The Institute of Mathematics, VAST (IM-VAST)

The International Research and Post-Graduate Training Centre for Mathematics (under the auspices of the UNESCO), IM-VAST

HSE University (The School and Workshop were organized as part of the International Academic Cooperation of HSE University).

Ivan Arzhantsev, HSE University
Dao Phuong Bac, Vietnam National University, Hanoi
Vo Quoc Bao, Institute  of Mathematics-VAST
Ivan Beldiev, HSE University
Sergey Gaifullin, HSE University
Phung Ho Hai, Institute  of Mathematics-VAST
Nguyen Thi Anh Hang, Thai Nguyen University of Education
Truong Thi Hien, Hong Duc University
Vu Tuan Hien, Vietnam National University, Hanoi
Dang Tuan Hiep, Da Lat University
Do Trong Hoang, Hanoi University of Science and Technology
Nguyen Van Ninh, Thai Nguyen University of Education
Veronika Kikteva, HSE University
Alexander Perepechko, HSE University
Ekaterina Presnova, HSE University
Nguyen Duy Tan, Hanoi University of Science and Technology
Dao Van Thinh, Institute  of Mathematics-VAST
Nguyen Thi Tra, Hanoi Pedagogical University 2
Tran Nam Trung, Institute  of Mathematics-VAST
Hoang Le Truong, Institute  of Mathematics-VAST
Kirill Shakhmatov, HSE University
Timofey Vilkin, HSE University
Hoang Ngoc Yen, Thai Nguyen University of Education
Yulia Zaitseva, HSE University
Alexander-Zheglov, Lomonosov Moscow State University

Registration

There is no participation fee.
If you plan to attend the activity please fill out the Online registration form

Contact E-mail: hltruong@math.ac.vn

Deadline for financial support: April 28, 2025.

Deadline for registration: May 7, 2025

Website: http://math.ac.vn/conference/SFVCAT2025

Ivan Arzhantsev, HSE University
Title: Examples of flexible varieties

Abstract: We will present different types of affine flexible varieties that were found during the last decade. The main aim of this talk is to introduce several techniques which allow to prove the flexibility property, and to apply these techniques to varieties under consideration.
Dao Phuong Bac, Vietnam National University, Hanoi
Title: On the cocharacter closedness and cocharacter closure of rational orbits for torus actions over valued fields

Abstract: Let $G$ be a linear algebraic group acting on an affine variety $V$, all are defined over a field $k$. In 2005, G. Roehrle et al proposed a geometric approach to study completely reducible subgroups due to J.-P. Serre via cocharacter closedness of rational orbits $G(k).v$. Here we say that the rational orbit $G(k).v$ of a rational point $v$ is cocharacter closed if this orbit contains the limit point (if exists) along any cocharacter of $G$. Now assume further that $k$ is a valued field, we may endow $G(k)$ and $V(k)$ with the $v$-adic topology induced from that of the base field $k$. The aim of this talk is to discuss the relationship between the cocharacter closedness and Hausdorff closedness of rational orbits, as well as the cocharacter closure and Hausdorff closure of these ones. More precisely, we claim the equality between the cocharacter closure and Hausdorff closure for torus actions. This is a joint work with Vu Tuan Hien and Vo Duy Hoang.

Vo Quoc Bao, Institute of Mathematics, VAST
Title: Cohomology of the differential fundamental group of algebraic curves

Abstract: Let $X$ be a smooth projective curve over a field $k$ of characteristic zero.  The differential fundamental group of $X$ is defined as the Tannakian dual
to the category of vector bundles with (integrable) connections on $X$. This work investigates the relationship between the de Rham  cohomology of a vector bundle with connection and the group cohomology of the corresponding representation of the differential fundamental group of $X$. Consequently, we obtain some vanishing and non-vanishing results for the group cohomology. This is joint work with Phung Ho Hai and Dao Van Thinh.

