Takuro Abe – Free arrangements, combinatorics and geometry

An arrangement is a finite set of hyperplanes in a vector space. An arrangement is said to be free if the logarithmic derivation module, i.e., the polynomial vector fields tangent to the hyperplanes in an arrangement is a free module. By Terao's factorization theorem, freeness is closely related to the Poincaré polynomial of the complement of an arrangement when the base field is the complex number field. It has been a long standing conjecture whether the freeness depends on the combinatorial data of an arrangement, that is still open. We introduce several new developments on this conjecture, including the division theorem, and combinatorial dependency of the addition-deletion theorems. Also, the recent topics on the relation between the Solomon-Terao algebras and the regular nilpotent Hessenberg varieties are explained.

Christin Bibby – The "generating function" of orbit configuration spaces

As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces, using the notion of twisted commutative algebras, which essentially categorify exponential generating functions. With this idea, we will describe a factorization of the orbit configuration space "generating function" into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it encodes all primary, secondary, and higher representation stability phenomena. This is joint work with Nir Gadish.

Dan Burghelea – A computer friendly Alternative to Morse Novikov (AMN) theory [cancelled]

Classical Morse Novikov theory considers pairs (M, ω), where M is a closed manifold, ω a Morse closed differential one form of cohomology class [ω] ∈ H^1(M; R) and relates algebraic topology invariants of (M, [ω]) (like Betti numbers, Novikov Betti numbers, monodromy, torsion, etc.) to the dynamics' elements (rest points, trajectories between rest points, closed trajectories) of X when ω is Lyapunov for X; in particular of X = grad_g ω, where g a Riemannian metric on M.

The limitations of the theory comes from the hypothesis (closed manifold, Morse closed one form) as well as the apparent computer unfriendly character of the intermediate invariants (Novikov field, Novikov homology, torsion). Despite the strong restrictions provided by hypotheses, the results of the theory can be used, among others, in the study of algebraic topology of some complements of hypersurfaces of algebraic varieties which provides pairs (M, ω) as above.

The AMN-theory extends considerably the class of pairs (M, ω) to M a compact ANR and ω a tame one Cech-cocyle, and derives similar results using computer friendly new topological invariants. The lecture will review the new invariants, formulates important results about them and explains why they are computer friendly, and why one expects applications to the topology of complements of hypersurfaces.

Graham Denham – Singular loci of configuration hypersurfaces

A finite graph determines a Kirchhoff polynomial, which is a squarefree, homogeneous polynomial in a set of variables indexed by the edges. The Kirchhoff polynomial appears in an integrand in the study of particle interactions in high-energy physics, which provides some incentive to study the motives and periods arising from the projective hypersurface cut out by such a polynomial.

From this perspective, work of Bloch, Esnault and Kreimer (2006) suggested that the more natural object of study is, in fact, a polynomial determined by a hyperplane arrangement, which is closely related to the basis generating polynomial of the associated matroid. I will describe joint work with Mathias Schulze and Uli Walther on the singular loci of such polynomials.

Gavril Farkas – Topological invariants of groups via Koszul modules

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, or the degree of growth and nilpotency class. For instance, we explicitly bound these invariants in terms of the first Betti number for the maximal metabelian quotients of the Torelli group or any non-fibred Kaehler group. Joint work with Marian Aprodu, Stefan Papadima, Claudiu Raicu and Jerzy Weyman.

Anthony Henderson – Braid groups of normalizers of reflection subgroups

Various results are known about the normalizers of reflection subgroups in a complex reflection group. At the most basic level, in certain situations it is known that the normalizer is a semidirect product of the reflection subgroup and a complementary subgroup. Ivan Marin gave a topological definition of a group which should be thought of as the braid group of such a normalizer, and raised the question of whether it has a similar semidirect product decomposition. In joint work with Thomas Gobet and Ivan Marin, we show that this holds in the Coxeter case but not in general. One corollary is a standard basis for the Hecke algebra associated to the normalizer of a standard parabolic subgroup in a finite Coxeter group.

Shunsuke Ichiki – Characterization of generic transversality

In this talk, the notion of generic transversality and its characterization are given. The characterization is also a further improvement of the basic transversality result and its strengthening which was given by John Mather.

András Cristian Lőrincz – Categories of equivariant perverse sheaves

In this talk, I will discuss some results concerning the category of perverse sheaves on stratified spaces. Under suitable finiteness conditions, such a category is equivalent to the category of finite-dimensional representations of a finite quiver. This is the case when one considers equivariant perverse sheaves on an algebraic variety with finitely many orbits under the action of an algebraic group. We describe such quivers explicitly for some irreducible representations of complex reductive groups and toric varieties. In these cases we use the results to find the explicit D-module structure of local cohomology modules supported in orbit closures.

