## The Manin-Peyre's conjectures for an infinite family of projective hypersurfaces in higher dimension

For a projective variety containing infinitely many rational points, a

natural question is to count the number of such points of height less

than some bound $B$. The Manin-Peyre's conjectures predict, for Fano

varieties, an asymptotic formula for this number as $B$ goes to

$+\infty$ in terms of geometric invariants of the variety. We will

discuss in this talk the Manin-Peyre's conjectures in the case of the

equation $$x_1y_2y_3\cdots y_n+x_2y_1y_3\cdots

y_n+\cdots+x_ny_1y_2\cdots y_{n-1}=0$$ for every $n \ge 2$.

## Feynman diagrams and a spectral sequence for the space of knots

## New guests at the MPIM

## Algebra up to homotopy

## Motivic obstruction to rationality of a very general cubic hypersurface in P^5

## Multiplicatively dependent points on curves and applications to algebraic dynamical systems

Bombieri, Masser and Zannier (1999) proved that the intersection of a curve defined over a number

field with the union of all proper algebraic subgroups of the multiplicative group $\mathbb{G}_m^n$

is a set of bounded height (unless this is false for an obvious reason). It is important to note that this

set is still infinite as the degree of the points is not bounded.

In this talk we present recent results on multiplicative relations of points on algebraic curves, when

## The Erdős–Kac theorem

the normal distribution with average loglog(n) and variance loglog(n).A proof via the Central Limit

Theorem will be given.

## On Lefschetz exceptional collections and quantum cohomology of Grassmannians

Given a Lefschetz exceptional collection on a variety $X$ one defines its residual subcategory as

the orthogonal to the rectangular part of the collection. In this talk we will discuss some conjectural

relations between the quantum cohomology of $X$ and the structure of the residual subcategory

motivated by homological mirror symmetry. We give examples of this relation when $X$ is an ordinary

or a symplectic isotropic Grassmannian.

## Topological partition functions and (iterated) integrals of modular forms

As a consequence of electric-magnetic duality, partition functions of four-dimensional gauge theories

can be expressed in terms of modular forms in many cases. I will discuss new results for the modularity

of topologically twisted partition functions of N=2 and N=4 supersymmetric theories, and in particular

how these partititon functions may involve (iterated) integrals of modular forms.

## Non-perturbative spectra, quantum curves and mirror symmetry

"Quantum curves'' have been all the rage for various subsectors of the geometry/mathematical physics community for the last few years; yet they might mean different things for different people. I will focus on one and only one angle of this story, due to Grassi--Hatsuda--Kashaev--Marino--Zakany: in their setting, "quantum curves" is the monicker of a precise connection between the spectral theory of a class of difference operators on the real line with trace-class resolvent, and the enumerative geometry (GW/DT invariants) of toric threefolds with trivial canonical bundle.

## Knots-quivers correspondence

I will present a surprising relation between knot invariants and quiver representation theory, motivated

by various string theory constructions involving BPS states. Consequences of this relation include the

proof of the famous Labastida-Marino-Ooguri-Vafa conjecture (at least for symmetric representations),

explicit (and unknown before) formulas for colored HOMFLY polynomials for various knots, new

viewpoint on knot homologies, a novel type of categorification, new dualities between quivers,

and many others.

## Mirror symmetry of branes and hyperbolic $3$-manifolds

We discuss the computation of normal functions between the van Geemen lines on the mirror quintic

Calabi-Yau threefold in a certain semi-stable degeneration limit. In this limit the normal functions are

described as elements of higher Chow groups. Physically this amounts to computing the domain wall

tension between certain B-branes on the mirror quintic in the large complex structure limit. By mirror

symmetry we expect that these normal functions/domain wall tensions have a geometric meaning on the

## Analyticity of Gross--Siebert Calabi--Yau families

Gross and Siebert gave an algorithm to produce from toric degeneration data a canonical formal

Calabi--Yau family. Siebert and I prove that this family is in fact the completion of an analytic

family. In particular, its nearby fibres are decent Calabi-Yau manifolds over the complex

numbers. Furthermore, the family is semi-universal, *i.e.* is in a sense locally the moduli

space of Calabi--Yaus. The key result on the route to analyticity is the computation of canonical

coordinates on the base by explicit integration of a holomorphic volume form over topological

## Comparing local and log GW invariants

Let $X$ be a smooth projective variety and let $D$ be a smooth nef divisor on it. In this collaboration

with Tom Graber and Helge Ruddat, we show that the genus $0$ local Gromov-Witten (GW) invariants

of the total space of $\mathcal{O}(-D)$ equal, up to a factor, the genus $0$ log GW invariants of $X$

with a single condition of maximal contact order along $D$.

## Global mirror symmetry

Conjecturally, global mirror symmetry connects the quantum cohomology of projective varieties which

are birational. In this talk, I will focus on the simplest case of a (dis-)crepant blow-up and explain the

construction of the corresponding global Landau-Ginzburg model

## Birational geometry of singular symplectic varieties and a global Torelli theorem

Verbitsky's global Torelli theorem has been one of the most important advances in the theory of holomorphic

symplectic manifolds in the last years. In a joint work with Ben Bakker (University of Georgia) we prove a

version of the global Torelli theorem for singular symplectic varieties and discuss applications. Symplectic

varieties have interesting geometric as well as arithmetic properties, their birational geometry is particularly

rich. We focus on birational contractions of symplectic varieties and generalize a number of known results

## Stability data, irregular connections and tropical curves

I will outline the construction of isomonodromic families of irregular meromorphic connections

on $\mathbb{P}^1$ with values in the derivations of a class of infinite-dimensional Poisson

algebras, and describe two of their scaling limits. In the ``conformal limit'' we recover a version

of the connections introduced by Bridgeland and Toledano-Laredo, while in the ''large complex

structure limit" the connections relate to tropical curves in the plane and, through work of

Gross, Pandharipande and Siebert, to tropical/GW invariants. This is joint work with

## Line defects in $\mathcal{N}=2$ QFT and framed quivers

I will discuss a certain class of line defects in four dimensional supersymmetric theories

with $\mathcal{N}=2$. Many properties of these operators can be rephrased in terms of

quiver representation theory. In particular one can study BPS invariants of a new kind, the

so-called framed BPS states, which correspond to bound states of ordinary BPS states with

the defect. Such invariants determine the IR vev of line operators. I will discuss how these

invariants arise from framed quivers. Time permitting I will also discuss a formalism to study

## Group actions on quiver varieties and application to branes

We study two types of actions on King's moduli spaces of quiver representations over a

field $k$, and we decompose their fixed loci using group cohomology in order to give

modular interpretations of the components. The first type of action arises by considering

finite groups of quiver automorphisms. The second is the absolute Galois group of a

perfect field $k$ acting on the points of this quiver moduli space valued in an algebraic

closure of $k$; the fixed locus is the set of $k$-rational points, which we decompose

## Tropical Hurwitz and GW numbers

Tropical geometry has been proved successful to study various types of enumerative

numbers, including Gromov-Witten invariants for toric surfaces and Hurwitz numbers

with at most two special points. In my talk I will try to give an overview on

some showcase results, recent developments (counting "real'' curves) and relations

to other approaches.

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