GTIIT workshop in combinatorics, sandpiles, tropical geometry, and number theory
Dates: 1–26 September 2026
Venue: Guangdong Technion – Israel Institute of Technology (GTIIT), Shantou, China
During September we will have a small informal workshop at GTIIT, with about ten participants coming at different times during the month.
The main topics will be combinatorics, sandpiles, tropical geometry, and elementary number theory, with related subjects very welcome. We expect to have many talks and, equally importantly, plenty of time for informal discussions and working sessions.
Format and topics
Each visitor will be invited to give a talk introducing their current research, and I will present several projects and open problems that I would particularly like to discuss with the participants. We expect approximately 2–3 hours of talks per day, leaving substantial time for discussions and joint work.
Among the topics I plan to discuss is a tropical approach to the Monge–Ampère equation, related to this recent preprint. It would be particularly useful if someone could give an introductory talk on the classical Monge–Ampère equation, its geometric meaning and applications, or on numerical methods for solving Monge–Ampère equations. If you would like to volunteer for such a talk, please let me know; I can suggest some papers and background material.
Another theme will come from sandpiles and 2-adic phenomena. We will discuss an unexpected connection with Chebyshev polynomials and their arithmetic, including divisibility and congruence phenomena, 2-adic properties, and related questions in elementary number theory. An introductory talk on Chebyshev polynomials and their number-theoretic aspects would therefore fit very naturally into the programme.
Abdul Quadir will talk about the limits of Directed Abelian Sandpiles on Cylinder. Guillaume Tahar will establish an important partial result concerning the asymptotic part of the conjecture about line arrangements decomposing the real projective plane into triangles (they form a particularly rigid class of objects in combinatorial geometry). He will also speak about the residual map associating to each meromorphic differential on a Riemann surface the configuration of its residues at the poles; singular behavior occurs along an arrangement of complex hyperplanes whose topological and combinatorial properties remain largely mysterious.
Most importantly, talks do not have to present finished work. Introductions to a subject, work in progress, computations, conjectures, open problems, incomplete arguments, and ideas that one would like to discuss with other participants are all particularly welcome.
Talks and discussions will be announced on this page as they are scheduled. The programme will therefore evolve throughout September.
Students are very welcome. If you know students or colleagues who might be interested in attending some of the talks or discussions, please send them this page (and I can add them to the corresponding WeChat group).
The workshop is deliberately informal. Many of the talks will be about work in progress, unfinished projects, open questions, preliminary ideas, or introductory material, rather than polished final results. The aim is to create an environment in which it is easy to develop ideas together.
Schedule, abstracts, materials
The schedule will be updated as talks and discussion sessions are confirmed. All talks are in E2-106, SC.
Monday, 31 August, 10:00. (Nikita Kalinin) Introduction to sandpiles. Open problems.
Monday, 31 August, 16:15. (Nikita Kalinin) Introduction to sandpile groups.
