### Preliminary Schedule

Mon
Tue
Wed Thu Fri
Random Matrices
and
Orthogonal Polynomials
Geometry and
Combinatorics
Moduli and
Hurwitz spaces
Commutative algebra
and Geometry
Analysis
and
Topology
9:00-9:20
Registration
and
coffee

Coffee

9:20-9:30
Pasechnik
9:30-9:50
Coffee

Coffee

Coffee

9:50-10:00 break
10:00-10:50 Zarembo Friedland Burman Piene Eremenko
10:50-11:00 break break break break break
11:00-11:50 Martinez-Finkelshtein Fomin Kazarian Ottaviani Saldanha
11:50-13:00 lunch
lunch lunch lunch lunch
13:00-13:50 Horozov Postnikov Lando
Welker Novikov
13:50-14:00 break break

Different activities
and
conference dinner
18:00 at Albanova.

break break
14:00-14:50 Tater
Bränden Stolin Kostov
14:50-15:05 coffee
coffee
coffee
coffee
15:05-15:55 Duits Degtyarev Forsgård Nenashev
15:55-16:05
TBA
break

16:05-16:25

Rojas
16:25-16:55 Mingle with wine and cheese

#### Titles and abstracts

Monday (Random matrices and orthogonal polynomials))

Zarembo 10.00 - 10.50
Title: Random Matrices in Quantum Field Theory
Abstract: Random matrices, or Matrix models, make ubiquitous appearance in many areas of physics,
in particular in Quantum Field Theory. Exact, non-perturbative results in Quantum Field Theory are very rare,
and one often has to rely on various approximation schemes, such as Feynman perturbation theory.
In some cases, however, the problem reduces to correlation function of random matrices
and can then be solved exactly without making any approximations.
Results of this type have been used to put conjectured relationship between Quantum Fields and String Theory to rigorous tests.
I will review how random matrices arise in Quantum Field Theory, going from simple examples based on the Gaussian matrix model
to more complicated, but still solvable matrix models arising in super-symmetric field theories via localization of path integrals.

Martinez-Finkelshtein 11.00 - 11.50
Title: Riding trajectories on a Riemann surface (Asymptotics of multiple orthogonal polynomials for a cubic weight)
Abstract: Polynomials defined by non-hermitian orthogonal relations (or multiple orthogonal polynomials, aka MOPS) with respect to a cubic weight, where the integration goes along non-homotopic paths on the plane, appear in some random matrix theory models. Their asymptotics is related to vector-valued measures on the plane that provide a saddle point for certain energy functionals in an electrostatic model in which the mutual interaction comprises both attracting and repelling forces ("critical vector measures"). For instance, the sequence of zero counting measures of these polynomials converges to the sum of the first two components of such a measure.
Critical vector measures can be characterized by a cubic algebraic equation (spectral curve)  whose solutions are appropriate combinations of the Cauchy transform of its components. In particular, these measures are supported on a finite number of analytic arcs that are trajectories of a quadratic differential globally defined on a three-sheeted Riemann surface.  The complete description of the so-called critical graph for such a differential (and its dynamics as a function of the parameters of the problem) is the key ingredient of the asymptotic analysis of the MOPS.
This is talk is based on the joint work with Guilherme L. F. Silva (U. Michigan, Ann Arbor).

Horozov    13.00 - 13.50
Title: Multiple orthogonal analogs of classical orthogonal polynomials and biorthogonal ensembles
Abstract: Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre have the property
that the polynomials of each system are eigenfunctions of a second order ordinary differential
operator. Using automorphisms of algebras we introduce their multiple orthogonal analogs, which
also have the property to be eigenfunctions of a differential (or difference) operator. They possess
many of the features of the classical orthogonal polynomials such as hypergeometric
representations, ladder operators, Rodrigues-like formulas, Pearson differential equations, etc.
Similarly they have applications to random matrices. Based on these polynomial systems, we
suggest a unified approach to a class of biorthogonal ensembles, which contains the recently studied
by A. Kuijlaars and L. Zhang products of Ginibre matrices, as well as the Borodin-Muttalib
ensembles.

