# Talk

Talks and seminars, possibly part of a conference or series.

## Geometry and large N asymptotics in Laughlin states

Posted in
Speaker:
Semyon Klevtsov
Affiliation:
Universität Köln
Date:
Tue, 2018-01-09 14:00 - 15:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Alisa Sedunova
Affiliation:
MPIM
Date:
Wed, 2017-11-29 14:15 - 15:15
Location:
MPIM Lecture Hall
Parent event:
Number theory lunch seminar

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

Posted in
Speaker:
Min Hoon Kim
Affiliation:
KIAS, Seoul/MPIM
Date:
Mon, 2017-11-27 16:30 - 18:00
Location:
MPIM Seminar Room
Parent event:
MPIM Topology Seminar

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.

## tba

Posted in
Speaker:
Shalini Bhattacharya
Affiliation:
Bar-Ilan University/MPIM
Date:
Wed, 2017-12-06 14:15 - 15:15
Location:
MPIM Lecture Hall
Parent event:
Number theory lunch seminar

## tba

Posted in
Speaker:
Giordano Cotti
Affiliation:
SISSA, Trieste/MPIM
Date:
Tue, 2017-12-05 14:00 - 15:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Tomasz Przezdziecki
Affiliation:
MPIM
Date:
Wed, 2017-11-29 16:30 - 18:00
Location:
MPIM Seminar Room
Parent event:
IMPRS seminar

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

Posted in
Speaker:
Kevin Destagnol
Organiser(s):
Université Paris Diderot, Paris 7/MPI
Date:
Tue, 2017-12-12 14:00 - 15:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Peter Teichner
Affiliation:
MPIM
Date:
Mon, 2017-11-20 16:30 - 18:00
Location:
MPIM Seminar Room
Parent event:
MPIM Topology Seminar

## New guests at the MPIM

Posted in
Speaker:
Sibasish Banerjee, Giordano Cotti, Mark Penney
Date:
Thu, 2017-11-23 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
MPI-Oberseminar

## Algebra up to homotopy

Posted in
Speaker:
Walker Stern
Affiliation:
MPI
Date:
Wed, 2017-11-22 16:30 - 18:00
Location:
MPIM Seminar Room
Parent event:
IMPRS seminar

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

Posted in
Speaker:
Affiliation:
Liverpool
Date:
Thu, 2017-11-23 10:30 - 12:00
Location:
MPIM Seminar Room

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

Posted in
Speaker:
Alina Ostafe
Affiliation:
University of New South Wales, Sydney
Date:
Wed, 2017-12-13 14:15 - 15:15
Location:
MPIM Lecture Hall
Parent event:
Number theory lunch seminar

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

Posted in
Speaker:
Efthymios Sofos
Affiliation:
University of Leiden/MPIM
Date:
Mon, 2017-11-27 14:00 - 15:00
Location:
MPIM Lecture Hall
The Erdős–Kac theorem states that the number of distinct prime factors of a positive integer n follows
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

Posted in
Speaker:
Maxim Smirnov
Affiliation:
Universität Augsburg/MPI
Date:
Thu, 2017-11-30 15:00 - 16:00
Location:
MPIM Lecture Hall
Parent event:
MPI-Oberseminar

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

Posted in
Speaker:
Jan Manschot
Affiliation:
Trinigy College Dublin
Date:
Thu, 2017-11-30 14:00 - 15:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Andrea Brini
Affiliation:
CNRS Montpellier/Imperial College
Date:
Thu, 2017-11-30 11:30 - 12:30
Location:
MPIM Lecture Hall

"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

Posted in
Speaker:
Piotr Sulkowski
Affiliation:
University of Warsaw
Date:
Thu, 2017-11-30 10:30 - 11:30
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Hans Jockers
Affiliation:
BCTP Bonn
Date:
Thu, 2017-11-30 09:00 - 10:00
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Helge Ruddat
Affiliation:
Universität Mainz
Date:
Wed, 2017-11-29 17:30 - 18:30
Location:
MPIM Lecture Hall

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

Posted in
Speaker:
Michel van Garrel
Affiliation:
Universität Hamburg
Date:
Wed, 2017-11-29 16:30 - 17:30
Location:
MPIM Lecture Hall

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 $D$.

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