Ivan Beldiev, HSE University
Title: Projective hypersurfaces of high degree admitting an induced additive action

Abstract:  An additive action is an effective regular action of the algebraic group~$\mathbb{G}_a^m$ on an algebraic variety~$X$. In the case when $X$ is a closed subvariety embedded in the projective space $\mathbb{P}^n$, we can study so-called induced additive actions, i.e. additive actions on $X$ that can be extended to a regular action on the ambient projective space.

The most well-studied case is when $X$ is a projective hypersurface. It is proved, for example, that the degree of a projective hypersurface in $\mathbb{P}^n$ admitting an induced additive action cannot be greater than~$n$. Also, for each $k$ from $2$ to $n$, there exists a non-degenerate hypersurface (i.e. a hypersurface that is not isomorphic to a projective cone over a hypersurface in a smaller projective space) $X\subseteq \mathbb{P}^n$ of degree $k$ that admits an induced additive action. It is known that such a hypersurface is unique up to isomorphism in the extremal cases $k = 2$ and $k = n$. We are going to study the case of hypersurfaces of high degrees. Namely, we are going to give a complete classification of such hypersurfaces for $k = n-1$, $n-2$ and $n-3$.

Sergey Gaifullin, HSE University
Title:  Flexibility of cylinders over affine varieties

Abstract:  Suppose we fix an LND $D$ on an algebra $B$. Let us denote by $A$ the kernel of $D$. In the talk we will consider a technique of lifting an LND of $A$, satisfying some condition to an LND of $B$. We call this technique by Bhatwadekar’s technique. Using this technique, we prove flexibility of cylinders over some classes of varieties.

Phung Ho Hai, Institute of Mathematics, VAST
Title: The fundamental group schemes of pinched schemes and local formal groups

Abstract: Let X be a proper, connected scheme over a perfect field. The fundamental group scheme \pi(X) of X classifies torsor on X under finite group schemes. The behavior of this group scheme is quite misterios due to its local part. We propose a new approach to compute \pi(X) of schemes X obtained from certain "simple schemes" (such as the projective spaces) by pinching at a modulus (i.e. a subscheme of dimension zero). The resulting group schemes turns out to be closely related to a class of irreducible, pointed, cocommutative Hopf algebras (also known as local formal groups) introduced some 50 years ago.

Nguyen Thi Anh Hang, Thai Nguyen University of Education
Title: Flexibility of affine cones over Mukai fivefolds of genus 6

Abstract. A point $p$ in an affine algebraic variety $V$ is called flexible if the tangent space $T_p V$ is spanned by tangent vectors to the orbits of actions of the additive group of the field $\mathbb{G}_a$ on $V$. If every smooth point of $V$ is flexible, we call $V$ a flexible variety. In this talk, we provide a new family of Mukai fivefolds of genus 6 with flexible affine cones. This talk is based on a joint work with Hoang Le Truong.

Do Trong Hoang, Faculty of Mathematics and Informatics, Hanoi University of Science and Technology
Title: Asymptotic regularity of invariant chains of edge ideals

Abstract: In this talk, we present a study of chains of nonzero edge ideals that are invariant under the action of the monoid Inc, consisting of all increasing functions on the set of positive integers. We prove that the sequence of Castelnuovo-Mumford regularity associated to such a chain eventually becomes constant, with the limiting value being either 2 or 3 . Moreover, we explicitly determine the point at which this constancy begins. These results provide further evidence for a conjecture concerning the asymptotic linearity of regularity in Inc-invariant chains of homogeneous ideals. Our proofs also reveal combinatorial properties of such edge ideal chains. These stem from a joint work with Tran Quang Hoa (Hue University of Education) and Nguyen Dang Hop (Institute of Mathematics, VAST).


Đặng Tuấn Hiệp, Dalat University
Title: Computational aspects of Schubert calculus

Abstract: Schubert calculus studies the intersection of Schubert varieties, addressing classical and modern enumerative problems. Computationally, it involves symbolic and combinatorial methods to compute products of Schubert classes in cohomology, quantum cohomology, and K-theory. Key algorithms include the Littlewood–Richardson rule, Pieri rule, and rim-hook algorithm for quantum cases. These computations are essential in solving geometric counting problems and are supported by software such as Maple, Macaulay2, SageMath, and Singular. Recently, the Python package SchubertPy has been developed to perform such computations with a focus on Grassmannians.

