User maciej starostka - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:04:48Z http://mathoverflow.net/feeds/user/14925 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115687/vortex-equations-on-cylinder Vortex equations on cylinder Maciej Starostka 2012-12-07T08:09:13Z 2012-12-07T16:21:46Z <p>Solutions to the vortex equations for a closed Riemann surface are well known (moduli space is a symmetric power). What do we know about solutions on surfaces with boundary or non compact surfaces? In particular I am interested in the case of a infinite cylinder $S^1 \times \mathbb{R}$.</p> http://mathoverflow.net/questions/103586/smooth-homotopy-on-exotic-r4 smooth homotopy on exotic R^4 Maciej Starostka 2012-07-31T07:48:15Z 2012-07-31T11:01:38Z <p>Take an exotic $\mathbb{R}^4$ i.e. $V = (\mathbb{R}^4,d)$ such that $V$ is not diffeomorphic to $\mathbb{R}^4$ with standard metric.</p> <p>Is it true (obvious?) that any two smooth maps $f_1, f_2: S^k \to V$ are equivalent via <strong>smooth</strong> homotopy?</p> <p>edited: (for any k)</p> http://mathoverflow.net/questions/100033/interesting-mathematical-documentaries/101918#101918 Answer by Maciej Starostka for Interesting mathematical documentaries Maciej Starostka 2012-07-11T06:57:04Z 2012-07-11T06:57:04Z <p>There is a document "Banach spaces" about S.Banach and other polish mathematicians from Lviv ( S.Ulam, J.P.Schauder)</p> <p>Unfortunetelly I don't know about any translation to english.</p> <p><a href="http://www.youtube.com/watch?v=EJgl_Z9Yz1Q" rel="nofollow">http://www.youtube.com/watch?v=EJgl_Z9Yz1Q</a></p> http://mathoverflow.net/questions/101077/perturbation-of-morse-function/101082#101082 Answer by Maciej Starostka for Perturbation of Morse function Maciej Starostka 2012-07-01T16:58:52Z 2012-07-01T16:58:52Z <p>Are you sure you cannot achieve what you want by simply changing coordinates? Answers are about adding general small function but you want to add a quadratic polynomial only on a neighborhood of $x_0$. </p> http://mathoverflow.net/questions/100698/group-action-on-spinc-4-manifold Group action on spin^c 4-manifold. Maciej Starostka 2012-06-26T16:33:59Z 2012-06-26T18:23:26Z <p> I'll try to be more precise.</p> <p>In paper N.Nakamura, "Bauerâ€“Furuta invariants under $Z_2$-actions" there is an assumption that $Z_2$ action "lifts to spin^c structure". What i think it means: $Spin^c$ structure is a principal $Spin^c$ bundle $\pi: P \to M$. A lift is (following Gottlieb) an action on $P$ such that $\pi$ is equivariant.</p> <p>1) What are the conditions under which a $Z_2$ action on $M$ lifts?</p> <p>2) What about other groups (different than $Z_2$)?</p> <p>I'll be greatful also for general references on this topic.</p> http://mathoverflow.net/questions/100054/cotetrad-spin-connection-and-dirac-operator/100078#100078 Answer by Maciej Starostka for Cotetrad, spin connection and Dirac operator Maciej Starostka 2012-06-20T06:21:55Z 2012-06-20T06:28:50Z <p>You have to be more precise. $A$ is an SO(5) connection but on what bundle? This is a formula for the Dirac operator which depends on local trivialization. I doubt that this defines a global operator in general (this works for a spin or $spin^c$ structures on TM and doesn't have to work on spin structure on every vetor bundle).</p> <p>I will recommend you chapters "spin structures on vector bundles" and "dirac opetors" form book D.Salamon,Spin geometry and Seiberg-Witten invariants.</p> <p>Hope it helped a little.</p> http://mathoverflow.net/questions/358/examples-of-great-mathematical-writing/90917#90917 Answer by Maciej Starostka for Examples of great mathematical writing Maciej Starostka 2012-03-11T17:21:13Z 2012-03-11T17:21:13Z <p>Simon Donaldson - Riemann Surfaces Great writing, deep understanding. I believe that noone have mentioned it because this topic is older than the book.</p> http://mathoverflow.net/questions/101077/perturbation-of-morse-function/101082#101082 Comment by Maciej Starostka Maciej Starostka 2012-07-01T21:34:51Z 2012-07-01T21:34:51Z You add a &quot;quadratic polynomial Q&quot;. This only make sense in a choosen local coordinates. http://mathoverflow.net/questions/100698/group-action-on-spinc-4-manifold Comment by Maciej Starostka Maciej Starostka 2012-06-26T18:03:38Z 2012-06-26T18:03:38Z I've edited the question. http://mathoverflow.net/questions/100698/group-action-on-spinc-4-manifold Comment by Maciej Starostka Maciej Starostka 2012-06-26T17:29:08Z 2012-06-26T17:29:08Z I mean it lifts to a principal Spin^c bundle.