. Thus uncountable axiom of choice is necessary to produce non-measurable sets. The proof of consistency is not given. |

Graphs and Eigenvalues Few years ago I thought about EF-games on graphs and came up with this thought. A proof that eigenvalues of the characteristic matrix of a graph is an invariant of graphs. |

Isoperimetric inequality A proof that among all sets (in R^n) of Lebesgue measure 1, ball minimizes the surface area. For non-specialists: An ice ball of volume 1 litre melts slower than any other ice shape which has volume 1 litre. This is also (as analytic sets) based on a presentation I gave on a course on geometric measure theory spring 2008. |

Compact vs Sequentially compact Examples which show that the two definitions of compact
1.Every sequence has a converging subsequence 2. Every open cover has a finite subcover have nothing in common on non-metrizable spaces. |

Last modified: Sun Feb 5 00:34:40 EET 2012