International conferenceTopology and geometry of group
actions
International laboratory
Algebraic topology and its applications
Faculty of Computer Science, NRU HSE
andLaboratory
Algebraic transformation groups
Faculty of Mathematics, NRU HSE
18-22 November 2020

Abstracts of talks
Abstracts are listed in alphabetical order of the speakers. The list of speakers can befound in the last page of this document (page 32).

Substitution complexes and nonrealizability example
Semyon Abramyan(joint with T. Panov)

HSE, Skoltech (Russia, Moscow)
semyon.abramyan at gmail.com

4

Abstract: We discuss the question of realisability of iterated
higher Whitehead products with a given form of nested bracket
by simplicial complexes, using the notion of the moment-angle
complex

ZK .

For an iterated higher Whitehead product

3

5

w

we describe a simplicial complex B ∆w that realises w (seeB∆rrµ ,µ ,µ s,µ ,µ s in Fig. 1). Furthermore, for a particular formof brackets inside w , we prove that B ∆w is the smallest complex
1

2

4

1

3

that realises w . For certain w , we show that ZK is homotopy

equivalent to a wedge of spheres and each sphere is a lift of2

an iterated higher Whitehead product. For example, for K in
Fig. 1, the corresponding moment-angle complex ZK is homo5 4
topy equivalent to S
S 6 3 S 7 S 8 (see [1], [2]).

p q_ _ p q_ _ _It is not true that if ZK splits into a wedge of spheres, then5each sphere is a lift of a Whitehead product. We present an
example of this situation (see [1]).

Figure 1:B∆rrµ ,µ ,µ s,µ ,µ s
1

2

3

4

5

References:[1] S. Abramyan.

plexes.

Sibirsk.

Iterated higher Whitehead products in topology of moment-angle comMat.

Zh. 60 (2019), no. 2, 243256 (Russian); Siberian Math.

J. 60(2019), no. 2, 185196 (English translation).

Higher Whitehead products in moment-angle complexes andsubstitution of simplicial complexes. Proc. Steklov Inst. Math. 305 (2019), 121.
[2] S. Abramyan, T. Panov.

1Commutative algebraic monoids and toric geometry
Ivan Arzhantsev1
HSE University (Moscow, Russia)
arjantsev@hse.ru

Abstract: An (ane) algebraic monoid is an irreducible (ane) algebraic variety S withan associative multiplicationµ : S  S Ñ S,

pa, bq ÞÑ ab,which is a morphism of algebraic varieties, and a unit element e

P S . Examples of ane

algebraic monoids are ane algebraic groups and multiplicative monoids of nite dimen-p q of an algebraic monoid
p q is an algebraic group. By a result of Rittatore, every
algebraic monoid S , whose group of invertible elements GpS q is an ane algebraic group,
is an ane monoid. An ane algebraic monoid S is called reductive if GpS q is a reductive
sional associative algebras. The group of invertible elements G S

S is open in S .

Moreover, G S

algebraic group.
By a group embedding we mean an irreducible ane variety X with an open embedding
G ãÑ X of an ane algebraic group G such that both actions by left and right multiplications of G on itself can be extended to G-actions on X . In other words, the variety X isa pG  Gq-equivariant open embedding of the homogeneous space pG  Gq{∆pGq, where∆pGq is the diagonal in G  G.
Any ane monoid S denes a group embedding GpS q ãÑ S . The converse statement
claims that for every group embedding G ãÑ S there exists a unique structure of anane algebraic monoid on S such that the group G coincides with the group of invertible
elements GpS q. This is proved in [5] under the assumption that G is reductive and in [4]for arbitrary G.
We study commutative ane algebraic monoids. The ground eld K is an algebraicallyclosed eld of characteristic zero. The theory of commutative reductive algebraic monoids,
i.e. algebraic monoids with an algebraic torus as the group of invertible elements, is nothing
but the theory of ane toric varieties.

We concentrate on non-reductive commutative

monoids. Here a crucial role plays the notion of a Demazure root of an ane toric variety.
Let us dene the rank of a commutative monoid S as the dimension of the maximalp q

torus of the group G S . There is a classication of commutative monoids of rank 0, n
1n
and n. In particular, this gives a classication of commutative monoid structures on A forn ¤ 2. More generaly, a classication of commutative monoid structures on normal anesurfaces is given in [2]. Every surface admitting a structure of a commutative monoid turns
out to be toric.

1 This research was supported by the Russian Science Foundation, grant 19-11-00172

2

3
In our main result in [1] we classify commutative monoid structures on A . For b, c

Z¥0 , b ¤ c, denote by Qb,c the polynomial
Qb,c px1 , y1 , x2 , y2 q 



d
¸

d
k



k 1
where c bd

ande, d, e P Z, 0 ¤ e
Qb,c px1 , y1 , x2 , y2 q 

Theorem.

1Pp  q y e bpdkq xdk 1 y k ,e b k 1

x1

1

2

2

b. Note that Qb,c px1 , y1 , x2 , y2 q  Qb,c py1 , x1 , y2 , x2 q

pxb1y2y1b x2 qd 1  pxb1 y2 qd 1  py1b x2 qd 1.
xb1e y1beEvery commutative monoid on A

3

is isomorphic to one of the following

monoids:

01

px1, x2, x3q  py1, y2, y3q

Notation

rk

3A
M A
M

2

M

y1 , x 2 y2 , x 3bx1 y1 , x1 y2 y1b x2 , xc1 y3

A
c

b

1

px1

A
b

A
b,c

M

x1 y1 , xb1 y2

A
b,c

3

3M

y1b x2 , xc1 y3

y1c x3y3 q 
y1c x3 ,Qb,c px1 , y1 , x2 , y2 q ,px1y1, x2y2, xb1xc2y3 y1b y2c x3q,
p x1 y 1 , x 2 y 2 , x 3 y 3 q

Moreover, every two monoids of dierent types or of the same type with dierent valuesof parameters from this list are non-isomorphic.
The proof is based on a classication of pairs of commuting homogeneous locally nilpo-rstent derivations of degree zero on a positively graded polynomial algebra K x1 , x2 , x3 .

Key words and phrases: algebraic group, algebraic monoid, graiding, toric variety, Demazure root.

References:

Commutative algebraic monoid structures onane spaces, Commun. Contemp. Math. 22:8 (2020), 1950064, 23 pp.

[1] I. Arzhantsev, S. Bragin, Yu. Zaitseva,[2] S. Dzhunusov, Yu. Zaitseva,

Commutative algebraic monoid structures on ane sur-faces, Forum Math, in press; arXiv:2007.06698, 17 pp.

[3] L. Renner. Linear Algebraic Monoids. Encyclopaedia Math. Sciences 134, Springer,
Berlin, 2005.
[4] A. Rittatore, Algebraic monoids and group embeddings, Transform. Groups 3:4 (1998),
375-396.
[5]

E. Vinberg,

On reductive algebraic semigroups, In:

Lie Groups and Lie Algebras,

E.B. Dynkin Seminar, AMS Transl. 169, 145-182, Amer. Math. Soc., 19953

B-root subgroups on ane spherical varieties
Roman Avdeev
HSE University (Moscow, Russia)
ravdeev@hse.ru

Abstract: In the study of automorphism groups of toric varieties, a key role is played by
one-parameter additive groups normalized by the acting torus. Such subgroups are called
root subgroups and each of them is uniquely determined by its weight, called the Demazure
root of the corresponding toric variety. Moreover, the set of all Demazure roots admits an
explicit combinatorial description in terms of the fan dening the toric variety, and thisdescription is especially simple in the case when the variety is ane.
In the setting of arbitrary connected reductive groups acting on algebraic varieties, anatural generalization of toric varieties is given by spherical varieties. A spherical variety
is an algebraic variety X equipped with an action of a connected reductive group G in sucha way that a Borel subgroup B of G has an open orbit in X. It turns out that a proper
generalization of root subgroups for spherical varieties is given by one-parameter additive
groups normalized by B, which we call B-root subgroups.

In the talk we shall discuss

B-root subgroups on ane spherical varieties, including basic properties, applications, andopen problems.
The talk is based on a work in progress joint with Ivan Arzhantsev.

On the cohomology of polyhedral products multiplicative results
Anthony Bahri
Rider University (USA)bahri@rider.edu

Abstract:

The problem of computing the cohomology of a polyhedral product has anextensive history. Some of the earliest results in this direction, for the case of momentangle complexes, appeared in the work of M. Franz, I. Baskakov, V. Buchstaber and T.
Panov and also S. López de Medrano. There is also a literature of results for other specicand general CW pairs which I shall review briey.
This talk is a report on the recent completion of a project initiated ten years ago. Our
goal is an explicit description of the additive and multiplicative structure of the cohomology
of a polyhedral product.

