## Geometry and large N asymptotics in Laughlin states

Laughlin states are N-particle wave functions, which successfully describe fractional quantum Hall effect (QHE) for plateaux with simple fractions. It was understood early on, that much can be learned about QHE when Laughlin states are considered on a Riemann surface. I will define the Laughlin states on a compact oriented Riemann surface of arbitrary genus and talk about recent progress in understanding their geometric properties and relation to physics. Mathematically, it is interesting to know how do Laughlin states depend on an arbitrary Riemannian metric, magnetic potential function, complex structure moduli, singularities -- for a large number of particles N. I will review the results, conjectures and further questions in this area, and relation to topics such as Coulomb gases/beta-ensembles, Bergman kernels for holomorphic line bundles, Quillen metric, zeta determinants.

## Integral points on elliptic curves over function fields

We give an upper bound for the number of integral points on an elliptic curve E over F_q[T] in terms of its conductor N and q. We proceed by applying the lower bounds for the canonical height that are analogous to those given by Silverman and extend the technique developed by Helfgott-Venkatesh to express the number of integral points on E in terms of its algebraic rank. We also use the sphere packing results to optimize the size of an implied constant. In the end we use partial Birch Swinnerton-Dyer conjecture that is known to be true over function fields to bound the algebraic rank by the analytic one and apply the explicit formula for the analytic rank of E.

## The bipolar filtration of topologically slice knots

The bipolar filtration of Cochran, Harvey and Horn presents a framework of the study of

deeper structures in the smooth concordance group of topologically slice knots. We show

that the graded quotient of the bipolar filtration of topologically slice knots has infinite rank

at each stage greater than one. To detect nontrivial elements in the quotient, the proof

simultaneously uses higher order amenable Cheeger-Gromov L^2 rho-invariants and infinitely

many Heegaard Floer correction term d-invariants. This is joint work with Jae Choon Cha.

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## Vertex algebras and the centre of certain universal enveloping algebras

## 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

restricted to certain proper subfields of the algebraic closure of $\mathbb{Q}$, complementing those of

Bombieri, Masser and Zannier (1999). Some of our initial motivation comes from studying multiplicative

relations in orbits of algebraic dynamical systems, for which we present several results.

## 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. That's both bizarre -- the two subjects have a priori little overlap, and share few-to-none joint practitioners -- and deep/beautiful/powerful to various degrees. I will first review part of the existing depth/beauty/power of this story, and then outline some work in progress aimed at applying mirror-symmetry techniques (in particular, the theory of mutations of potentials of Galkin--Usnich) to the computation of quantum mechanical spectra.

## 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

quintic Calabi-Yau threefold for suitable A-branes. As we discuss, the number theoretic structure of the

computed normal functions indicates that the relevant A-branes correspond to hyperbolic $3$-manifolds.

## 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

cycles that we construct from tropical $1$-cycles in the base of the SYZ-fibration.

## 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 $D$.