Tater     14.00 - 14.50
TitleSpectral and resonance properties of the Smilansky model
AbstractWe analyze the Hamiltonian proposed by Smilansky to describe
irreversible dynamics in quantum graphs and studied further by Solomyak and others. We derive a weak-coupling asymptotics
of the groung state and add new insights by finding the discrete spectrum numerically in the subcritical case. Furthermore, we
show that the model then has a rich resonance strcture.
This is a joint work with P. Exner and V. Lotoreichik.

Duits  15.00 - 15.50
Title: Random Tilings and non-hermitian orthogonality
Abstract: This talk will report on some recent progress on a new approach for analyzing random tilings
of large planar domains using polynomials that satisfy non-hermitian orthogonality relations.
Two special (and classical) examples that will be discussed are lozenge tilings of a hexagon and domino tilings
of the aztec diamond. In these particular cases Jacobi polynomials with non-standard parameters appear.
The talk is based on joint work (in progress) with Arno Kuijlaars.

Tuesday (geometry and combinatorics)

Pasechnik, 9.20-9.50
Title: An efficient sum of squares nonnegativity certificate for quaternary quartic
Abstract: We give an embarrassingly simple construction of a representation of a
nonnegative 4-ary homogeneous polynomial f (i.e. a form) of degree 4 as a ratio p/q of a sum of squares forms
of degrees 8 and 4, respectively. This allows an efficient check of
nonnegativity of f using semidefinite optimization. No similar results
were known for forms of degree greater than 2 in more than 3
variables.

Friedland, 10.00 - 10.50
TitleTensors and entanglement in quantum physics

Abstract: Tensor, or multi-arrays with at least three indices, are ubiquitous in modern applications, mainly due to data explosion.

While matrices, (two indices), are well understood and widely used, tensors pose theoretical and numerical challenges.
Tensors also arise naturally in quantum physics, when dealing with d-particle systems.
In this talk,  we will describe several fundamental results and problems in tensors:  tensor ranks,
low rank approximation of tensors, spectral and nuclear norm of tensors, and their relation to the entanglement and nonseparability
in quantum information theory.

Fomin, 11.00 - 11.50
Title: Webs, invariants, and clusters
Abstract: Let V be a 3-dimensional complex vector space endowed with a volume form. The special linear group SL(V) naturally acts on collections of vectors, covectors, and matrices. A powerful tool for constructing and manipulating polynomial invariants of such actions is provided by the combinatorial machinery of tensor diagrams, which includes Kuperberg's diagrammatic calculus of webs. In joint work with Pavlo Pylyavskyy, we used these techniques to describe and study cluster structures on classical rings of SL(V)-invariants.

Postnikov
13.00 - 13.50
Title: Purity and separation for oriented matroids.
Abstract: Leclerc and Zelevinsky, in their study of quasi-commuting quantum Plucker coordinates, introduced the notions of strongly and
weakly separated collections of subsets in {1,..,n}.   A key feature of such collections, called the purity phenomenon, asserts that every
maximal by inclusion strongly/weakly separated collection has the same cardinality.   Recently, several other instances of the purity
phenomenon were discovered.   The study of weak separation has close links with the theory of cluster algebras and with the positive
Grassmannian.
In this paper, we extend these notions in the general context of oriented matroids and, in particular, vector configurations.   For an
oriented matroid M, we define M-separated collections.   We show that maximal by size M-separated collections are in bijections with
1-element liftings of M, which in case of realizable matroids correspond to fine zonotopal tilings.
We introduce the class of pure oriented matroids for which the purity phenomenon holds.     We show that an oriented matroid of rank 3 is
pure if and only if it is isomorphic to a positroid.    A graphical oriented matroid is pure if and only if it corresponds to an
outer-planar graph.   We give a conjectural description of pure oriented matroids in terms of forbidden minors and prove it in many
cases.
This is a joint work with Pavel Galashin.

Bränden      14.00 - 14.50
Title: Zeros of mixed characteristic polynomials
Abstract: One of the key ingredients in the recent solution to the Kadison-Singer
problem is to bound the zeros of so called mixed characteristic polynomials.
These polynomials are also related to the van der Waerden conjecture on permanents,
and matching polynomials. We will present open problems and some new results on zeros of such polynomials.