The result, for eld coecients and general CW pairs, has atransparency sucient to allow for direct computation. The focus in this lecture will beon the multiplicative results. This is joint work with M. Bendersky, F.R. Cohen and S.
Gitler.

4

Coloring by 15 colors and small covers in dimension 4Ðore Barali¢
Mathematical institute SANU, Belgrade, Serbia
djbaralic@mi.sanu.ac.rs

Abstract: In general, for an n-dimensional simple polytope P , when n¡ 3 it is not known

whether it admits the structure of a small cover or a quasitoric manifolds.

However,the following estimation is known [1] for the chromatic number of a simple polytope P
admitting a small cover (quasitoric manifold)n ¤ χpP q ¤ 2n  1,

p q is the minimal number of colors necessary to paint the facets of P
regularly. In dimension 2, χpP q is equal 2 or 3, while in dimension 3, χpP q is equal 3 or 4.
in sense that χ P

However, in higher dimensions we do not know if the upper bound in the upper inequality
is sharp or not.
In the talk, we construct a simple 4-polytope admitting a small cover and whose chro24 1, the predicted upper bound. The same polytope also admitts

matic number is 15

 ¥ 4 does there always exist a
simple n-polytope P admitting a small cover such that χpP q  2n  1?
a quaitoric manifold. Now an natural question arises: For n

Key words and phrases: small cover, quasitoric manifold, simple polytope, chromatic number.

References:[1] V. Buchstaber, T. Panov. Torus Actions and Their Applications in Topology and Combinatorics. University Lecture Series, 24. AMS, 2002.

Singularities of T n-action on the Grassmannians Gn,2and their universal resolution
Victor M. Buchstaber
Steklov Mathematical Institute (Moscow, Russia)
buchstab@mi-ras.ru

Abstract: The talk is devoted to the classical problem about the structure of the standard
action of the compact torus Tnon the complex Grassmann manifold Gn,2 , which is closely

related to series of problems in algebraic topology, algebraic geometry and mathematicalphysics.

We use the known moment map µ : Gn,2 Ñ ∆n,2 which gives the projection µ̂ : Xn Ñ∆n,2 , where ∆n,2 is the hypersimpex and Xn  Gn,2 {T n .
5In the base of the proposed approach is the notion of the singular point of the orbit
T n for which the set Yn
Xn singXn is an open algebraic manifold, where

space Gn,2

{



z

singXn is the set of singular points. We discuss the relation between the set singXn andthe set of critical points of the moment map as well as with the set of the points from Gn,2
which have the non-trivial stationary subgroup.
Our main result is the construction of the smooth, compact manifold Fn called the uni- Ñ
p p qq p qPversal space of parameters by which we dene continuous, surjective map Gn : ∆n,2
Fnn1
Yn is a dieomorphism and µ̂ Gn x, fx.
Gn,2 T such that the map Gn : Gn Yn
1singXn .
By the same construction it is given the description of the sets Gn z for all z

p

{

qÑ

Note that this description essentially uses the fact that our resolution of singularities of thespace Xn
Gn,2 T n is functorial for the canonical equivariant embeddings of the manifolds

{Gn1,2 and Gn,1 into the manifold Gn,2 .
The talk is based on joint results with Svjetlana Terzi¢.

Key words and phrases: torus action, Grassmann manifolds, singular points.

Torsion free small covers
Li CaiXi'an JiaotongLiverpool University (China)
Li.Cai@xjtlu.edu.cn

Abstract: It is well known that if a small cover is a pullback of the linear model in thesense of [1], then it is torsion free. In this talk we will show the the converse is also true:
if a small cover is toresion free then it must be a pullback of the linear model. This is a
joint work with Suyoung Choi and Hanchul Park.
Details can be found in [2].

Key words and phrases: small cover, pullback of the linear model
References:[1] M. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions,
Duke Math. J., 62 (1991), 417451.
[2] L. Cai, S. Choi and H. Park, On the KO -groups of toric manifolds, to appear (2019).
arXiv 1903.08423.

6

The cohomology of GelfandZeitlin bersJerey Carlson
Fields Institute (Toronto, Canada)
j.carlson@imperial.ac.uk

Abstract:

GelfandZeitlin systems are a well-known family of examples in symplectic

geometry, singular Lagrangian torus brations whose total space is a coadjoint orbit of aunitary group and whose base space is a certain convex polytope. They are easily denedin terms of matrices but do not t into the familiar framework of integrable systems with
non-degenerate singularities, and hence are much studied.
Despite the prominence of Gelfand-Zeitlin systems, not much has been known about
the topology of their bers. In this talk, we compute their integral cohomology rings.
This represents joint work with Jeremy Lane.

Topology of orbit spaces of complexity one torus actions
in non-general position
Vladislav Cherepanov
HSE University (Moscow, Russia)
vilamsenton@gmail.com

Abstract: In this talk, I am going to discuss the results of the paper [1] which is devoted
to orbit spaces complexity one torus actions on smooth manifolds.
main result for actions not in general position.

I will present theIt turns out that for such action, thehomology groups of the orbit space can be arbitrary in dimensions 3 and higher and, giventhat the action is Hamiltonian, homology groups of lower dimensions are trivial. I will also
propose concrete examples of manifolds with such actions  namely, manifolds of isospectral
Hermitian matrices and regular semisimple Hessenberg varieties.
Preliminary knowledge will be given during the talk. For further details, see the reference list.

Key words and phrases: torus actions, complexity one, Hessenberg varieties.
References:[1] A. Ayzenberg, V. Cherepanov, Torus actions of complexity one in non-general posi-tion, to appear in Osaka Journal of Mathematics, preprint: arXiv:1905.04761 [math.AT].
[2] A. Ayzenberg, Torus actions of complexity one and their local properties, Proc. of
the Steklov Institute of Mathematics 302:1 (2018), 1632, preprint arXiv:1802.08828.
[3] V. M. Buchstaber, T. Panov, Toric Topology, Math. Surveys Monogr., 204, AMS,
Providence, RI, 2015.

7

4[4] V. M. Buchstaber, S. Terzi¢, Topology and geometry of the canonical action of T
5on the complex Grassmannian G4,2 and the complex projective space CP , Moscow Mathematical Journal, 16:2 (2016), 237273 (preprint: arXiv:1410.2482).
[5] V. M. Buchstaber, S. Terzi¢, The foundations of (2n,k)-manifolds, preprint:
arXiv:1803.05766.
[6] V. Cherepanov, Orbit spaces of torus actions on Hessenberg varieties, preprint:
arXiv:1905.02294[7] Y. Karshon, S. Tolman, Topology of complexity one quotients, preprint
arXiv:1810.01026v1[8] A. Ayzenberg, M. Masuda, Orbit spaces of equivariantly formal torus actions,preprint: arXiv:1912.11696

Gromov width of symplectic toric manifolds associatedwith graph associahedra
Suyoung Choi
Ajou University (Republic of Korea)
schoi@ajou.ac.kr

Abstract: A graph associahedron is a nestohedron corresponding to a simple graph. It
can be regarded as a Delzant polytope, and, hence, there is the associated symplectic toric
manifold. The Gromov width of a symplectic manifold is an invariant that measures the
maximal size of a ball which can be symplectically embedded. We give an explicit formulafor the Gromov width of symplectic toric manifolds obtained from connected simple graphs.
This talk is based on a joint work with Taekgyu Hwang.

Geometrization of 3-manifolds realizable as small covers
Nikolai Erokhovets

2

Steklov Mathematical Institute of Russian Academy of Sciences (Moscow, Russia)
erochovetsn@hotmail.com

Abstract: Geometrization conjecture of W. Thurson (nally proved by G. Perelman) saysthat any oriented closed 3-manifold which is prime with respect to the connected sum oper-

ation, can be cut along tori so that the interior of each of the resulting manifolds has a geometric structure with nite volume of one of the 8 types. M. W. Davis and T. Januszkiewicz
in Example 1.21 of [1] considered a special class of 3-manifolds, which are small covers over
3-polytopes. They sketched a proof of Thurston's conjecture for manifolds from this class
2 This work is supported by the Russian Science Foundation under grant 20-11-19998

8

using Andreev's theorem.