(Wednesday, 2 September, 10:00, Andrey Kupavskii) Intersection theorems
I'll discuss some of the basic results in extremal set theory: Erdos-Ko-Rado theorem, Katona's theorem, as well as some of the classical methods.(Wednesday, 2 September, 16:15, Guillaume Tahar) Simplicial arrangements and the geometry of planar cubic curves
In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. Combining this theorem with a rigidity result on regular simplicial arrangements, we obtain that Grünbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points. This is a joint work with Dmitri Panov.(Thursday, 3 September 10:15, Maksim Klimenko) Covering integer points of euclidean ball by subspaces
Let $C \subset \mathbb{R}^n$ be an central-symmetric convex body, and let $g(C)$ denote the minimum number of proper linear subspaces required to cover lattice points inside $C$. In 2002, Bárány, Harcos, Pach, and Tardos established general lower and upper bounds for this quantity. In talk we discuss the connection between this problem and lattice packing densities, and improved upper bounds which rely on this link. Also we will talk about lower bounds and related algorithms(Thursday, 3 September 16:15, Elizaveta Iarovikova) Intersecting families of linear spaces
We discuss large families of k-dimensional subspaces of an n-dimensional space over a finite field such that any two subspaces have intersection of dimension at least t. If n>2k the largest possible family consists of all subspaces that contain a fixed t-dimensional space, if n<2k, the largest example is the family of all subspaces of some fixes (2k-t)-subspace. If n=2k, both example are possible, moreover they are dual. We are interested, how large can these families be if they are different from examples discussed above. We will discuss several results in this field and several approaches such as spread-approximations method and spectral analysis of the Grassman scheme.(Friday, 4 September, 10:15, Nikolai Terekhov) Weak Saturation and the Rank of a Closure System
The weak saturation number of $K_t$ in $K_n$ is the minimum number of edges in an $n$-vertex graph $F$ such that the missing edges can be added one at a time in some order, with each added edge creating a new copy of the clique $K_t$. Despite the simplicity of this definition, determining these numbers for all $t$ requires a linear-algebraic method, and no combinatorial proof is known. We will present a more general and natural perspective on weak saturation problems. This framework unifies a number of related questions and helps explain why linear-algebraic methods are so effective in weak saturation numbers.Monday, 7 September, 10:00. 2-adic phenomena in sandpiles-1.
Monday, 7 September, 16:00. 2-adic phenomena in sandpiles-2. solution
(Tuesday, 8 September, 10:00, Abdul Quadir) Algebraic and Dynamical Structures in Directed Abelian Sandpiles on Cylinder
Self-organized criticality (SOC) describes how slowly driven dissipative systems can spontaneously organize into a critical state without external fine-tuning, exhibiting scale-free avalanches and long-range correlations. The sandpile model provides a paradigmatic realization of SOC, in which the addition of individual grains, followed by local threshold-driven relaxation, generates avalanches across a broad range of scales. The directed Abelian sandpile model on a cylinder provides a setting for exploring the interplay between critical dynamics, algebraic structure, and topology. For a cylinder of longitudinal length \(n\) and circumference \(L\), recurrent configurations can be characterized through a finite Abelian sandpile group, whose structure can be resolved using the Smith normal form. Natural maps between groups of different system sizes lead to algebraic hierarchies described by inverse and direct systems, along with their corresponding projective (profinite) and inductive limits. The compatibility of these constructions, sufficient conditions for the commutation of the corresponding limiting procedures, and possible obstructions in the general case will be discussed. Connections between these algebraic structures and recurrent configurations, periodic and randomly driven dynamics, avalanche statistics, and random-walk behavior will also be presented. The hierarchy across system sizes further provides a possible route toward algebraic coarse-graining and renormalization, linking SOC dynamics, finite Abelian groups, and scale transformations within a common framework.Tuesday, 8 September, 16:00. (Abdul Quadir) Algebraic and Dynamical Structures in Directed Abelian Sandpiles on Cylinder
(Wednesday, 9 September, 10:00, Mikhail Shkolnikov) Basics of Tropical Relaxation