Degtyarev   15.00 - 15.50
Title: Lines on smooth K3-surfaces
Abstract: I will settle a conjecture on the maximal number of lines in a sextic surface in P^4 (42 lines)
or octic surface in P^5 (36 lines). I will also discuss other polarizations of K3 surfaces. The asymptotic maximum is 24 lines.

Rojas   16.00 - 16.50
TitleSharper Topological Bounds for Near-Circuit Polynomials
Abstract: Suppose A is a subset of Z^n of cardinality n+3 with non-defective A-discriminant. We show that a polynomial f supported on A
has at most O(n^2) connected components for its positive zero set Z. The best previous bound (for just the number of non-compact
connected components) was exponential in n. Our bound is based on a more refined look at the singularities of Z as f varies along
certain monomial curves.

Wednesday (Moduli and Hurwitz spaces)

Lando 9.00 - 9.50
Title: Polynomials: Real Hurwitz Numbers and Colored Jones
Abstract: A construction due to D.~Bar-Natan, M.~Kontsevich, and E.~Witten (around 1990) allows one to associate
a knot invariant to any semisimple Lie algebra. The Lie algebra $\sl(\C,2)$ is the simplest
such Lie algebra. The corresponding knot invariant is known to be the aggregate of colored Jones polynomials.
   However, this class of knot invariants is far from being understood completely. In particular, a construction due

to V.~Vassiliev ascribes to a Lie algebra knot invariant a function on chord diagrams. Such functions are called
{\it weight systems}. The weight system corresponding to the Lie algebra $\sl(2)$ takes any chord diagram
to a polynomial in a single variable~$c$, the (quadratic) Casimir element of the Lie algebra.
The explicit values of this weight system are not known even for those chord diagrams where each
chord is a diameter (equivalently, whose intersection graph is a complete graph).
    There are hints and partial results showing that the coefficients of the corresponding sequence of polynomials

are related to Hurwitz numbers of real polynomials. A conjectural explicit formula for the polynomials
in this sequence will be suggested.

Burman  10.00 - 10.50
Title: On Hurwitz-Severi numbers
Abstract: For a point p \in CP^2 and a triple (g,d,\ell) of non-negative integers
we define a Hurwitz--Severi number H_{g,d,\ell} as the number of generic irreducible plane curves of genus g
and degree d+\ell having an \ell-fold node at p and at most ordinary nodes as singularities at the other points,
such that the projection of the curve
from p has a prescribed set of local and remote tangents and lines
passing through nodes. In the cases d+\ell >= g+2 and d+2\ell >= g+2 >
d+\ell we express the Hurwitz-Severi numbers via appropriate ordinary Hurwitz numbers.
The remaining case d+2\ell<g+2 is still widely open.

Kazarian 11.00 - 11.50
Title: Symplectic Geometry of Topological Recursion
Abstract: The topological recursion is a becoming more and more popular recursive procedure leading
to computation of correlators in a wide range of problems in mathematical physics, combinatorics,
and Gromov-Witten theory. It is usually formulated in terms geometry of the so called spectral curve
and the algebra of meromorphic forms on it. We provide a formulation of this procedure
in quite different terms, namely, the initial data of the recursion is a lagrangian subspace
in an infinitely dimensional symplectic space and the result of the recursion is the potential of the problem.
This point of view simplifies many considerations related to topological recursion.

Baryshnikov     13.00 - 13.50
Title: Enumerative combinatorics of Stokes sets
Abstract: Stokes sets, a stratification of the space of univariate polynomials describing the Stokes
phenomena in the theory of oscillating integrals, can be described using a combinatorial object
popularized recently as accordion complex. In this talk I will remind the setp and describe
some recent enumerative results.