It turned out that some statements in their exposition need

to be corrected. We show that for this one needs to use the notion of almost Pogorelov
polytopes, which is not mentioned there. Thus, in fact, [1, Example 1.21] led to a problem.
We give a solution of this problem (see [2]).
We use details from the theory of 3-manifolds following [3]. A connected sum of closed

 pN1zint BiqYf pN2zint B2q, where Ni the boundary BBi is endowed with the orientation induced
from Ni zint Bi , and f : B B1 Ñ B B2 is an orientation-reversing homeomorphism. Is can beoriented 3-manifolds N1 and N2 is dened as N1 #N2 :for embedded 3-balls Bi

shown that the homeomorphism type of N1 #N2 does not depend on the choice of Bi andf . An orientable 3-manifold N is called prime if it cannot be decomposed as a non-trivial
33
connected sum of two manifolds, that is, if N  N1 #N2 , then N1  S or N2  S . A 3manifold N is called irreducible if every embedded 2-sphere in N bounds a 3-ball in N . Anorientable, irreducible 3-manifold is prime. Conversely, if N is an orientable closed prime
3-manifold, then either N is irreducible or N  S 1  S 2 . The sphere S 3 is irreducible.
Theorem (Kneser-Haken-Milnor) Let N be a compact oriented closed 3-manifold.
There exist oriented prime 3-manifolds N1 , . . . , Nm such that N  N1 # . . . #Nm . More11over, if N  N1 # . . . #Nm , and N  N1 # . . . #Nn , with oriented prime 3-manifolds Ni
1
and Ni , then m  n and (possibly after reordering) there exist orientation-preserving
1
homeomorphisms Ni Ñ Ni .
By a polytope we mean a combinatorial convex 3-polytope. Boundaries of polytopes arein bijection with combinatorial types of simple spherical graphs with all facets bounded bysimple cycles and any nonempty intersection of two facets being either a vertex or an edge.
A k -belt of a 3-polytope is a cyclic sequence of k -facets such that facets are adjacent if
and only if they follow each other, and no three facets have a common vertex. A simple 33
polytope P dierent from the simplex ∆ is ag, if it has no 3-belts. For a ag 3-polytope P
its real moment-angle manifold RZP and any orientable small cover M(that is,

πi pX q  0 for i ¥ 2).

pP, Λq is aspherical

Any aspherical compact oriented closed 3-manifold isirreducible. A connected sum of simple 3-polytopes P and Q along vertices is a polytope
such that its boundary arises from boundaries of P and Q if we cut the corresponding
vertices of P and Q, glue the complements to the arising triangles inBP and BQ along

their boundaries and delete the 3-cycle from the graph. The following result is known.

Theorem 1. 1) Any simple 3-polytope P can be uniquely decomposed into a connected
sum along vertices of 3-polytopes, such that each polytope is either ∆3 or a ag 3-polytope.
2) For RZP this corresponds to the prime decomposition into a connected sum of copies of
S 3 , S 1  S 2 , and manifolds RZQ of ag 3-polytopes Q.
3) For any orientable small cover M pP, Λq this corresponds to the prime decomposition
into a connected sum of copies of RP 3 and small covers over ag 3-polytopes.
A Seifert bered manifold is a compact 3-manifold N together with a decomposition of
N into disjoint simple closed curves (called Seifert bers) such that each Seifert ber hasa tubular neighborhood that forms a standard bered torus. The standard bered torus
corresponding to a pair of coprime integers pa; bq with a ¡ 0 is obtained from the cylinder
D2  I , if we identify its bases by the rotation by the angle 2πb{a. It has a natural beringby circles.

9A 2-dimensional closed submanifold is called incompressible, if its inclusion induces

injection on π1 . A 3-manifold N is atoroidal if any map T

Ñ N from a torus to N whichp q Ñ π1pN q can be homotoped into the boundary of N .

induces a monomorphism π1 T

Orientable hyperbolic manifold of nite volume is atoroidal. The following theorem wasproved independently by JacoShalen and Johannson.

Theorem. (JSJ-decomposition) Let N be a closed compact orientable irreducible 3manifold. There exists a (possibly empty) collection of disjointly embedded incompressible
tori T1 , . . . , Tm such that each component of N cut along T1 , . . . , Tm is atoroidal or Seifertbered. Any such collection of tori with a minimal number of components is unique up to
isotopy.
An almost Pogorelov polytope is a ag 3-polytope with any 4-belt surrounding a facet.
If P is not a k -prism, then all its quadrangles are disjoint and Andreev's theorem implies
p is realizable as a right-angledthat if we shrink them to points, the arising polytope P
3polytope of nite volume in the Lobachevsky (hyperbolic) space H . The quadrangles of
P correspond to ideal vertices of Pp. Details see in [4]. A connected sum of ag 3-polytopes
P and Q along quadrangles is a polytope such that its boundary arises from boundaries of
P and Q if we glue the complements to quadrangles in B P and B Q along the boundaries
and delete the arising 4-cycle. Our main result is the following.
Theorem 2. 1) Any simple ag 3-polytope P  I 3 can be uniquely decomposed into a
connected sum along quadrangles of k -prisms k ¥ 5 and almost Pogorelov polytopes suchthat for any two prisms glued along quadrangles their bases are not glued along edges.
2) For RZP this corresponds to the minimal JSJ-decomposition: each 4-belt gives a collection of incompressible tori, each k -prism corresponds to a Seifert bered manifold

poriented surface with holesq  S 1
and carries a metric of H2  R with nite volume, and each almost Pogorelov polytope

corresponds to a hyperbolic manifold of nite volume.
3) For any orientable small cover over P this corresponds to the minimal JSJ-decomposition:
each 4-belt gives 1 or 2 incompressible tori, each k -prism corresponds to a Seifert bered
manifold and carries a metric of H2  R with nite volume, and each almost Pogorelov
polytope corresponds to a hyperbolic manifold of nite volume.
4) For non-orientable small covers this gives a collection of incompressible tori and Kleinbottles, such that for their complement each k -prism corresponds to a manifold with a
H2  R metric of nite volume, and each almost Pogorelov polytope corresponds to a hyperbolic manifold of nite volume.
Thus, for real moment-angle manifolds and small covers over simple 3-polytopes the3
3
2
2
following 5 Thurston's geometries arise: S for P  ∆ (simplex), S  R for P  ∆  I
332(3-prism), R for P  I (4-prism), H  R for pieces corresponding to k -prisms, k ¥ 5,
H3 for pieces corresponding to almost Pogorelov polytopes dierent from k -prisms. In
2particular, if P is a k -prism, k ¥ 5, then the manifold has a geometry of H  R, and if
P is a Pogorelov polytope, that is an almost Pogorelov polytope without quadrangles, thenthe manifold is a compact hyperbolic manifold.

10

Key words and phrases: 3-manifold, Geometrization Conjecture of Thurston, prime manifold, JSJ-decomposition, small cover, almost Pogorelov polytope.

References:[1] M. W. Davis and T. Januszkiewicz, Convex polytopes,

tions, Duke Math. J. 62:2 (1991), 417451.

Coxeter orbifolds and torus ac-[2] N. Erokhovets, Geometrization of 3-manifolds realizable as small covers, preprint arXiv(in preparation).
[3] M. Aschenbrenner, S. Friedl, H. Wilton. 3-manifold groups. EMS Series of Lectures inMathematics, 2015.
[4] N. Yu. Erokhovets, Three-dimensional right-angled polytopes of nite volume in the Lo-

bachevsky space: combinatorics and constructions, Proc. Steklov Inst. Math. 305 (2019),

78134.

The cohomology rings of smoothcomplex and real toric varieties
Matthias Franz
University of Western Ontario (London, Ontario, Canada)mfranz@uwo.ca
Abstract:

We phrase our results in the language of toric varieties. They equally holdfor partial quotients of moment-angle complexes and real toric spaces.

p q has been deter-

Additively, the integral cohomology of a smooth toric variety XΣ C
mined in [1]. The result is an isomorphism of graded Z-modulesH  XΣ pCq; Z TorH pBL;Zq Z, ZrΣs





( )

r s is the StanleyReisner ring of Σ and L the torus acting on XΣpCq.

where Z Σ

Contrary to a claim made by BuchstaberPanov, the cup product in cohomology does
not always correspond to the canonical product on the Tor term. It does so if one inverts

2, but in general there are twisting terms that we describe explicitly in terms of the fan Σ,see [3].

For coecients in Z2 , an additive isomorphism analogous to ( ) holds for the real locus

p q

p q

of XΣ C , which is the smooth real toric variety XΣ R . This implies that smooth toricvarieties are M-varieties. As before, the product is twisted, and we give an explicit formulafor it [4].
In the complex case we also describe the morphism induced in cohomology by a toricp q Ñ XΣ1 pCq. Once again, it corresponds to the canonical map between

morphism XΣ C

the Tor terms if 2 is invertible, and to a deformation of it in general [5]. (This part is joint
work with Xin Fu.)

