Mahmut Levent Doğan (Ruhr University, Bochum)
Title: The Gaussian Approximation Method for Volumes of Spectrahedra
Abstract: In this talk, we introduce a method for computing asymptotic formulas and approximations for the volumes of spectrahedra, based on the Gaussian approximation method of Barvinok and Hartigan. The method gives an approximate volume formula based on a single convex optimization problem of minimizing − log det P over the spectrahedron. Spectrahedra can be described as affine slices
of the convex cone of positive semi-definite (PSD) matrices, and the method yields asymptotic
formulas whenever the number of affine constraints is sufficiently dominated by the dimension
of the PSD cone.
We give three main applications of this method. First, we prove that spectrahedra possess a remarkable feature not shared by polytopes, central sections of the set of density matrices (the quantum version of the simplex) all have asymptotically the same volume.
Further, we apply this method to explicitly compute the asymptotic volume of central sections of the set of density matrices. Lastly, we consider what we call the “multi-way Birkhoff spectrahedron” and obtain an explicit asymptotic formula for its volume.
Ignacio Garcia-Marco (Universidad de La Laguna, Spain)
Title: Coloring Cayley graphs of groups
Abstract: In 1978, Babai asked whether all minimal Cayley graphs of finite groups have bounded chromatic number. In this talk, we show that every minimal Cayley graph of a generalized dihedral or nilpotent group has chromatic number at most three, while four colors are sometimes necessary for solvable groups. We will also discuss related problems and open questions.
This is joint work with Kolja Knauer.
Harald Helfgott (Université Paris Cité, France)
Title: Diameter bounds for linear algebraic and permutation groups
Abstract: Let $G$ be a finite simple group and $A$ a subset of $G$ generating $G$. We would like to bound the diameter of the
Cayley graph $\Gamma(G,A)$ independently of $A$. The best bounds in the literature are of the type $O\left((\log |G|)^{(\log n)^{3+\epsilon}}\right)$ (Helfgott-Seress, 2011--14), for $G=\Alt(n)$, and $O((\log |G|)^{O(n^4)})$ (Bajpai-Dona-Helfgott), for classical simple groups of rank $n$. As one can see, for classical simple groups such as $\PSL_{n+1}(\mathbb{F})$, we are still exponentially behind: we would like to have $O(\log n)^C$ in the exponent, not $n^4$.
The talk will have two halves: (a) we will go over the current, improved version of Bajpai-Dona-Helfgott, (b) we will discuss some parallels involving buildings and the theory of the field with one element.
Selvi Kara (Bryn Mawr College, USA)
Title: Independence Polynomials of Graphs and the Edge Ideals
Abstract: Graphs give rise naturally to squarefree monomial ideals through their edge ideals, and many algebraic invariants of these ideals reflect combinatorial properties of the graph. In this talk, I will discuss how the independence polynomial P_G(x), the generating function for independent sets of a graph G, encodes surprisingly rich information about the Hilbert series of the corresponding edge ideal. In particular, I will explain how P_G(x) determines the h-polynomial, and how the special value P_G(−1) together with the multiplicity of −1 as a root reveals the top coefficient and degree of that polynomial. I will illustrate these ideas through familiar examples and a simple suspension construction that lets us track these invariants cleanly.
Anargyros Katsampekis (University of Ioannina, Greece)
Title: Toric ideals and their splittings.
Abstract: Let A be a vector configuration in $\mathbb{Z}^{m}$ such that the affine semigroup $\mathbb{N}A$
is pointed, and let $I_A$ be the corresponding toric ideal.The toric ideal $I_A$ is splittable if it has a toric splitting; namely, if there exist toric ideals $I_{A_1}$ and $I_{A_2}$ such that $I_A=I_{A_1}+I_{A_2}$ and $I_{A_i} \neq I_A$ for all $1\leq i \leq 2$.
We provide a necessary and sufficient condition for a toric ideal to be splittable in terms of $A$. Special attention is given to the case in which $I_A$ is the toric ideal of a graph. In this case, we provide techniques for finding toric splittings. The talk is based on joint work with Apostolos Thoma.
Dimitra Kosta (University of Edinburgh, UK)
Title: On the strongly robust property of toric ideals.
Abstract: A toric ideal is called strongly robust when the Graver basis is a minimal system of generators. In the talk, I will explain how to build a strongly robust simplicial complex which determines the strongly robust property of toric ideals. I will then discuss our results on the strongly robust property which include the case of monomial curves as well as codimension 2 toric ideals and configurations in general position. This is joint work with A. Thoma and M. Vladoiu.
Özgür Kişisel (Middle East Technical University, Türkiye)
Title: On nests and large components of random real algebraic curves
Abstract: abstract_kişisel
Greta Panova (University of Southern California, USA)
Title: Ehrhart positivity and skew plane partitions
Abstract: Computing linear extensions and order polynomials of posets is in general a hard problem with no explicit formulas and nice structure and properties. When the poset is a Young diagram of a straight shape, the number of linear extensions is given by the hook-lnegth formula, yet the corresponding number of plane partitions (order polynomial of that poset) does not have a product formula. In the absence of such nice formulas we will show a general approach to proving that the order polynomials of posets corresponding to skew Young diagrams have positive coefficients. We will also discuss some other cases related to matroids, lattice paths and plane partitions. Joint work with Luis Ferroni and Alejandro Morales.