Laurentiu Maxim – Defect of Euclidean distance degree

Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for nearest point problems of the algebraic variety. It is well known that the latter is an upper bound for the former. While this bound may be tight, many varieties appearing in optimization, engineering, statistics, and data science, have a significant gap between these two numbers. In this talk, I will explain how to compute the difference of these two invariants (usually referred to as the defect of ED degree) for a smooth complex projective variety, by using classical techniques in Singularity Theory. (Joint work with Jose Rodriguez and Botong Wang.)

Florian Pop – Variants of the Grothendieck–Teichmüller group

The Grothendieck–Teichmüller group GT was introduced by Drinfel’d and Ihara, in an attempt to answer the question about giving a combinatorial/topological description of the absolute Galois group G_Q of the field of rational numbers. Unfortunately, the question whether GT=G_Q is wide open. I will discuss old and new results about GT, including its “birational variant” which turns out to be equal to G_Q, thus giving a combinatorial/topological description of G_Q. Finally, time permitting, I will mention work in progress on a “hyperplane arrangements” variant of GT and its relation with G_Q.

Antoni Rangachev – Zariski equisingularity and Lipschitz geometry

In this talk I will describe an approach to showing that Zariski equisingular families of hypersurfaces are Lipschitz equisingular in all dimensions. It is based on the well-known observation that Zariski equisingular families are birational to families of quasi-ordinary singularities having the same characteristic exponents. A singularity is called quasi-ordinary if there exists a finite map from it to an affine space whose discriminant is contained in a normal crossing divisor. I will show that two germs of quasi-ordinary singularities are biLipschitz equivalent if and only if they have the same normalized characteristic exponents. This is joint work with Hussein Mourtada and Bernard Teissier.

Mario Salvetti – On the K(π,1)-conjecture for affine Artin groups

We discuss a recent joint paper with Giovanni Paolini where we gave a proof of the well-known K(π,1)-conjecture for Artin groups of affine type.

Shinichi Tajima – An implementation of the Suwa method for computing versal unfoldings of holomorphic foliations

In the 1980's, Tasuo Suwa investigated singularities of holomorphic foliations and dveloped a theory of unfoldings. He gave in particular a method for computing versal unfoldings of codimension one local foliations. In this talk, we consider the Suwa method from the point of view of computational complex analysis. We show that, by utilizing Grothendieck local duality on residues, the Suwa method can be realized as an algorithm that works on computer algebra systems. Key to our approach is the use of local cohomology. This is a joint work with Katsusuke Nabeshima.

Mihai Tibar – Concentration of curvature and Lipschitz invariants of holomorphic functions of two variables

By combining analytic and geometric viewpoints on the concentration of the curvature of the Milnor fibre, we find some new (discrete) Lipschitz invariants which supplement the (continuous) invariants discovered by Henry and Parusinski in 2003. This reports on joint work with Laurentiu Paunescu.

Uli Walther – Lyubeznik numbers and topology

In 1993, G. Lyubeznik introduced for each commutative local ring A = R/I containing its residue field a set of integer valued invariants by writing it as a quotient of a regular local ring and appealing to (his) results in local cohomology. In particular, for a projective variety X one can associate these numbers to the local ring of the cone over X at the vertex. As one would expect, this connects the corresponding Lyubeznik numbers to properties of X. I will discuss the construction of these numbers (including the basic definitions of local cohomology), survey what is known about them in terms of topology, and discuss the latest developments.

Ruibin Zhang – Equivalence between a category of framed tangles and a category of infinite dimensional representations of quantum SL(2)

We introduce a certain category of framed tangles, and prove an equivalence of it with the full subcategory of quantum SL(2) representations with objects being the tensor products of infinite dimensional projective Verma modules with powers of the 2-dimensional irreducible module. Both categories are shown to be equivalent to the Temperley-Lieb category of type B, which has been widely studied in representation theory. The results generalise the well-known finite dimensional Schur-Weyl duality between quantum SL(2) and the Temperley-Lieb algebra, which underlies the quantum group theoretical construction of the Jones polynomial. This talk is based on joint work with Kenji Iohara and Gus Lehrer.