11

The main technical tools of our theory are the notion of a homotopy Gerstenhaber
algebra and the homotopy Gerstenhaber formality of DavisJanuszkiewicz spaces [2]. The
latter generalizes the well-known dga formality result due to the speaker and Notbohm
Ray.

Key words and phrases: toric variety, partial quotient of moment-angle complex, real toric
space, homotopy Gerstenhaber algebra

References:[1] M. Franz, The integral cohomology of toric manifolds, Proc. Steklov Inst. Math. 252(2006), 5362[2] M. Franz, Homotopy Gerstenhaber formality of DavisJanuszkiewicz spaces,
arXiv:1907.04782[3] M. Franz, The cohomology rings of smooth toric varieties and quotients of moment-angle

complexes, arXiv:1907.04791, to appear in Geom. Topol.

[4] M. Franz, The mod 2 cohomology rings of real toric spaces and smooth real toric vari-eties, arXiv:2008.08961[5] M. Franz, X. Fu, Cohomology of smooth toric varieties: naturality, in preparation.

On the GKM correspondence in low dimensions
Oliver Goertsches
University of Marburg (Germany)
goertsch@mathematik.uni-marburg.de

Abstract: The GKM correspondence associates to a torus action of GKM type a labelledgraph encoding the one-skeleton of the action. In the toric symplectic case, this graph isnothing but the one-skeleton of the momentum polytope. Two natural questions arise: in
how far is the space acted on determined by the graph, and which graphs are realizable
by GKM actions? We will discuss three low-dimensional (i.e., 6 and 8) results on thesequestions, obtained in joint work with Panagiotis Konstantis and Leopold Zoller:• Every abstract 3-valent GKM graph that bers over a 2-valent one is realized as2
the projectivization of an equivariant rank 2 vector bundle over a 4-dimensional T manifold. [2]• The dieomorphism type of a simply-connected, compact 6-dimensional integer GKM
T 2 -manifold is encoded in the GKM graph. [1]• In dimension 8, this is no longer true: we construct GKM T 3 -actions with identical
GKM graph on two 8-dimensional compact manifolds with dierent homotopy type.
[3]

12

References:[1] O. Goertsches, P. Konstantis, and L. Zoller, GKM theory and Hamiltonian non-Kähler

actions in dimension 6. Adv. Math. 368 (2020), 107141.

[2] O. Goertsches, P. Konstantis, and L. Zoller, Realization of GKM brations and new

examples of Hamiltonian non-Kähler actions, preprint arXiv:2003.11298.

[3] O. Goertsches, P. Konstantis, and L. Zoller, GKM manifolds are not rigid, preprint
arXiv:2007.13620.

Universal simplicial complexes inspired by toric
topology
Jelena Grbi¢
University of Southampton (UK)j.grbic@soton.ac.uk

Abstract: Let k be the eld Fp or the ring Z. In this talk I'll discuss combinatorial and

p q

p q

n
n
topological properties of the universal complexes X k
and K kwhose simplices arencertain unimodular subsets of k . I'll describe their f -vectors, show that they are shellable
but not shifted, and mention their applications in toric topology and number theory.
nn
As a main result I'll show that X k , K k
and the links of their simplicies are

p q

p q

homotopy equivalent to a wedge of spheres specifying the exact number of spheres in thecorresponding wedge decompositions. This is a generalisation of Davis and Januszkiewicz'snnresult that K Z
and K F2 are n
2 -connected simplicial complexes.
This is joint work with Djordje Barali¢, Ale² Vavpeti£, and Aleksandar Vu£i¢.

p q

p q

p  q

13

Branched coverings of manifolds and nitetransformation groups
Dmitry V. Gugnin
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
dmitry-gugnin@yandex.ru

Abstract: In this talk I will focus on various problems concerning smooth manifolds X andtheir nite smooth transformation groups G such that the orbit space X{G is a smoothable

topological manifold. Certainly, we consider non-free actions of nite groups. The case ofan arbitrary nite transformation group is very dicult. Here I consider two special cases:(A) a nite number of commuting involutions, and (B) transformation groups of odd order.

Theorem α. (Transfer for group actions. Folklore) Suppose X is a nite polyhe-dron, and G - a nite group acting on X simplicially. Then for any eld K of characteristics
zero or p, p|G|, pq  1, the induced map f  : H  pX {G; Kq Ñ H  pX; Kq is an isomorphism
onto the G-invariant subring H  pX; KqG .
There is also the well known variation of this theorem, where X is any paracompact(for us it will be enough to consider the case of metric compacts of nite dimension), andthe singular cohomology is replaced by ech cohomology. And, certainly, for ENR spacesech cohomology is canonically isomorphic to singular cohomology.
mConsider a standard sphere S
x21 x22 . . . x2m 1 1 (of nonzero dimension),and denote by τ x1 , . . . , xm , xm 1
x1 , . . . , xm , xm 1 the standard involution of the

p
qp

p



 q

q

sphere, that interchanges the Northern and the Southern Poles. Take a direct product ofk spheres M m S m1 S m2 . . . S mk of arbitrary dimensions. On each factor there acts





 
¤ ¤

the corresponding involution τi , 1
i k . We get an action of k commuting involutionsmon M , and, therefore, the action of the group Gk
Zk2 . Let us consider the subgroup



P



Ḡk Gk of orientation preserving elements g Gk . This subgroup has index 2, so Ḡk is
k 1
isomorphic to Z2. The following result was obtained in [4].
Theorem 1. (G., 2019) In the above notation the orbit space M m Ḡk is the topologicalsphere S m . Moreover, the projection f : M m
M m Ḡk can be realized by the followingÑ

explicit formula:{{χ : M m Ñ S m,

χpx1,1 , . . . , x1,m1 , x1,m1 1 , x2,1 , . . . , x2,m2 , x2,m2 1 , . . . , xk,1 , . . . , xk,mk , xk,mk 1 q :

px1,1, . . . , x1,m ,bx2,1, . . . , x2,m , . . . , xk,1, . . . , xk,m ; x1,m 1  x2,m 1    xk,m 1q .
x21,1 . . . x2k,m
x21,m 1 x22,m 1    x2k,m 1
In addition, the branched covering χ : M m Ñ S m is globally C ω -smooth and nondegenerate
1

2

1

k

1

k

2

2

k

k

at the points of local homeomorphism.
It turns out that the number pk  1q of commuting involutions in this theorem is a
sharp estimate. Recall, that the rational cup-length LpX q of a connected ENR space
X is the maximal integer m such that there exist m homogeneous cohomology classes
a1 , a2 , . . . , am P H  pX; Qq of positive degrees with nonzero product a1 a2 . . . am  0.
14

Theorem 2. (G., 2020) Let X be a smooth (PL, TOP) closed connected orientablemanifold of dimension m ¥ 3. If the rational cup-length LpX q is equal to k ¥ 3, then forany pk  2q commuting involutions τ1 , . . . , τk2 acting on X and preserving orientation, theorbit space X {Zk2 2 has a nonzero rational Betti number Bq pX {Zk2 2 q (in ech cohomology)
for some 1 ¤ q ¤ m  1. In particular, the orbit space X {Z2k2 is not a Q-homology sphere.
Let us focus on the nite transformation groups of odd order. The following lemma isquite easy (modulo the celebrated Feit-Thompson theorem stating that any group of odd
order is solvable), but I have not found it in literature.

Lemma 1. Let G be a nite group of odd order, and K  R be an arbitrary subeld.
Then for any nite-dimensional linear space V over K and any linear representation G on
V the following equation holds dimpV q  dimpV G q mod 2.
mTake a smooth oriented closed connected manifold X . Suppose it has some rationalm
Betti number Bq pX q, 1 ¤ q ¤ m  1, which is odd. Then by lemma 1 there is no action ofmany nite group G of odd order such that the orbit space X {G is a Q-homology sphere.
For example, we have the following

Proposition 1. (G., 2020) Consider an m-dimensional torus T m . If m is not apower of 2, then there is no action of any nite group G of odd order such that the orbit
space T m {G is the sphere S m .
22
2k
For T there is a classical action of Z3 with the orbit space S . For T , k ¥ 2, thequestion is still open.

p q ¤ i ¤ dimpZ q, the Stiefel-Whitney classes of a smoothable

Let us denote by wi Z , 1
topological manifold Z .

