20 10, 2021

Homological Mirror Symmetry for Hypersurface Singularities

2021-10-21T12:35:26+03:00October 20th, 2021|Joint ICMS & IMSA Seminar|

In this talk, I will briefly introduce homological mirror symmetry for certain hypersurface singularities. I will introduce basic definitions, Berglund-Hubsch duality of invertible polynomials, and some known results. Then I will discuss several examples in detail.

8 10, 2021

More instances of Steenbrink Spectra in Singularity Categories

2021-10-16T13:48:03+03:00October 8th, 2021|Joint ICMS & IMSA Seminar|

Now that we have set up the Frobenius Manifold of a singularity, we can see 2 more places where the steenbrink spectrum of a singularity appears in its singularity category. Namely, the non-commutative mixed hodge structure of a singularity and the dimensional properties of the category. The former appearance being somehow natural, and the latter somewhat mysterious. In this talk we'll conduct a surface level investigation of these appearances.

8 10, 2021

Windows and the BGMN conjecture

2021-10-26T14:31:25+03:00October 8th, 2021|Joint ICMS & IMSA Seminar|

Let $C$ be a smooth projective curve of genus at least 2, and let $N$ be the moduli space of semistable rank-two vector bundles of odd degree on $C$. We construct a semi-orthogonal decomposition in the derived category of $N$ conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form $D(C_d)$ where $C_d$ are $d$-th symmetric powers of $C$, and the semi-orthogonal complement to these blocks is conjecturally trivial. In order to prove our result, we use the moduli spaces of stable pairs over $C$. Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall. This is a joint work with J. Tevelev.

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