Cosmin Pohoata (Emory University, USA)
Title: A Lovász-Kneser theorem for triangulations
Abstract: In a highly influential paper from 1978, Lovász used topological methods to determine the chromatic number of the Kneser graph of the set of k-element subsets of a set with n elements. In this talk, we will discuss the Kneser graph of the set of triangulations of a convex n-gon and a recent proof that the chromatic number of this graph is n-2. Joint work with Anton Molnar, Michael Zheng and Daniel Zhu
Daria Poliakova (University of Hamburg, Germany)
Title: Cylindrical projections and parallel transport
Abstract: Let us call a projection of posets P -> Q cylindrical if its fibers are paths and the top/bottom elements of these paths embed the Hasse diagram of Q into that of P. For cylindrical projections, I will introduce parallel transport between fibers and explain how path-independence of this parallel transport (aka flatness) allows for elementary inductive proofs of lattice properties. Applications include both well-known lattices coming from polytopes (permutahedron, associahedron, multiplihedron) and some previously unknown examples, most remarkably the permutoassociahedron. I will also discuss connections with order dimension theory. The talk is based on joint work with Vincent Pilaud.
Francisco Santos (Universidad de Cantambria, Spain)
Title: Recent progress on the Lonely Runner Conjecture(s)
Abstract: The lonely runner conjecture (LRC) is the following statement formulated by Jörg Wills in 1968:
If $n+1$ ``runners’’ move along a circle of length one, all starting at the same point, each with its own (constant) velocity then for each individual runner $i$ there is a time $t_i$ at which she at distance at least $1/(n+1)$ from all the others.
In its shifted version (sLRC) the runners are allowed to start each at a different position and the velocities are assumed distinct. (Otherwise, giving all runners the same velocity and distributing their starting points uniformly produces an easy counter-example). The conjectures have been approached from different perspectives, and as of 2025 they were proved until $n=6$ in the original version (Barajas and Serra, 2008) and $n=3$ in the shifted version (Cslovjecsek, Malikiosis, Naszódi, Schymura, 2022).
In 2017 Malikiosis and Schymura developed a connection of the conjectures to the convex geometry of
certain zonotopes, a connection that was later (2025) used by them together with myself to prove that for each number $n$ of runners, if the conjecture holds for integer velocities adding up to at most $\binom{n+1}{2}^{n-1}$ then it holds for arbitrary velocities. A similar finiteness result, with a worse and less explicit bound, had been proved by Tao in 2018.
This new bound has then been used to prove the conjectures for additional values of $n$:
- With Alcántara and Criado (2026) I have proved the shifted version for $n=4$.
- Rosenfeld (2025+) devised a method based on analysing prime factors of candidate velocities. Using it he proved $n=7,8$ and the method was extended by Trakulthongchai to prove $n=9$ (2026).
In a related development, together with Blanco and Criado I have found counter-examples to the shifted conjecture starting at $n=5$, the first value that was not proven true (2026+).
Jozsef Solymosi (University of British Columbia, Canada)
Title: On the number of pairwise touching cylinders in R^d.
Abstract: John E. Littlewood posted the question ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.'' Bozoki, Lee, and Ronyai constructed a configuration of 7 mutually touching unit cylinders. The best-known upper bounds show that at most 10 unit cylinders in R^3 can touch one another. We consider this problem in higher dimensions, and obtain exponential (in d) upper bounds on the number of mutually touching cylinders in R^d. Our method is fairly flexible, and it makes use of the fact that cylinder touching can be expressed as a combination of polynomial equalities and non-equalities. (Joint work with Josh Zahl)
Balázs Szendrői (University of Vienna, Austria)
Title: The projective coinvariant algebra
Abstract: The coinvariant algebra, the quotient of the coordinate ring of
(A^1)^n=A^n by the ideal generated by positive degree invariant
polynomials, plays a basic role in algebraic combinatorics and the
representation theory of the symmetric group S_n, equipping its regular
representation with a graded algebra structure. Using the coordinate
ring of (P^1)^n in its Segre embedding, I will introduce a degeneration
of the coinvariant algebra, the projective coinvariant algebra, which
gives a bigraded structure on the regular representation of S_n with
interesting Frobenius character that generalises a classical result of
Lusztig and Stanley. I will also show how this algebra contains bigraded
versions of partial coinvariant algebras, coming from coordinate rings
of all possible Segre embeddings corresponding to partitions of n.
Connections to an interesting deformation of the cohomology rings of
partial flag varieties will also be discussed. At the end of the talk, I
will comment on potential generalisations to other types (in the sense
of Lie theory) moving away from the classical type A story.
Elias Tsigaridas (Inria and Sorbonne Université, France)
Title: Multivariate generalizations of Combinatorial Nullstellensatz and Schwartz–Zippel Lemma on Multi-Grids
Abstract: We consider the problem of estimating the number of zeros
of a polynomial $p$ of degree $d$ in $n$ variables on a multi-grid
$S = S_1 \times \cdots \times S_m \subset \mathbb{C}^n$,
where $S_i \subseteq \mathbb{C}^{\lambda_i}$ is a finite set and
$\lambda = (\lambda_1, \dots, \lambda_m)$ is a partition of $n$.
We present a multivariate generalization of Alon's Combinatorial
Nullstellensatz to certify that there is at least one point
in $S$ that is not a zero of $p$.
Subsequently, we present a multivariate Schwartz-Zippel lemma,
to bound the number of zeros of a $\lambda$-irreducible polynomial $p$ in $S$.
These polynomials have a zero set that does not contain the
Cartesian product of positive-dimensional varieties.
If time permits, we will present a symbolic algorithm to
identify $\lambda$-reducible polynomials
(whose zero set contains a Cartesian product of hypersurfaces)
and how to recover the Szemer\'{e}di-Trotter theorem
over $\mathbb{C}$ (up to $\varepsilon$).
Joint work with Mahmut Levent Doğan and Alperen Ergür.