Theorem 3. (G., 2020) Let X be a smooth closed connected manifold of dimensionm ¥ 3, considered with a smooth eective action of a nite group G of odd order. Suppose
the orbit space X {G is a smoothable topological manifold. Denote by f : X Ñ X {G the
canonical projection. Then the induced map in mod 2 cohomology f  : H  pX {G; Z2 q ÑH  pX; Z2 q is an isomorphism onto the subring of invariants H  pX; Z2 qG and, moreover,f  pwi pX {Gqq  wi pX q, 1 ¤ i ¤ m.
In the paper [1] Alexander proved the classical
Theorem β . Let X m be a PL oriented closed connected manifold, mmm

exists a PL nite-fold branched covering f : XÑS .

¥ 2. Then there

But in Alexander's construction the degree n of such a branched covering is always
greater than m!. On the other side the is a classical lower bound for such an n, namely,n L X m . This is a particular case of the celebrated Berstein-Edmonds theorem (see [2]).

¥ pq

A long standing question is that for any dimension m one can construct an n-fold PLmbranched covering X
S m with n m. It is true for m 2, 3, 4, and is still open form 5 even for the torus T m .

¥

Ñ





In the celebrated work of Chernavskii [3] it was obtained a good behavior properties
for branched coverings of manifolds in TOP category.
This work is supported by the Russian Science Foundation under grant 20-11-19998.

Key words and phrases: nite transformation groups, characteristic classes, branched coverings.

15

References:[1] J. W. Alexander,

Note on Riemann spaces, Bull.

Amer.

Math.

Soc.

26:8 (1920),

370372.
[2] I. Berstein, A. L. Edmonds, The degree and branch set of a branched covering, Invent.
Math. 45:3 (1978), 213220.
[3] A. V. Chernavskii, Finite-to-one open mappings of manifolds, Mat. Sb.

65:3 (1964),

357369.
[4] D. V. Gugnin, Branched coverings of manifolds and nH -spaces, (Russian) Funktsional.
Anal. i Prilozhen.

53:2 (2019), 6871; translation in Funct.

Anal. Appl.

53:2 (2019),

133136.

Permutation bases for the cohomology rings of regularsemisimple Hessenberg varieties
Megumi Harada
McMaster University (Toronto)
Megumi.Harada@math.mcmaster.ca

Abstract: Recent work of Shareshian and Wachs, Brosnan and Chow, and Guay-Paquet
connects the well-known Stanley-Stembridge conjecture in combinatorics to the dot actionof the symmetric group on the cohomology rings

H  pHesspS, hqq of regular semisimple

Hessenberg varieties. In particular, in order to prove the Stanley-Stembridge conjecture, itsuces to construct for any Hessenberg function h a permutation basis of H
whose elements have stabilizers isomorphic to reection subgroups.
outline several recent results which contribute to this goal.

In this talk I will

Specically, in some special

cases, we give a new, purely combinatorial construction of classes in H
form permutation bases for subrepresentations in H

 pHesspS, hqq

 pHesspS, hqq which pHesspS, hqq. Our techniques use theGoresky-Kottwitz-MacPherson theory in equivariant cohomology.

Special cases of ourconstruction have appeared in past work of Abe-Horiguchi-Masuda, Timothy Chow, and
Cho-Hong-Lee. This is a report on joint work with Martha Precup and Julianna Tymoczko.

16

Holomorphic foliations and transversal realsubmanifolds in Kähler manifolds
Hiroaki Ishida
Kagoshima University (Japan)
ishida@sci.kagoshima-u.ac.jp

Abstract: This talk is based on the ongoing joint work with Hisashi Kasuya (Osaka University). In this talk I will explain a new construction of a pair of a real submanifold and aholomorphic foliation in a Kähler manifold equipped with a torus action. This construction
provides non-invariant complex structures on complact Lie groups and a generalization of
LVM theory.

Key words and phrases: torus action, moment map, holomorphic foliation.

On homology of Lie algebras over commutative rings
Sergei Ivanov
St. Petersburg University (Russia)
ivanov.s.o.1986@gmail.com

Abstract:

In this talk I will tell you about ve dierent types of homology of a Lie

algebra over a commutative ring.

These ve types are naturally isomorphic over elds.

However, they become non-isomorphic over commutative rings, even over Z. We study
connections between these types of homology, in particular, we construct some spectral
sequences, that connect them. As an auxiliary result we prove that the Koszul complex ofa module M over a principal ideal domain that connects the exterior and the symmetricpowers 0
Λn M
M Λn 1 M
S n 1M M
S nM
0 is pure acyclic.

Ñ

Ñ

b



Ñ  Ñ



b

Ñ

Ñ

Key words and phrases: Homology, Lie algebras, derived functors, Chevalley-Eilenberg
complex.

References:[1] S.O. Ivanov, F. Pavutnitskiy, V. Romanovskii, A. Zaikovskii,

algebras over commutative rings, arXiv:2010.00369.

17

On homology of LieEquivariant orbit preserving dieomorphisms
Yael Karshon
University of Toronto (Canada)
karshon@math.toronto.edu

Abstract:

Let a torus act smoothly on a manifold.

Assume that the manifold is con-nected and the action is faithful. Then every orbit-preserving equivariant dieomorphism
is obtained from an invariant smooth map from the manifold to the torus by acting by thevalues of this function.
This theorem  in a more general setup of torus actions on orbifolds  appeared inHaeiger and Salem's 1991 paper Actions of tori on orbifolds.

The crucial step of theproof was attributed to Gerald Schwarz. Together with Gerald Schwarz, we close a gap inthis proof.

Ring of conditions and cohomology ring of toric bundles
Askold Khovanskii
University of Toronto (Canada)
askold@math.toronto.edu

Abstract: Schubert developed a special calculus in order to solve various problems on
computation of intersection numbers of generic algebraic subvarieties of complementary
dimensions in complex Grassmannians.

In 1980-th De Concini and Procesi developed atheory of rings of conditions which generalized Schubert calculus for a wide class of notnecessarily compact homogeneous spaces.
In the talk I will describe rings of conditions for so-called horospherical homogeneous
spaces. This description relays on computation of cohomology rings of toric bundles, i.eof smooth equivariant compactications of torus principal bundles over a smooth (notnecessarily algebraic) compact base.
The talk is based on join paper with Johannes Hofscheier and Leonid Monin.

References:[1] Johannes Hofscheier, Askold Khovanskii, Leonid Monin.

Cohomology rings of toric

bundles and the ring of conditions. arXiv:2006.12043 [math.AG]18

Two-dimensional Golod complexes
Daisuke Kishimoto
Kyoto University (Japan)
kishi@math.kyoto-u.ac.jp

Abstract: Although Golodness of a triangulation of a closed surface is characterized by its
neighborliness through the fat-wedge ltration of a polyhedral product as in [1], Golodnessof a general two-dimensional simplicial complex is not characterized by its neighborliness.
I will talk about the characterization of Golodness of a general two-dimensional simplicial
complex by introducing a new notion called vertex-breakability. Our characterization applies to give a simple example of a Golod complex whose moment-angle complex is not asuspension.
This is joint work with Kouyemon Iriye [2].

Key words and phrases: Stanley-Reisner ring, Golod property, neighborly complex, polyhedral product, fat-wedge ltration.

References:[1] K. Iriye and D. Kishimoto,

Golodness and polyhedral products for two-dimensional

simplicial complexes, Forum Math. 30 (2018), no. 2, 527532.

[2] K. Iriye and D. Kishimoto, Two-dimensional Golod complexes, arXiv:2010.06753.

Khovanov homology via immersed curves
Artem Kotelskiy
Indiana University Bloomington (USA)artofkot@iu.edu

Abstract:
Khovanov homology is a powerful combinatorially dened knot invariant.

We will

describe a novel geometric approach to Khovanov homology, where the topological input
is a two-sphere S intersecting a knot K in 4 points. The geometric outcome is that ther p qs is equal to the minimal intersection number

dimension of Khovanov homology dim Kh K

of a pair of specically constructed immersed curves on the dividing 4-punctured sphere

S . We will also discuss applications of this result to tangle replacement questions, suchas mutation invariance of KhpK; Qq and the Cosmetic Crossing Conjecture. This is joint
work with Liam Watson and Claudius Zibrowius.

19

Gale dual of a GKM graph and its application to theextension problem of torus actions
Shintaro Kuroki
Okayama University of Science (Japan)
kuroki@xmath.ous.ac.jp

Abstract:LetpΓ, α, ∇q be a GKM graph with an axial function α : E Ñ H 2pBT nq,

where E is the set of edges of Γ. In the paper [1], we dene the group of axial functions

ApΓ, α, ∇q which is isomorphic to the nite rank abelian group without torsions. We have`r : E Ñ H 2 pBT ` q of theproved that if ApΓ, α, ∇q  Z , then there exists an extension α2nr, ∇q is a GKM graph; moreover, this is theaxial function α : E Ñ H pBT q such that pΓ, α11maximal extension among axial functions α such that pΓ, α , ∇q becomes a GKM graph.
However, in general, to compute the rank of ApΓ, α, ∇q is complicated because we need tomake and solve many (at most number of edges) systems of equations.