We we’ll cover an operational minimum on relaxation in tropical sandpiles, including singularities and deformations of tropical series on planar convex domains needed to state the stochastic approach to numerically solving Monge-Ampère equation.(Wednesday, 9 September, 16:00, Mikhail Shkolnikov) Faster Solvers, Boundary Conditions, and the question of Zeno
During this time together, we will try to distribute the labour in a project of adopting the setup discussed in the morning to an industrial-level solver of the MA nonlinear PDE, as well as review the status of related open problems.(Thursday, 10 September, 16:00, Jacob Shubin) Algebraic Approaches to Intersecting Families of Finite Sets
Using the Johnson association scheme and the representation theory of the symmetric group, we describe the relevant eigenspaces and eigenvalues. As an application, we derive the Erdős–Ko–Rado theorem via Hoffman's bound.(Friday, 11 September, 9:30, Nikolai Terekhov) Entropy and $(k,L)$-Systems
For an integer $k$ and a set of integers $L$, a $(k,L)$-system is a family of $k$-element subsets such that the size of the intersection of any two distinct members belongs to $L$. We will show that the asymptotic behavior, as $n\to\infty$, of the maximum size of a $(k,L)$-system on an $n$-element ground set is closely related to almost entropic vectors and information inequalities. In particular, Shannon-type information inequalities will lead to a general upper extending all previously known general bounds.(Friday, 11 September, 11:00, Georgy Sokolov) Introduction to Erdos Matching Conjecture
One of the most famous open problem in extremal set theory asks for the largest possible size of a family of k-element subsets of an n-element set, that does not contain s pairwise disjoint sets. The problem is usually refered as the Erdos Matching Conjecture, since Erdos made a very probable conjecture about the answer. In the lecture I will give an overview of the problem and related results and discuss a method, based on some clever averaging, that was used to prove the conjecture in some regimes.(Friday, 11 September, 16:00, Guillaume Tahar) Resonance hyperplanes arrangements in the moduli theory of differential forms
Given a stratum of meromorphic one-forms on a Riemann surface with prescribed orders of zeros and poles, the residual map assigns to each differential the collection of its residues at the poles. In this talk, we will review the topological features of the residual map: the characterization of its singular locus, the topology of its isoresidual fibers, and, if time allows, its monodromy.In the complex vector space of residue configurations, the resonance arrangement is the union of the hyperplanes defined by the vanishing of a nontrivial partial sum of residues. This complex hyperplane arrangement is closely related to the braid arrangement, but its topological and combinatorial properties remain largely unknown. We show that, over each open flat of the stratification induced by the resonance arrangement, the restriction of the residual map is a topological fibration. This result follows from a geometric interpretation of meromorphic differentials in terms of translation structures, which we will explain for strata with one or two zeros. This is joint work, mostly with Dawei Chen, Quentin Gendron, Miguel Prado, and Nick Salter.(Monday, 14 September, 10:00, Nikita Kalinin) One-dimensional tropical series
(Tuesday, 14 September, 14:00, Nikita Kalinin) Two-dimensional tropical series and Monge–Ampere equation
(Wednesday, 16 September, 14:00, Peter Petrov) An invitation to p-adic numbers: Monsky´s theorem
In this talk we shall prove the famous theorem of Paul Monsky about equiareal triangulations of square. We shall discuss the Sperner lemma, then will be a brief introduction to p-adic numbers, and finally, we will see some generalizations and open problems about the result.(Thursday, 17 September, 10:00, Nikita Kalinin) Main statement of the tropical Monge-Ampere
(Friday, 18 September, 14:00, Peter Petrov) An invitation to p-adic numbers: Monsky´s theorem-2, (see details here)
(Monday, 21 September, 14:00, Higinio Serrano, online, zoom) The Hard-Core Farey Gas
