• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Multiplicity of the Lyashko-Looijenga Map for Laurent Polynomials

Student: Elizaveta Shuvaeva

Supervisor: Sergei Lando

Faculty: Faculty of Mathematics

Educational Programme: Mathematics (Bachelor)

Final Grade: 10

Year of Graduation: 2020

It is well-known that the multiplicities of the Lyashko-Looijenga map on the strata of compactified Hurwitz spaces (spaces of meromorphic functions on complex curves) is related to the Hurwitz numbers corresponding to the functions with prescribed branching types in these spaces. The multiplicity of the Lyashko-Looijenga map can be computed if we know the cohomology classes that are Poincaré dual to the homology classes of the strata in the compactified Hurwitz space, and this computation, in turn, is significantly simplified if the primitive strata in this space intersect transversally. We restrict our attention to the Hurwitz spaces $H_{1,n}$ ——- spaces of functions with $2$ poles (i.e. Laurent polynomials), one of which is simple, while the other has order $n$. We explore a suitable compactification of $H_{1,n}$ that arises from its connection to certain moduli spaces. For the case of $n = 3$, we use this compactification to compute the cohomology classes of the strata along with showing that the intersection of the primitive strata is transversal. We also use this computation to find the multiplicities of the Lyashko-Looijenga map on the generic stratum, the caustic and the Maxwell stratum.

Full text (added June 1, 2020)

Student Theses at HSE must be completed in accordance with the University Rules and regulations specified by each educational programme.

Summaries of all theses must be published and made freely available on the HSE website.

The full text of a thesis can be published in open access on the HSE website only if the authoring student (copyright holder) agrees, or, if the thesis was written by a team of students, if all the co-authors (copyright holders) agree. After a thesis is published on the HSE website, it obtains the status of an online publication.

Student theses are objects of copyright and their use is subject to limitations in accordance with the Russian Federation’s law on intellectual property.

In the event that a thesis is quoted or otherwise used, reference to the author’s name and the source of quotation is required.

Search all student theses