In this talk I will introduce the Gale dual pΓ, ρ, ∇q of the complexity one GKM graphpΓ, α, ∇q, i.e., Γ is an pn 1q-valent graph and the axial function is α : E Ñ H 2pBT nq.
Here the label ρ : E Ñ Z is dened byρpe1 qαpe1 q    ρpen 1 qαpen 1 q  0
for every vertex p and its out going edges te1 , . . . , en 1 up Ep q. Then, I will explain thefollowing theorem by using examples:pq be an pn 1q-valent complexity one GKM graph and pΓ, ρ, ∇qpq  Zn 1 if and only if pρpe1q, . . . ρpen 1qq P ApΓ, α, ∇qp
for some p P V and te1 , . . . , en 1 up Ep q.
Theorem. Let Γ, α, ∇be its Gale dual. Then, A Γ, α, ∇pqApΓ, α, ∇q onto the vertex p.

Here A Γ, α, ∇ p in Theorem is the restriction of the elements of group of axial functions

This is a joint work with Tomoo Matsumura (see the paper [2]).

Key words and phrases: Extension problem of torus actions, GKM graph, Gale dual.
References:[1] S. Kuroki,

Upper bounds for the dimension of tori acting on GKM manifolds, J. of

Math. Soc. Japan, 71 (2019), 483513.
[2] S. Kuroki, T. Matsumura, Gale duality of GKM graphs, in preparation.

20

Innite transitivity of regular automorphisms on
Calogero-Moser spaces
Karine Kuyumzhiyan
HSE University (Moscow, Russia)
karina@mccme.ru

Abstract: Consider an algebraic variety X and the group of all its regular automorphisms.
For which X any m its (smooth) points can be moved to any other m (smooth) points via aregular automorphism? Such varieties are quite rare. In the work [3] by Berest-EshmatovEshmatov the simplest Calogero-Moser space Cn was studied from this point of view. Thisvariety holds an action of the group isomorphic to the unimodular automorphism group of
the free associative algebra on two variables, this is a subgroup of its regular automorphismgroup.

Berest-Eshmatov-Eshmatov proved 2-transitivity of this action and conjectured

that this action is innitely transitive. In the talk, we will discuss the anatomy of Cn andsketch the proof of innite transitivity.
For details on innitely transitive actions, see [1] and [2]. Details of the proof of innitetransitivity for Calogero-Moser spaces can be found in [4].

Key words and phrases: Calogero-Moser space, innitely transitive action, automorphismgroup.

References:[1]

I. Arzhantsev,

H. Flenner,

S. Kaliman,

F. Kutzschebauch, andM. Zaidenberg,

Flexible varieties and automorphism groups, Duke Math. J. 162 (2013), no. 4, 767823,
arXiv:1210.6937
I. Arzhantsev, K. Kuyumzhiyan, M. Zaidenberg, Innite transitivity, nite generation, and Demazure roots, Advances in Mathematics, 351 (2019), 1-32 arXiv:1803.10620
[2]

[3]

Y. Berest,

A Eshmatov, andF. Eshmatov, Multitransitivity of Calogero-Moser spaces,

Transform. Groups 21 (2016), no. 1, 3550[4] K. Kuyumzhiyan,Innite transitivity for Calogero-Moser spaces, Proceedings of theAmerican Mathematical Society. No. 148 (2020), 3723-373.

21

On the torus orbit closures in ag varieties
Eunjeong Lee
IBS Center for Geometry and Physics (Republic of Korea)
eunjeong.lee@ibs.re.kr

Abstract: Let G be a semisimple Lie group over C, B a Borel subgroup of G, and T amaximal torus of G contained in B . The left multiplication of T on G induces an action{of T on the ag variety G B . It has been known that the closure of a generic torus orbit{{

in G B , which means that the closure contains all the T -xed points in G B , is a smoothprojective toric variety whose fan is associated with the Weyl chamber. In this talk, we{

consider a not necessarily generic torus orbit closure in G B , and we study how to describe

the fan of a torus orbit closure using the matroid maps, which we call retractions. This
talk is based on joint work with Mikiya Masuda and Seonjeong Park.

Key words and phrases: ag varieties, toric varieties, Coxeter matroids.

On polyhedral products over minimally non-Golod
complexes
Ivan Limonchenko
University of Toronto (Canada)iylim@mail.ru

Abstract:

A simplicial complex is called minimally non-Golod if it is not Golod itself(that is, Tor-algebra of its face ring over a certain eld has a non-trivial product or Massey
bracket), but removing of any vertex yields a Golod complex. The results obtained recently
by Amelotte and by Grbic, Ilyasova, Panov, Simmons show that this property of a complex
is closely related to those of its (real) moment-angle-complex (being a connected sum of
products of spheres with two spheres in each product), its Pontryagin algebra (being a onerelator algebra), and the commutator subgroup of its Right-Angled Coxeter group (being
a one-relator group), when the simplicial complex is ag.
In this talk I will recall their results, point out that a similar relationship is likely toexist in a wide class of non-ag minimally non-Golod triangulated spheres, and nallyintroduce some examples showing that non of the above mentioned properties hold for ageneral minimally non-Golod complex.

22

On simple orbifolds
Zhi Lü
Fudan University (Shanghai, China)
zlu@fudan.edu.cn

Abstract: In this talk I will give the notion of simple orbifolds. We will pay more attention
on a special class of simple orbifolds, called simple handlebodies, each of which is a simple
orbifold whose underlying spaces is a handlebody (i.e., the tubular neighborhood of thenwedge of some circles in R ) and can be cut into a simple polytope along some speciccodimension-one suborbifolds. We generalize the notions of ag and belt in the setting of
simple polytopes to the setting of simple handlesbodies, with a dierence. Furthermore, we
study the relations among topology, geometry, combinatorics and the orbifold fundamentalgroups of simple handlebodies. This is a joint work with Lisu Wu.

Key words and phrases: simple handlebody, agness, asphericality, orbifold fundamentalgroup.

Invariants of the cohomology ringsof the permutohedral varieties
Mikiya Masuda
Osaka City University Advanced Mathematical Institute (Japan)
mikiyamsd@gmail.com

Abstract: The permutohedral variety X of complex dimension ntoric variety associated to the permutohedron of dimension n

 1 is a smooth compact

 1. The natural action ofpq

the symmetric group Sn on X induces the action on the cohomology H X; Q and theSring structure of its Sn -invariants H X; Q n is explicitly described by Klyachko [1]. In
S
this talk I will discuss the ring structure of the invariants H X; Q λ under the restricted

p

q

p

q

action of a Young subgroup Sλ . This talk is based on joint work with T. Horiguchi, J.
Shareshian and J. Song.

Key words and phrases: permutohedral variety, Young subgroup, invariants.
References:[1] A. A. Klyachko.

Orbits of a maximal torus on a ag space, Functional Analysis and

Its Applications, 19(1) (1985), 6566.

23

Polyhedral products, loop homology, and right-angled
Coxeter groups
Taras Panov
Lomonosov Moscow State University (Russia)
tpanov@mech.math.msu.su

Abstract: Using results on the topology of polyhedral products, we link distinct conceptsof homotopy theory and geometric group theory.

On the homotopical side, we describe

the Pontryagin algebra (loop homology) of the moment-angle complex ZK . On the group-1

theoretical side, we describe the structure of the commutator subgroup RCK of a rightangled Coxeter group RCK , viewed as the fundamental group of the real moment-angle
complex RK .

For a ag simplicial complex K , we present a minimal generator set for pΩZK q and for the commutator subgroup RCK1 , and specify a1necessary and sucient combinatorial condition for H pΩZK q and RCK to be a free or one-

the Pontryagin algebra H

relator algebra (group). We also give homological characterisations of these properties. For

1 , this is given by a condition on the homology group H2 pRK q, whereas for H pΩZK q
RCK
this can be stated using the bigrading of the homology of ZK .
Parts of this talk are joint works with Jelena Grbic, Marina Ilyasova, George Simmons,
Stephen Theriault, Yakov Veryovkin and Jie Wu.

Uniqueness of a toric structure on a Fano Bott manifold
Seonjeong Park
KAIST (Republic of Korea)
psjeong@kaist.ac.kr

Abstract: A smooth Fano variety is a smooth projective variety X whose anticanonicaldivisorKX is ample. In this talk, we consider the conjecture that two smooth Fano toricvarieties are isomorphic if there exists a c1 -preserving isomorphism between their integral
cohomology rings. This conjecture is related to McDu 's question which asks whether asmooth Fano toric variety may have more than one toric structure.

