Personal webpage of Valentina Kiritchenko

Faculty of Mathematics

April 25, Wednesday, 15:30, room 211

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Ivan Nikitin

Defining Grassmannians by Plücker relations

I will introduce Grassmannians using detailed description of Plücker embedding. I will also tell about properties of Grassmannians regarded as complex algebraic varieties. Though Plücker embedding is studied in the first year linear algebra course, I hope my talk will help to understand it better. No preliminary knowledge is required.

April 18, Wednesday, 15:30, room 211

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Grigory Solomadin (Moscow State University)

K-theory of toric varieties, convex chains and scissors-congruence

The complex K-theory ring $K(X)$ of a (nice) compact topological space $X$ is by definition the Grothendieck ring of the semigroup of vector bundles (up to isomorphism) over $X$ with respect to the Whitney sum and tensor product. Choosing a topological group $G$ and taking into account different good actions of $G$ on $G$-vector bundles over $X$ one obtains the ring $K_{G}(X)$ of equivariant K-theory. For a smooth projective toric variety $X^{2n}$ and complex torus $T^n$ the ring $K_{T^n}(X)$ has the embedding to the McMullen polytopal algebra $L(\Z^n)$. Forgetting the action structure on vector bundles leads to the embedding of the ring $K(X)$ to the coinvariants of the translation group acting on $L(\Z^n)$, i.e. ring generated by scissors-congruence classes of convex polytopes in $\R^n$. In the talk, we will discuss these embeddings.

April 11, Wednesday, 15:30, room 211

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Yury Rud’ko

Euler characteristic of determinantal varieties

Determinantal variety is a specific type of algebraic variety that is quite nice to study. I will introduce it and calculate its Euler characteristic for a special simple case. The tecnique is based on the Bernstein-Koushnirenko theorem and requires some toric geometry and singularity theory (all of it will be introduced during the talk!)

April 4, Wednesday, 15:30, room 211

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Artem Kalmykov

Determinantal representation of plane curves

By definition, a determinantal representation of a plane curve C is a matrix triple (A_0,A_1,A_2) such that the zero set of the equation det(x_0 A_0+x_1 A_1+x_2 A_2)=0 is C. In the talk I am going to present the proof (due to V. Vinnikov) of the theorem stating that every smooth plane curve over complex numbers admits such a representation. We will also discuss an analogous result over real numbers, in particular, various relations between the signature of the defining matrices and the topological type of the curve C(\R).