Truong Thi Hien, Hong Duc University
Title: On the set of associated radicals of powers of monomial ideals

Abstract: Let $I$ be a monomial ideal in a polynomial ring. In this paper, we study the asymptotic behavior of the set of associated radical ideals of the (symbolic) powers of $I$. We show that both $\asr(I^s)$ and $\asr(I^{(s)})$ need not stabilize for large value of $s$. In the case $I$ is a square-free monomial ideal, we prove that $\asr(I^{(s)})$ is constant for $s$ large enough. Finally, if $I$ is the cover ideal of a balanced hypergraph, then $\asr(I^s)$ monotonically increases in $s$. Joint work with N.T. Hang.

Vu Tuan Hien, Faculty of Mathematics, VNU University of Science
Title: Cocharacter closedness and complete reducibility of subgroups.

Abstract: Let $G$ be a reductive group over a field $k$. Motivated by the complete
reducibility in group representation theory, we have the notion of completely reducible algebraic groups and its relative version. A subgroup $H$ of $G$ is called $G$-completely reducible ($G$-cr) over $k$ if whenever $H$ is contained in a $k$-defined $R$-parabolic subgroup $P$ of G, there exists a $k$-defined $R$-Levi subgroup of $P$ containing $H$.  In 2005, G. Roehrle et al initiated a geometric approach to complete reducibility via cocharacter closedness and multi-conjugate actions. The aim of this talk is to discuss some results on completely reducible subgroups and cocharacter closures of conjugate actions over a valued field.

Veronika Kikteva, HSE University
Title:  On the connectedness of the automorphism group of an affine toric variety
We obtain a criterion for the automorphism group of an affine toric variety to be connected, stated in combinatorial terms and in terms of the divisor class group of the variety. We describe the component group of the automorphism group of a nondegenerate affine toric variety. In particular, we show that the number of connected components of the automorphism group is finite.

Nguyen Van Ninh, Thai Nguyen University of Education
Title: Classify local algebras of modality two

Abstract: In 1999, B. Hassett and Yu. Tschinkel introduced a correspondence between generically transitive actions of the commutative unipotent algebraic group $G_a^n$ and finitedimensional local algebras. The modality $\bmod (R)$ of local algebra $R$ is the modality of the action of $G_a^n$ on $\mathbb{P}^n$. In this talk, we classify local algebras of modality two.

Alexander Perepechko, HSE University
Title: Cylinders on weak del Pezzo surfaces

Abstract:  Weak del Pezzo surfaces are characterized by the anticanonical divisor being nef but not necessarily ample. Their anticanonical polarizations admit ADE singularities and are called singular del Pezzo surfaces.
We will survey known cylinder constructions for weak del Pezzo surfaces. They allow to study generic flexibility of affine cones over both weak and singular del Pezzo surfaces depending on a chosen polarization.

Ekaterina Presnova, HSE University
Title: Affine monoids

Abstract: An affine algebraic monoid is an irreducible affine algebraic variety X with an associative multiplication μ: X × X → X, which is a morphism of algebraic varieties and admits a unit element. The group of invertible elements G(X) of an algebraic monoid X is an algebraic group, which is Zariski open in X. We construct affine monoids on toric varieties, there G is semidirect product of G_a^k and  torus T. We study affine monoids with active group of invertible elements.


Dao Van Thinh, Institute of Mathematics, VAST
Title:Stratified cohomology on hyperelliptic curves

Abstract: Let $k$ be an algebraically closed field of characteristics $p>0$ and $X$ be a smooth connected projective variety over $k$. Write $\mathcal{D}_X$ for the sheaf differential operators on $X$ (in the sense of Grothendieck), and a $\mathcal{D}_X$-module is called a stratified bundle (or $F$-bundle, where $F$ stands for the Frobenius map). Now fix a stratified bundle $E$, we mainly concerned with the case $E$ is coherent as an $\mathrm{O}_X$-module.

