| Sergey Finashin | Jan 12 | New Invariants of Manifolds in Dimensions 3 and 4 (definition and the main properties of the invariants, examples, motivations and applications) |
| Huseyin Azcan | Fundamental Quandle of a link and its representations (continuation) | |
| Huseyin Azcan | Nov 24 | On the crossing number of links whose determinant is different from the unity |
| Mustafa Korkmaz | Symplectic Lefschetz fibrations with arbitrary fundamental groups |
| Sahin Kocak | Mar 31 | Topological Quantum Fields Theories in the sense of Quinn F.Quinn generalized Atiyah's TQFT set-up in such a way that it could be applied to objects other than manifolds where the "boundaries" needn't be actual boundaries. For example, for a CW-pair (X,A), the subcomplex A could be interpreted as a boundary, giving rise to a so-called Euler-type TQFT. The axioms are almost the same as the Atiyah's, but the categories upon which the theory is defined are more general and include the classical case. |
| Ozgur Kisisel | Vassiliev invariants and the Kontsevich integral |
| Ebru Keyman | Dec 16 | On Birman's Conjecture Birman conjectured that the Vassiliev map from SNn to ZBn is injective. I will talk about the developments to prove this conjecture. |
| Huseyin Azcan | Topological Quantum Field Theory
The talk is based on a paper of Blanchet, Vogel et. al. I will discuss a definition and give an example of a quantization functor. | |
| Mohan Bhupal | Nov 11 | A partial order on the group of contactomorphisms of R^{2n+1} via generating functions |
| Sergey Finsahin | 4-manifolds with inequivalent symplectic forms and 3-manifolds with
inequivalent fibrations
Review of the paper by C. McMullen and C. Taubes. |
| Huseyin Azcan | Dec 11 | Fundamental groups of the virtual knots (Part II) |
| Muhiddin Uguz | The moduli space of the G-invariant solutions to SW-equations
In this talk I will explain the G-invariant solutions of the Seiberg-Witten equations when G is a cyclic group acting on a manifold M, preserving the metric and the orientation. G is assumed to have a lift to principle Spinc bundle which gives rise to Seiberg-Witten equations in question. It will be shown that when the dimension b+G of the G-fixed points of harmonic two forms is positive, for a generic choice of an element in this fixed point set, the moduli space of invariant solutions of Seiberg-Witten equations is a compact, smooth and oriented manifold. In case b+G is zero, it will be shown that there exist a unique singularity which has a compact neighborhood homeomorphic to a cone on a certain projective space. Using the latter case, a version of the theorem of Fintushel and Stern which gives a necessary condition for a Seifert homology 3-sphere occurs as the boundary of a negative definite four manifold whose first cohomology contains no 2-torsion, will be proven. | |
| Sergey Finashin | Nov&13 | Introduction to Calabi-Yau manifolds and the Mirror Symmetry (based primarily on the lecture of Kontsevich) |
| Huseyin Azcan | Fundamental groups of the virtual knots |
| Mustafa Korkmaz | Jul 3 | On the number of vanishing cycles in a Lefschetz fibration |
| Ebru Keyman | Arrow diagrams and finite type invariants | |
| Sergey Finashin | Apr 17 | New results of Kronheimer, Fintushel-Stern and Akbulut in 4-dimensional topology |
| Huseyin Azcan | Quantum invariants of the real and virtual knots | |
| Huseyin Azcan | Feb 27 | Virtual knot theory and quantum link invariants (after Kauffman) |
| Alex Degtyarev | Variations of knotted graphs. The geometric n-equivalence techniques (after Goussarov) | |
| Muhiddin Uguz | Structure of the moduli space for G-invariant SW-solutions near singularities |
| Sahin Kocak | Dec 12 | Vassiliev Invariants and Chord Diagrams |
| Ebru Keyman | Virtual Knots and Braids, and Virtual Knot Invariants
Every knot diagram on S2 can be uniquely coded (up to an isotopy of the sphere) by its Gauss Diagram but not every Gauss Diagram comes from a knot diagram. To overcome this difficulty, virtual crossings for a knot diagram has been introduced. The isotopy classes of the virtual knot diagrams are obtained by virtual Reidemeister moves and the classical knot invariants extend to virtual knot invariants. We can also define the virtual braid group which is actually a quotient of the braid permutation group BPn. We distinguish the virtual knot invariants which factorize through BPn and which do not. | |
| Turgut Onder | (Z/2r)-Actions on Spin 4-manifolds: Some Applications
In some earlier talks J.Bryan's results about Z/2r-actions on Spin 4-manifolds were presented. In this talk, some applications of these results will be emphasized, e.g., results about the genus bounds (which sharpen the earlier estimates that have used Furuta's results) and the fixed point sets of involutions on rational cohomology K3's. | |
| Sergey Finashin | Nov 7 | Links and 3-dimensional manifolds: from the skein-theory, TQFT
and quantum-polynomials to the finite type invariants (a survey)
3-dimensional topology attracts nowadays not less attention then 4-dimensional topology and surely deserves to be discussed in our seminar. I only mention the names of Jones, Witten and Kontsevich, whose works (in particular) on this subject were rewarded by Fields medals. The latest development in Low-dimensional topology suggests that there may be soon a discovery of a deep relationship between 3- and 4-dimensional topology (although these subjects look currently quite different in their philosophy). This is one more reason to discuss the most popular modern trends in 3-dimensional topology and the link theory. |
| Sergey Finashin | Review of the recent work of Viro-Goussarov-Polyak on finite type invariants of links and virtual links | |
| Muhiddin Uguz | Structure of the moduli space for G-invariant SW-solutions near singularities |
| [ Geometry / Topology Seminar Group | Mathematics Department ] |