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I will explain how to bound from above and below the expected Betti numbers of a random subcomplex in a simplicial complex and get asymptotic results under infinitely many barycentric subdivisions. This is a joint work with Nermin Salepci. It complements previous joint works with Damien Gayet on random topology.

52Cxx ; 60C05 ; 60B05 ; 55U10

Multi angle  Real curves and a Klein TQFT
Georgieva, Penka (Auteur de la Conférence) | CIRM (Editeur )

The local Gromov-Witten theory of curves studied by Bryan and Pandharipande revealed strong structural results for the local GW invariants, which were later used by Ionel and Parker in the proof of the Gopakumar-Vafa conjecture. In this talk I will report on a joint work in progress with Eleny Ionel on the extension of these results to the real setting.

14N35 ; 53D45

Multi angle  On real algebraic knots and links
Orevkov, Stepan (Auteur de la Conférence) | CIRM (Editeur )

I will present the following results on real algebraic spatial curves:
(1) (joint with Mikhalkin) Classification of smooth irreducible spatial real algebraic curves of genus 0 or 1 up to degree 6 up to rigid isotopy.
(2) (joint with Mikhalkin) Classification of smooth irreducible spatial real algebraic curves with maximal encomplexed writhe up to (not rigid yet) isotopy.
(3) Classification of smooth spatial real algebraic curves of genus 0 with two irreducible components up to degree 6 up to rigid isotopy, in particular, the first (as far as know) example of two spatial real algebraic curves which are isotopic, have equal degree, genus and encomplexed writhe of each irreducible component but not rigidly isotopic.
I will present the following results on real algebraic spatial curves:
(1) (joint with Mikhalkin) Classification of smooth irreducible spatial real algebraic curves of genus 0 or 1 up to degree 6 up to rigid isotopy.
(2) (joint with Mikhalkin) Classification of smooth irreducible spatial real algebraic curves with maximal encomplexed writhe up to (not rigid yet) isotopy.
(3) Classification of smooth spatial real algebraic curves of genus 0 with ...

14P25

It is a theorem of Hilbert that a real polynomial in two variables that is nonnegative is a sum of 4 squares of rational functions. Cassels, Ellison and Pfister have shown the existence of such polynomials that are not sums of 3 squares of rational functions. In this talk, we will prove that those polynomials that may be written as sums of 3 squares are dense in the set of nonnegative polynomials. The proof is Hodge-theoretic.

11E25 ; 14Pxx ; 14D07 ; 14M12

Let $X$ be an algebraic subvariety in $(\mathbb{C}^*)^n$. According to the good compactifification theorem there is a complete toric variety $M \supset (\mathbb{C}^*)^n$ such that the closure of $X$ in $M$ does not intersect orbits in $M$ of codimension bigger than dim$_\mathbb{C} X$. All proofs of this theorem I met in literature are rather involved.
The ring of conditions of $(\mathbb{C}^*)^n$ was introduced by De Concini and Procesi in 1980-th. It is a version of intersection theory for algebraic cycles in $(\mathbb{C}^*)^n$. Its construction is based on the good compactification theorem. Recently two nice geometric descriptions of this ring were found. Tropical geometry provides the first description. The second one can be formulated in terms of volume function on the cone of convex polyhedra with integral vertices in $\mathbb{R}^n$. These descriptions are unified by the theory of toric varieties.
I am going to discuss these descriptions of the ring of conditions and to present a new version of the good compactification theorem. This version is stronger that the usual one and its proof is elementary.
Let $X$ be an algebraic subvariety in $(\mathbb{C}^*)^n$. According to the good compactifification theorem there is a complete toric variety $M \supset (\mathbb{C}^*)^n$ such that the closure of $X$ in $M$ does not intersect orbits in $M$ of codimension bigger than dim$_\mathbb{C} X$. All proofs of this theorem I met in literature are rather involved.
The ring of conditions of $(\mathbb{C}^*)^n$ was introduced by De Concini and Procesi in ...

14M25 ; 14T05 ; 14M17

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