In the first half of the talk, following Ogus, I will recall the stratified cohomology of $X$ with coefficient $E$ (another name: the cohomology of the infinitesimal site). This cohomology theory was initiated by Grothendieck when he attempted to develop a $p$-adic cohomology in characteristic $p>0$. Soon later, he realized that it would be better to add the "divided powers" structure (i.e. crystalline site). While people abandon the stratified cohomology theory, we still want to study it because of its relation to the group cohomology of the stratified fundamental group, and to the étale cohomology of $p$-torsion sheaves, both of which demand rigorous exploration.
The second part of the talk is about computations on the stratified cohomology of a (not necessarily coherent) stratified sheaf on hyperelliptic curves in characteristic 3 . This will open up some questions on the non-vanishing of the stratified cohomology and illuminate "how bad the theory is."
(joint work with Vo Quoc Bao and Phung Ho Hai)

Nguyen Thi Tra, Hanoi Pedagogical University 2
Title: Dimensions of Zassenhaus filtration subquotients of right angled Artin groups

Abstract: We determine the restricted Lie algebra associated with the Zassenhauss filtration of a right-angled Artin group. We also provide the formular to define the dimensions of Zassenhaus filtration subquotients of this groups.

Hoang Le Truong, Institute of Mathematics, VAST
Title: Flexibility of Affine Cones over Complete Intersections of Three Quadrics

Abstract: In this talk, we present a smooth family of complete intersections of three quadrics in P^7 that includes fibers with flexible affine cones as well as fibers without flexible ones. This is a joint work with N.T.A. Hang.

Roman Stasenko, HSE University
Title: On Chistophersen's problem
Abstract: Let $A$ be an abelian local algebra over the field $\mathbb{C}$ of dimension $n$. In the beginning of 21th century Jan Christophersen ask the question, is it true that  the dimension of group $\aut (A)$ is more than $n-1$ and equality should only happen when A is a hypersurface. This problem is still opened nowadays.
In 1996 S.Yau proved, that for graded artinian local algebras the following bound is true:
 $$\dim \aut (A)\geqslant \dim A- \dim \operatorname{Soc}(A).$$
 Then in 2013 A. Perepechko give another lower bound:
 $$\dim \aut (A)\geqslant \dim (\mathfrak{m}/ \mathfrak{m}^2)\cdot \dim \operatorname{Soc} (A).$$
 Let $S$ be a reductive algebraic group and let $\mathfrak{g}$ be an algebra. The homomorphism $\Phi:S\rightarrow \operatorname{Aut} (\mathfrak{g})$ is called the $S$-structure on the algebra $\mathfrak{g}$. $S$-structure can be viewed as another type of graduation on $\mathfrak{g}$.
The aim of the talk is to give some remarks to this problem and, using the upper bounds, prove their analogies for the $S-$graded local algebras. One of the part of the talk, will be the answer on the question, is it true, that $S-$graded local algebra has  $S$-graded Lie algebra of derivations?

Kirill Shakhmatov, HSE University
Title:  Flexibility of affine cones over intersections of two quadrics

Abstract:  Consider a smooth complete intersection $X$ of two quadrics in $P^(n+2)$ for $n\geq3$. It is known that $X$ is a Fano variety with Picard number 1 and index $n-1$. In this talk we will discuss flexibility of affine cones over $X$

Timofey Vilkin, HSE University
Title:  On flexibility of trinomial varieties

Abstract:  Trinomial varieties are affine varieties given by some special system of equations consisting of polynomials with three terms. In this talk, we discuss some properties of trinomial varieties related with flexibility and prove sufficient conditions to be flexible for an arbitrary trinomial variety.