To a strictly convex $C^3$ arc $\Gamma$ whose normals fill a unimodular cone we associate a statistical-mechanical ensemble built on the Stern--Brocot tree. After normalizing the normal directions to $[0,1]$ by an $SL_2(\mathbb{Z})$ change of coordinates, the vertices of the tree index Farey sectors, each weighted by a geometric quantity $\delta_\Gamma(I)$ --- the size of the tangent-triangle cap cut from $\Gamma$ over the sector. Declaring two sectors incompatible when they are comparable in the tree, the admissible configurations are the finite antichains, and the finite-volume partition function is $$ \Xi_\Lambda(\lambda;s) = \sum_{\substack{A \subseteq \Lambda \\ A \text{ an antichain}}} \lambda^{|A|} \prod_{I \in A} \delta_\Gamma(I)^{s}, $$ a polynomial in $\lambda$ with coefficients entire in $s$. I will explain how this hard-core polymer gas encodes the approximation of $\Gamma$ by lattice polygons: the single-particle sum is a Farey zeta function of the curve. The main point is the analytic and geometric meaning of its abscissa of convergence, which sits at $s = 2/3$ and marks a genuine transition in the ensemble: a change from configurations dominated by finitely many coarse sectors to configurations proliferating sectors at every scale. I will develop the tree recursion (cavity equations) behind these statements and, time permitting, the primon-gas analogy that gives the model its statistical-mechanical name.(Tuesday, 2 September, 14:00, Mikhail Shkolnikov) Averaging out the lattice as a way to pass from tropical to affine
The talk will be focused on a few things proven and several suggestions made in a recently published paper with Nikita Kalinin where a particular connection between tropical and affine geometries is introduced. The main construction consists of averaging a tropical distance series of a convex domain over the space of underlying tropical structures leading to equiafine geometric entities. I will spend some time to go through the main steps of the average’s convergence statement which consist of combining some classical tools from the geometry of numbers due to Minkowski and Siegel. Then we will pass to the problem of determining the limit structure of level sets, and compute the value of the average in the centre of the unit disc (in fact, so far this is the only nontrivial value we know explicitly). At the end, I will enumerate unresolved questions and conjectures inviting everyone take part in walking in this direction. (see details [here](https://sigma-journal.com/2026/071/))(Wednesday, 23 September, 14:00, Peter Petrov) An invitation to p-adic numbers: Monsky´s theorem-3
Igusa zeta function, p-adic integration, etc.(Thursday, 24 September, 16:00, Faith Shadow Zottor) On Y-coordinate of Pell equations which are Fibonacci numbers
Let $d \geq 2$ be an integer which is not a square. We show that if $(F_n)_{n\geq 0}$ is the Fibonacci sequence and $(X_m, Y_m)_{m\geq 1}$ is the $m$th solution of the Pell equation $X^2 -dY^2 = \pm 1$, then the equation $Y_m = F_n$ has at most two positive integer solutions $(m,n)$ except for $d=2$ when it has three solutions $(m,n)=(1,2),(2,3),(3,5)$.(Friday, 25 September, 10:00, Max Karev) Interpolation between complex and real Hurwitz numbers: progress report
(Friday, 25 September, 16:00, Faith Shadow Zottor) A $p$-adic ($p\equiv 3 \pmod 4$) depth-$5$ supercongruence for Gaussian $p$-th power sums over a square
For an odd prime $p$, define $$ G_n(p)= \sum_{a=1}^{p-1}\sum_{b=1}^{p-1}(a+bi)^n \in\mathbb Z[i]. $$ We study the $p$-adic valuation of these Gaussian power sums for $n\leq p, n=r(p-1)$ and show that it is governed by the interaction of the fourfold symmetry of the square, ordinary power-sum congruences, and Bernoulli numbers. If $p\equiv3\pmod4$ and $p\ge7$, then we prove an unexpectedly deep supercongruence $$ G_p(p)\equiv -\frac{p^5}{12}(p-1)^2(p-2)(1-i)B_{p-3} \pmod{p^6}. $$Organiser: Nikita Kalinin (GTIIT)
Mathematical cinema
Mathematical cinema
(in the sense of Ernesto Lupercio)
should not be a screen
where finished mathematics
is displayed.
It should be a place
we enter.
A place
we play in.
A workshop.
A seminar.
A theater—
where we inhabit
a way
of thinking.
Draw the pictures!
Compute examples!
Make conjectures.
Try
the wrong
proof.
Let it
fail.
Find out
why
it fails.
Change
the representation.
Argue
at the blackboard.
Ask
the naïve question.
Compute.
Use AI.
Push
beyond
what we can immediately see—
then
return.
Return
to the object.
Again.
Until
its structure
becomes
our own.
The aim
is not primarily
to produce papers.
The aim
is to
transmit
and practise
ways of seeing.
So that another mathematician
can learn the moves,
develop the taste,
recognize the obstruction,
know
where to push
and where not to push—
and eventually
continue the mathematics
without its originator.
Proof
remains essential.
But the polished sequence
THEOREM.
PROOF.
is only
one trace—
one footprint—
left behind
by the larger activity.
In mathematical cinema,
papers come
as a side effect.
Durable records.
Deposits
left by a living practice.
Because
its real product
is not the paper.
Its real product
is mathematicians—
capable
of seeing,
questioning,
trying,
failing,
seeing again—
and carrying
the mathematics
further.