We show that theconjecture above is true for Fano Bott manifolds. This implies that a toric structure ona Fano Bott manifold is unique. This is a joint work with Yunhyung Cho, Eunjeong Lee,and Mikiya Masuda.
Details can be found in [1].

Key words and phrases: Bott manifold, Fano variety, toric action.
References:[1] Y. Cho, E. Lee, M. Masuda, S. Park, Unique toric structure on a Fano Bott manifold,preprint arXiv:2005.02740.

24

Automorphisms of ane surfaces,
the Markov numbers, and the Thompson group T
Alexander Perepechko
IITP RAS, MIPT, HSE (Moscow, Russia)
a@perep.ru

Abstract:

We describe the correspondence between the unit circle and the divisor atinnity for some ane algebraic surfaces, which induces the correspondence between theThompson group T and the automorphism groups of considered surfaces.
In particular, we generalise the Markov surface and describe automorphism groups ofthe obtained surfaces. Finally, we raise the question about integer point orbits on thesesurfaces.

The answer to it might be useful in studying the Uniqueness conjecture forMarkov numbers.
Details can be found in [1] for the Markov numbers and the Uniqueness conjecture, in[3] and [4] for the Thompson group T, and in [2] for the surface transformations that weuse.

Key words and phrases: ane surface, automorphism group, boundary divisor, Markov
number, Thompson group.

References:[1] M. Aigner. Markov's Theorem and 100 Years of the Uniqueness Conjecture. Springer
International Publishing 2013, DOI:10.1007/978-3-319-00888-2.

Birational transformations of weighted graphs,
in: Ane algebraic geometry, 107147. Osaka Univ. Press, 2007. Corrigendum, in: Ane[2] H. Flenner, S. Kaliman, M. Zaidenberg,

algebraic geometry, 3538. CRM Proc. Lecture Notes, 54, Amer. Math. Soc., Providence,
RI, 2011.
[3] A. Fossas, PSL(2,Z) as a Non-distorted Subgroup of Thompson's Group T, Indiana University Mathematics Journal 60 (2010).
[4] E. Ghys, V. Sergiescu, Sur un groupe remarquable de diéomorphismes du cercle, Comment. Math. Helv. 62, no.2 (1987), 185239, DOI:10.1007/BF02564445.

25

Some topological properties of monotone complexity
one spaces
Silvia Sabatini
University of Cologne, Germany
sabatini@math.uni-koeln.de

Abstract:
In symplectic geometry it is often the case that compact symplectic manifolds withlarge group symmetries admit indeed a Kähler structure.

For instance, if the manifold

is of dimension 2n and it is acted on eectively by a compact torus of dimension n in a
Hamiltonian way (namely, there exists a moment map which describes the action), thenit is well-known that there exists an invariant Kähler structure.

These spaces are also

called complexity-zero spaces, where the complexity is given by n minus the dimension of
the torus. In this talk I will explain how there is some evidence that a similar statement
holds true when the complexity is one and the manifold is monotone (the latter being
the symplectic analog of the Fano condition in algebraic geometry), namely, that every
monotone complexity-one space is simply connected and has Todd genus one, properties
which are also enjoyed by Fano varieties.
These results were largely inspired by a conjecture posed by Fine and Panov in 2010,
by work of Lindsay and Panov of 2019, and it is joint with Isabelle Charton and Daniele
Sepe.
Some of the details can be found in [1].

Key words and phrases: torus action, Hamiltonian spaces, complexity one, monotone.
References:[1] S. Sabatini, D. Sepe, On topological properties of positive complexity one spaces, Transformation Groups 9 (2020).

On LS-category and topological complexity of some
ber bundles and Dold manifolds
Soumen Sarkar
Indian Institute of Technology Madras (India)
soumensarkar20@gmail.com

Abstract: In this talk, I'll extend the product formula for LusternikSchnirelmann category and topological complexity to some classes of ber bundles. Then I'll compute theseinvariants for Dold manifolds. This is a joint work with B. Naskar.

26

On the maximal extension problem for GKM-graphs
Grigory Solomadin
RUDN University (Moscow, Russia)
grigory.solomadin@gmail.com

Abstract: In 1998, M. Goresky, R. Kottwitz and R. MacPherson [1] introduced a specialclass ofpS 1qk -actions on any smooth manifold M 2n with vanishing odd cohomology. They

computed the respective equivariant cohomology ring in terms of so-called GKM-graphs.
kRoughly speaking, an n, k -type GKM-graph is an n-regular connected graph with Z -plabels on its edges.

q

In 2001, V. Guillemin and C. Zara [2] introduced GKM-graphs as

independent combinatorial objects and studied the associated ring.

GKM ber bundles

were introduced by V. Guillemin, S. Sabatini and C. Zara [3] in 2012.

This subclass of

GKM-graphs consists of combinatorial GKM-analogues for torus-equivariant ber bundles.
In 2019, S.Kuroki [4] dened the free abelian group for any GKM-graph and called it thegroup of axial functions. He showed that this group is isomorphic to the group generated
by the edge labels of the maximal extension for a given GKM-graph. In particular, for agiven GKM-manifold the rank of the respective axial function group is an upper boundfor the dimensions of the tori that appear in the extensions of the given GKM-action.
The explicit computation of the axial function group is an open problem. It was recently
conjectured that the group of axial functions has rank n for any 4-linearly independentn-regular GKM-graph.
In the talk (joint work with S. Kuroki) for any n-regular GKM ber bundle (satisfying
some mild conditions) we give the criterion for the axial function group to have rank n.

pn 1q1 r, n 1q-type GKM-graph with axial function group of rank
1 for any integers n ¡ 1, r ¡ 0.

In addition, we disprove the above conjecture by presenting a new example of anlinearly independentnpn

Key words and phrases: torus action, GKM-graph, GKM ber bundle.
References:[1] M. Goresky, R. Kottwitz, R. MacPherson, Equivariant cohomology, Koszul duality, andthe localization theorem, Invent. Math. 131 (1998), 2583.
[2] V. Guillemin, S. Sabatini, C. Zara, Cohomology of GKM ber bundles, J. Alg. Comb.

35 (2012), 1959.
[3]

V. Guillemin, C. Zara,

1-skeleta, Betti numbers, and equivariant cohomology, Duke

Math. J. 2 (2001), 283349.
[4] S. Kuroki, Upper bounds for the dimension of tori acting on GKM manifolds, J. Math.
Soc. Japan 71:2 (2019), 483513.

27

Homotopy classication of 4-dimensional toric orbifolds
Jongbaek SongKIAS (Seoul, Republic of Korea)
jongbaek.song@gmail.com

Abstract:

A toric orbifold [1] is a 2n-dimensional compact orbifold equipped with a
T n -action whose orbit space is a simple polytope. It generalizes the

locally standard

category of (quasi)toric manifolds to the orbifold category. This talk aims to study certain
CW-complexes that model 4-dimensional toric orbifolds and investigate their homotopy
theory. We show that a 4-dimensional toric orbifold without 2-torsion in its cohomology ishomotopy equivalent to the wedge of a Moore space and a CW-complex consisting of evendimensional cells. As a corollary, we prove that the integral cohomology ring distinguishes
the homotopy type of such orbifolds. This is a joint work with Xin Fu and Tseleung So.

Key words and phrases: toric orbifold, homotopy type, cohomological rigidity.
References:[1] M. W. Davis and T. Januszkiewicz Convex polytopes, Coxeter orbifolds and torus actions, Duke Mathematical journal 62:2 (1991), 417451.

Realization of cohomology modulo torsion.
Donald Stanley
University of Regina (Canada)
Donald.Stanley@uregina.ca

Abstract: A classical problem in homotopy theory asks which graded rings are the cohomology of a space.

We simplify this problem by asking which graded rings are thecohomology modulo torsion of a space.

We use polyhedral products to show that any

graded symmetric algebra over Z, modulo a monomial ideal is the cohomology modulo
torsion of a space.
This is joint work with Larry So.