Thursday (commutative algebra and geometry)

Piene      10.00 - 10.50
TitleEuclidean projective geometry: reciprocal polar varieties and focal loci
AbstractEuclidean orthogonality and “distance” can be defined in a given (real or com-
plex) projective space with respect to a quadric, or a quadric in the hyperplane at
infinity. Using this, one defines the Euclidean normal bundle and reciprocal polar
varieties of a given, possibly singular, projective variety. These polar varieties are
related to the classical concepts of focal loci and caustics of reflections. In fact,
the focal locus is the branch locus of the end point map of the Euclidean normal
bundle. There has been recent interest in computing, or bounding, the degree of
the focal locus. I will give examples of such computations, especially in the case of
curves, surfaces, and toric varieties.

Ottaviani  11.00 - 11.50
Title: How many Waring decompositions has a polynomial ?
Abstract: Any homogeneous polynomial can be decomposed as the sum of powers of linear
homogeneous forms. We get a Waring decomposition when the number of summands is minimal.
Several interesting polynomials have a unique Waring decomposition, giving a canonical
form. We investigate the number of Waring decompositions, giving the complete
classification for generic polynomials of subgeneric rank,
which is joint work with Chiantini and Vannieuwenhoven. A variant with powers of
homogeneous forms of higher degree has been considered in a joint work with Froeberg and
Shapiro. Here we have asymptotically sharp results and several open problems remain.

Welker     13.00 - 13.50
Title
Subracklattices of finite groups
Abstract: We introduce a news class of lattices associated to (finite) groups
and study their combinatorial structure. For that we need to consider a group as a rack,
a structure with a binary selfdistributive operation, given in case of groups by conjugation.

Stolin      14.00 - 14.50
TitleOn groups of prime square power order
Abstract: We discuss certain results about the groups in the title related to fundamental properties of prime numbers.

Forsgård 15.00 - 15.50
TitleSingularities of Discriminants
Abstract: We will discuss singularities of A-discriminants in terms of the Horn--Kapranov uniformization.
Applications to the study of dual defect toric varieties will be given. This is joint work with J. Maurice Rojas.

Friday (Analysis and topology)

Eremenko 10.00 - 10.50
TitleCircular pentagons and real solutions of Painleve VI equations.
Abstract:  We study real solutions of a class of Painlev´e VI equations. To
each such solution we associate a geometric object, a one-parametric
family of circular pentagons. We describe an algorithm which permits
to compute the numbers of zeros, poles, 1-points and fixed points of
the solution on the positive ray and their mutual position. The
monodromy of the associated linear equation and parameters of the
Painlev´e VI equation are easily recovered from the family of pentagons.

Saldanha    11.00 - 11.50
Title: Spaces of locally convex curves
Abstract: We present several results concerning the homotopy type of spaces of closed locally convex curves in spheres.

Novikov  13.00 - 13.50
TitleWilkie's conjecture for restricted elementary functions (Binyamini and Novikov)
Abstract:
Let X be a set definable in some o-minimal structure. The Pila-Wilkie theorem (in its basic form) states that the number
of rational points in the transcendental part of X grows sub-polynomially with the height of the points.
The Wilkie conjecture stipulates that for sets definable in $R_\exp$, one can sharpen this asymptotic to polylogarithmic.
I will describe a complex-analytic approach to the proof of the Pila-Wilkie theorem for subanalytic sets.
I will then discuss how this approach leads to a proof of the "restricted Wilkie conjecture", where we replace $R_\exp$
by the structure generated by the restrictions of $\exp$ and $\sin$ to the unit interval. If time permits
I will discuss possible generalizations and applications

Kostov  14.00 - 14.50
Title: On the partial theta function
Abstract: The partial theta function $\theta :=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ is connected with
univariate real section-hyperbolic polynomials.
In the talk we explain this link as well as some properties of the spectrum of $\theta$,
i.e. the set of values of the parameter $q$ for which $\theta (q,.)$ has a multiple zero.

Nenashev  15.00 - 15.50
TitleOn a class of commutative algebras associated to graphs
Abstract: I will talk about a class of commutative algebras counting spanning forests and trees,
introduced by A.Postnikov and B.Shapiro. These algebras are defined for an arbitrary non-directed graph,
their Hilbert series are specializations of the Tutte polynomial of this graph.
I will discuss combinatorial properties of the original algebras and some generalizations.

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