Posters

Sunil Chebolu and Papa A.Sisspkho – Zero-sum-free tuples and hyperplane arrangements

A vector (v₁, v₂, . . . , v_d) in Z^d_n is said to be a zero-sum-free d-tuple if there is no non-empty subset of its components whose sum is zero in Z_n. We let G^d_n denote the set of all such tuples and we denote the cardinality of this collection by α^d_n. Using the natural action of Aut(Z_n) on G^d_n, we obtain results on the numbers α^d_n including an exact formula when d > n/2, divisibility results, and some bounds in the general case. When n is a prime, we relate this problem to counting points in the complement of a certain hyperplane arrangement and we show that the characteristic polynomial of the hyperplane arrangement captures this number for sufficiently large primes and more generally for integers that are relatively prime to some determinants. We also show how the numbers α^d_n arise naturally in the study of Mathieu subspaces in products of finite fields.

Brian Hepler – Deformation formulas for parameterizable hypersurfaces

We investigate one-parameter deformations of functions on affine space which define parameterizable hypersurfaces. With the assumption of isolated polar activity at the origin, we are able to completely express the Lê numbers of the special fiber in terms of the Lê numbers of the generic fiber and the characteristic polar multiplicities of the comparison, a perverse sheaf naturally associated to any reduced complex analytic space on which the shifted constant sheaf is perverse. This generalizes the classical formula for the Milnor number of a plane curve in terms of double points as well as Mond’s image Milnor number. We also recover results of Gaffney and Bobadilla using this framework. We obtain similar deformation formulas for maps from C² to C³, and provide an ansatz for obtaining deformation formulas for all dimensions within Mather’s nice dimensions

Kazumasa Inaba – Join theorem for mixed hypersurface singularities

Assume that mixed function maps g : Cⁿ → C and h : C^m → C admit tubular Milnor fibrations. Let X_t and Y_t be the Milnor fibers of g and h respectively. We denote the join of X_t and Y_t by X_t ∗ Y_t. In this poster, we show that the existence of a homotopy equivalence a:X_t ∗Y_t →Z_t, where Z_t is the Milnor fiber of g+h.

Yuta Kambe – Computation of singular points and homology of the Hilbert scheme via its Bialynicki-Birula decomposition

Bialynicki-Birula introduced a cell decomposition of a smooth projective variety X such that a 1-dimensional algebraic torus acts on X with finitely many fixed points. Nowadays that decomposition is called Bialyniski-Birula decomposition (BB decom- position for short). Thanks to a recent study of Drinfeld or Jelisiejew and Sienkiewicz, we can define BB decomposition of a singular variety with a 1-dimensional algebraic torus action. In my poster, I will introduce a criterion of singularity on fixed points via BB decomposition. As an example, I will deal with Hilbert schemes that are typical varieties having singular points.

Wataru Komine – How model completeness apply to real algebraic geometry?

The Ax-Grothendieck theorem states that if a polynomial map Cⁿ → Cⁿ is an injection, then it is surjective. The theorem has a simple proof by using completeness of a theory of algebraically closed field. A key of the proof is that the theorem is represented by first-order sentence. A theory of real closed field (RCF) also has completeness and model completeness. We present whether we can use completeness and model completeness of RCF for given claims in real algebraic geometry.

Phil Tosteson – Representation stability and braid Milnor fibers

The of Milnor fiber, F_n, associated to the nth braid arrangements carries an action of the alternating group, A_n, and a mondromy action by the roots of unity μ₍n₂₎. We discuss how methods from representation stability can be used to determine the the homology H_i(F_n, Z) for n ≫ 0, as a representation of these groups.

Takeki Tsuchiya – On properness of polynomial maps F(x, y) = (f(x, y), xy)

We state a condition such that the polynomial map C² → C² defined by F(x, y) = (f(x, y), xy) is proper. As a corollary, we obtain a condition of F(x, y) = (P(x, y), Q(x, y)), Q a quadratic, being proper.

So Yamagata – Combinatorics of the discriminantal arrangement

Let us fix n hyperplanes {H⁰₁ , . . . , H⁰_n} of C^k in general position and consider the space S of parallel translates of n hyperplanes which is defined as n-tuples (H₁, . . . , H_n) such that H_i ∩ H⁰_i = ∅ or H_i = H⁰_i for all i. One can identify S with n-dimensional affine space Cⁿ in such a way that (H⁰₁ , . . . , H⁰_n) corresponds to the origin. For a fixed general position arrangement consider the closed subset D of S which is the collection of parallel translations which fails to be general position. This subset D can be regarded as a hyperplane of S ≃ Cⁿ. The collection of hyperplanes D is called the Discriminantal arrangement and was introduced by Yu. I. Manin and V. V. Schechtman in 1986. We show that the description of codimension 2 intersections of hyperplanes of the Discriminantal arrangement can be characterized as a quadric in Grassmannian. This poster is based on the joint work with S. Sawada and S. Settepanella.