28

The universal space of parameters for T n-action
on the Grassmannians Gn,2Svjetlana Terzi¢
University of Montenegro (Montenegro)
sterzic@ucg.ac.me

Abstract: It is a classical problem to study the structure of the standard action of thecompact torus Tnon the Grassmann manifold Gn,2 . This problem is closely related withseries of problems in algebraic topology, algebraic geometry and mathematical physics, as
it will be demonstrated in this talk as well.

pGn,2, T nq that is a smooth manifold with corners
Un  ∆n,2  Fn and the continuous surjection G : Un Ñ Gn,2 {T n , which implies that the
We recently constructed the model fororbit space Gn,2 is homeomorphic to the quotient space of Un by the map G. Here ∆n,2

is the hypersimplex which in this context appears as an image of the standard momentmap µ : Gn,2Rn and Fn is a new smooth compact manifold. This manifold we call theÑuniversal space of parameters.
In the talk we explain our explicit construction of the universal space of parameters Fn .
This construction essentially uses the techniques of wonderful compactication coming fromalgebraic geometry and singularity theory. Moreover, we show that our universal space ofparameters coincide with the Deligne-Mumford compactication of the moduli space of n1
- pointed Riemann sphere CP .
The talk is based on joint work with Victor M. Buchstaber.

Key words and phrases: torus action, Grassmann manifolds, universal space of parameters.

The homotopy type of the based loops on a
moment-angle complex
Stephen Theriault
University of Southampton (UK)
S.D.Theriault@soton.ac.uk

Abstract: An interesting problem is to determine the homotopy type of a moment-angle
complex. An appealing application of this is given by Buchstaber and Panov, who showedthat the problem is equivalent to determining the homotopy type of the complement ofa corresponding complex coordinate subspace arrangement. A variety of cases have been
worked out but the problem gets dicult quickly. In this talk motivation is given as to why
it is also interesting to consider the homotopy type of the based loops on a moment-angle
complex, and show that methods from homotopy theory (with no immediate connection
to toric topology) can be used to work out additional cases.

29Four-dimensional Generalized-Kähler structures andT-duality
Yury Ustinovskiy
New York University (USA)yu3@nyu.edu

Abstract: A hyperkahler manifold is a Riemannian manifold of dimension 4k and holonomy group contained in Sp(k). Such manifolds are always Kahler and Ricci-at and thus areexamples of Calabi-Yau manifolds. Complete asymptotically at hyperkahler 4-manifolds
are also known as gravitational instantons and are of a special interest in physics. In 1978,
Gibbons and Hawking suggested an ansatz allowing to construct an innite family of com1
plete asymptotically at Hyperkahler 4-manifolds with S -symmetry, unifying a large zoo
1of examples known before. The presence of S -symmetry is crucial for this ansatz, as itallows to reduce a complicated nonlinear system of PDEs to a single Laplace equation.
In this talk we explain how this construction can be modied to produce a much more
exible geometric structure of a Generalized Kähler (GK) manifold. By denition, a GK
manifold carries two integrable complex structures I and J and a (not necessarily Kähler)bi-Hermitian metric, satisfying certain compatibility conditions. We provide an exhaustive
1
local ansatz for GK 4-manifolds with S -symmetry. As an essential part of our construction,
we will use the operation of T-duality, which provides an involution on the set of all GK
1structures with a free S -action.
This talk is based on a joint project with Jerey Streets and more details can be foundin [1].

Key words and phrases: GK structures, S 1 -symmetry, Gibbons-Hawking ansatz, moment
map.

References:[1] J. Streets, Y. Ustinovskiy The Gibbons-Hawking ansatz in generalized Kähler geometry,
preprint arXiv:2009.00778

Polyhedral products and commutator subgroups ofright-angled Artin and Coxeter groups
Yakov Veryovkin
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
verevkin_j.a@mail.com

Abstract: We construct and study polyhedral product models for classifying spaces ofright-angled Artin and Coxeter groups, general graph product groups and their commutator30

subgroups.

By way of application, we give a criterion of freeness for the commutator

subgroup of a graph product group, and provide an explicit minimal set of generators forthe commutator subgroup of a right-angled Coxeter group.
This is a joint work with Taras Evgenievich Panov.
The author is the winner of the 2019 Young mathematics competition in Russia.
This work is supported by the Russian Science Foundation under grant 20-11-19998.
Details can be found in the work [1].

References:
Taras Panov, Yakov Veryovkin, Polyhedral products and commutator subgroups of right-

angled Artin and Coxeter groups, Russian Academy of Sciences Sbornik Mathematics
207(11). DOI: 10.1070/SM8701.

A combinatorial approach to the exponents of Moore
spacesJie WuHebei Normal University (China)
wujie@hebtu.edu.cn

Abstract:This talk gives a report on the combinatorial approach to the exponents ofMoore spaces using Cohen's program.

By working out the combinatorics of the Cohenr 1rgroups, we obtain that the power map pon the loop space of the mod p Moore space
P 2n 1 pr projecting to the atomic piece T 2n 1 pr is null homotopic for p
3, n
1
and r

p q
t u
¡
¡
¡ 1. This gives a shorter proof on the exponents of Moore spaces using the Cohengroups. This is a joint work with Fred Cohen and Roman Mikhailov.
Details can be found in the work [1]. This work is one of our eorts for attacking theBarratt conjecture on the exponents of homotopy groups. Some other recent works on this
topic can be found in [2,3]. The detailed descriptions on the combinatorial approach tothe homotopy theory of loop suspensions can be found in the books [4,5].

Key words and phrases: homotopy groups, Moore spaces, Cohen groups.
References:[1] F. Cohen, R. Mikhailov and J. Wu, A combinatorial approach to the exponents of Moore
spaces, Math. Z. 290 (2018), no. 1-2, 289-305.
[2] F. Pavutnitskiy and J. Wu, A simplicial James-Hopf map and decompositions of theunstable Adams spectral sequence for suspensions, Algebr. Geom. Topol. 19 (2019), no.
1, 77-108.

[3] H. Ruizhi and J. Wu, Combinatorics of double loop suspensions, evaluation maps andCohen groups, J. Math. Soc. Japan Vol. 72, No. 3 (2020) pp. 847-889.
31

[4] J. Wu, On maps from loop suspensions to loop spaces and the shue relations on the

Cohen groups, Mem. Amer. Math. Soc. 180 (2006), no. 851, vi+64 pp.

[5] P. Selick and J. Wu, On natural coalgebra decompositions of tensor algebras and loop

suspensions, Mem. Amer. Math. Soc. 148 (2000), no. 701, viii+109 pp.

On the Picard groups of regular semisimple Hessenberg
varieties
Haozhi Zeng
Huazhong University of Science and Technology (China)
zenghaozhi@hust.edu.cn

Abstract: Regular semisimple Hessenberg varieties are a family of smooth subvarieties of{the full ag varieties G B . This family contains the full ag variety and Permutohedral
variety. In this talk we will discuss on the Picard groups of regular semisimple Hessenberg
varieties. This is joint work with Hiraku Abe and Naoki Fujita in progress.

Key words and phrases: second Betti number, Permutohedral varieties, Hessenberg functions.

References:

The cohomology rings of regular semisimple
Hessenberg varieties for h  php1q, n, . . . , nq, J. Comb. 10 (2019), 2753.

[1] H. Abe, T. Horiguchi, and M. Masuda,[2] J. Carrell, and D. I. Lieberman, Holomorphic vector elds and Kähler manifolds, Invent.
Math. 21 (1973), 303309.
[3] F. De Mari, C. Procesi, and M. A. Shayman, Hessenberg varieties, Trans. Amer. Math.
Soc., 332 (1992), no.2, 529534.

32

List of speakers
Abramyan Semyon, 1

Kuroki Shintaro, 20

Arjantsev Ivan, 2

Kuyumzhiyan Karine, 21

Avdeev Roman, 4
Lü Zhi, 23
Bahri Anthony, 4

Lee Eunjeong, 22

Barali¢ Ðore, 5Limonchenko Ivan, 22

Buchstaber Victor, 5Masuda Mikiya, 23

Cai Li, 6
Carlson Jerey, 7

Panov Taras, 24

Cherepanov Vladislav, 7Park Seonjeong, 24

Choi Suyoung, 8

Perepechko Alexander, 25

Erokhovets Nikolai, 8Sabatini Silvia, 26
Sarkar Soumen, 26Franz Matthias, 11

Solomadin Grigory, 27

Goertsches Oliver, 12Song Jongbaek, 28

Grbi¢ Jelena, 13

Stanley Donald, 28

Gugnin Dmitry, 14
Terzi¢ Svjetlana, 29
Harada Megumi, 16

Theriault Stephen, 29

Ishida Hiroaki, 17Ustinovskiy Yury, 30

Ivanov Sergei, 17
Veryovkin Yakov, 30

Karshon Yael, 18
Khovanskii Askold, 18

Wu Jie, 31

Kishimoto Daisuke, 19
Zeng Haozhi, 32

Kotelskiy Artem, 19

33