Hoang Ngoc Yen, Thai Nguyen University of Education
Title: On sectional genera and Cohen-Macaulay rings

Abstract: My talk is based on joint work with H.L. Truong. Let ( $R, \mathfrak{m}$ ) be a commutative Noetherian local ring of dimension $d$, where $\mathfrak{m}$ is the maximal ideal. Let $I$ be an $\mathfrak{m}$ primary ideal of $R$. It is well-known that there are integers $\mathrm{e}_i(I, R)$, called the Hilbert coefficients of $M$ with respect to $I$, such that
$$
\ell_R\left(R / I^{n+1}\right)=\mathrm{e}_0(I, R)\binom{n+d}{d}-\mathrm{e}_1(I, R)\binom{n+d-1}{d-1}+\cdots+(-1)^d \mathrm{e}_d(I, R)
$$
for all $n \gg 0$. Here $\ell_R(N)$ denotes the length of an $R$-module $N$. In 1987, A. Ooishi ([1]) introduced the notion of sectional genera in commutative rings. Let
$$
\operatorname{sg}(I, R)=\ell_R(R / I)-\mathrm{e}_0(I, R)+\mathrm{e}_1(I, R)
$$
and call it the sectional genus for $R$ with respect to $I$. In this talk, we provide characterizations of a Cohen-Macaulay local ring in terms of sectional genera, the CohenMacaulay type, and the second Hilbert coefficients for certain primary ideals.

References
[1] A. Ooishi, $\Delta$-genera and sectional genera of commutative rings, Hiroshima Math. J., 27 (1987), 361-372.

Yulia Zaitseva, HSE University
Title:  On projective hypersurfaces with an additive action

Abstract:  We investigate projective hypersurfaces that admit an induced additive action, meaning an effective action of the vector group with an open orbit, which can be extended to an action on the ambient projective space. Hassett and Tschinkel established a relationship between commutative local Artinian unital algebras and additive actions on projective spaces. This framework can be utilized to explore additive actions on projective hypersurfaces. I will discuss recent results including the uniqueness of actions and the normality of such hypersurfaces. The talk is based on joint works with Ivan Arzhantsev and Ivan Beldiev.

Ivan Arzhantsev, HSE University
Ivan Beldiev, HSE University
Tran Do Minh Chau,  Thai Nguyen University of Education
Doan Trung Cuong,  Institute of Mathematics, VAST
Le Van Dinh, FPT University, Vietnam
 Hoang Phi Dung, Posts and Telecommunications Institute of Tecnology, Vietnam
Sergei Gaifullin, HSE University
Nguyen Thi Anh Hang, Thai Nguyen University of Education
Nguyen Thu Hang, Thai Nguyen University of Sciences
Pham My Hanh,  An Giang University
Truong Thi Hien, Hong Duc University
Le Tuan Hoa, Institute of Mathematics, VAST
Do Trong Hoang, Hanoi University Sciences and Techology
Nguyen Van Hoang, University of Transport and Communications
Nguyen Dang Hop, Institute of Mathematics, VAST
Mikhail Ignatev, HSE University
Do Van Kien, Hanoi Pedagogical University 2
Veronika Kikteva, HSE University
Ha Minh Lam, Institute of Mathematics, VAST
Pham Hong Nam, Thai Nguyen University of Sciences
Nguyen Van Ninh, Thai Nguyen University of Education
Aleksandr Perepechko, HSE University
Ekaterina Presnova, HSE University
Pham Hung Quy, FPT University, Vietnam
Nguyen Thanh Tam, Hung Vuong University
Tran Dai Tan, Institute of Mathematics, VAST
Phan Thi Thuy, Hanoi National University Education
Nguyen Thi Tra, Hanoi Pedagogical University 2
Tran Nam Trung, Institute of Mathematics, VAST
Hoang Le Truong, Institute of Mathematics, VAST
Nguyen Bich Van, Institute of Mathematics, VAST
Anton Shafarevich, HSE University
Kirill Shakhmatov. HSE University
Roman Stasenko, HSE University
Do Hoang Viet, Institute of Mathematics, VAST
Timofei Vilkin, HSE University
Hoang Ngoc Yen, Thai Nguyen University  of Education
Iuliia Zaitseva, HSE University
Alexander-Zheglov, Lomonosov Moscow State University