| Karim Adiprasito | (1) Hopf conjecture: The conjecture concerns the Euler characteristic of aspherical manifolds. It predicts that for a 2n-dimensional closed manifold M, (-1)nχ(M) ≥ 0. The problem connects combinatorics, group theory, and algebraic geometry in deep and poorly understood ways.
(2) Reconstruction problems in graph theory: Key open questions include the graph reconstruction conjecture (whether a graph is determined by all its single-edge deletions) and related problems about reconstructing cellulations of spheres from their 1-skeletons. These highlight our limited understanding of high-dimensional graph embeddings.
(3) Quantum gravity and renormalization: I am interested in renormalization in general relativity and models of quantum gravity. This area brings together many branches of mathematics and is closely related to combinatorial problems, such as counting triangulations of manifolds (Regge calculus). | ECA2022_S3I6.pdf |
| Michael Albert | Enumeration of Permutation Classes: In many permutation classes, the enumeration sequences of subclasses are surprisingly restricted: although one might expect many different sequences when adding a new restriction, often only o(f(n)) distinct sequences appear for a class with f(n) permutations of size n. The main open question is whether this phenomenon occurs in the class of 321-avoiding permutations. | ECA2022_S4I1PP.pdf |
| Noga Alon | (1) The P versus NP problem, formulated as a question in Circuit Complexity (whose solution could be even stronger than solving the P versus NP problem).
(2) Finding a proof of the Four Color Theorem that is not computer-aided, possibly by using algebraic tools.
(3) Deciding if the maximum possible Shannon capacity of a graph with independence number 2 is bounded. Equivalently: deciding if there is a finite constant C such that the maximum possible number of vertices in a complete graph whose edges can be colored with k colors with no monochromatic triangle is at most Ck (this is sometimes called the Schur–Erdős problem). | ECA2021_S3I2.pdf |
| George E. Andrews | Conjectured Rogers–Ramanujan–type partition identities where the relevant generating function is not a modular form. See, for example, the work of Kanade and Russell. Some of these conjectures involve modular forms, while others do not. Researchers including Bringmann, Mahlburg, Jennings-Shaffer, and Rosengren have made progress studying these results, but there is no overarching theory comparable to the one available when the generating function is modular. These conjectures are reminiscent of what is now called the "Big Göllnitz Theorem", which, despite having at least five proofs, remains as mysterious as when Goellnitz originally discovered it. | ECA2021_S3I12.pdf |
| Edward A. Bender | P vs. NP problem, possibly via combinatorial approaches. | ECA2022_S3I1.pdf |
| Francesco Brenti | (1) Combinatorial Invariance Conjecture: Determine whether Kazhdan-Lusztig polynomials are combinatorially invariant, i.e., depend only on the Bruhat graph of the Coxeter group element pair.
(2) Combinatorial Interpretations of Kazhdan-Lusztig Polynomials: Find explicit combinatorial models or interpretations for the coefficients of Kazhdan-Lusztig polynomials.
(3) Combinatorial Interpretations of Plethysm Coefficients: Provide a combinatorial interpretation for the coefficients arising when expanding the plethysm of two Schur functions into the Schur basis. | ECA2024_S3I1.pdf |
| Maria Chudnovsky | (1) The Erdős–Hajnal Conjecture: This conjecture states that for every graph H there exists an ε(H)∈(0, 1) such that every n-vertex graph with no induced subgraph isomorphic to H contains a clique or a stable set of size at least nε(H).
(2) Hadwiger’s Conjecture: This conjecture states that any graph that does not contain the complete graph on t+1 vertices as a minor is t-colorable.
(3) A combinatorial algorithm to find the largest clique in a perfect graph that runs in time polynomial in the number of vertices of the input. While finding a largest clique in a perfect graph is already known to be polynomial-time solvable (a result of Grötschel, Lovász, and Schrijver), their algorithm relies on techniques from combinatorial optimization, such as the ellipsoid method. The goal is to find an algorithm that can be formulated entirely in terms of the graph itself. | ECA2021_S3I4.pdf |
| David Conlon | (1) Diagonal Ramsey Numbers for 3-Uniform Hypergraphs: Estimate the Ramsey number r3(t) of the complete 3-uniform hypergraph Kt(3). It is known that 2ct² ≤ r3(t) ≤ 22c′t, but the exponential gap remains wide open. Erdős offered \$500 for showing that the upper bound is essentially correct.
Euclidean Ramsey Problem.
(2) Characterize finite sets X ⊂ ℝd that are Ramsey under isometric embeddings. While such sets must be spherical, it is unknown whether every spherical set is Ramsey. The problem remains open even for cyclic quadrilaterals.
(3) Sidorenko’s Conjecture: Determine whether every bipartite graph H satisfies Sidorenko’s conjecture, which asserts that graphs of fixed edge density contain asymptotically at least as many copies of H as a random graph. Many cases are known, but the full conjecture remains unresolved. | ECA2024_S3I5.pdf |
| Alan Frieze | Random Graphs and Ramsey Theory: Some accessible open problems include proving that Gn,cn conditioned on minimum degree three is Hamiltonian with high probability, and determining the likely cover-time of two-dimensional random geometric graphs. A more ambitious challenge is to improve Spencer’s lower bound on diagonal Ramsey numbers. | ECA2024_S3I4.pdf |
| Ira Gessel | (1) Formal power series for de Bruijn’s formula: Let Σn≥1 an xn/n! be the compositional inverse of √(2x - 2 log(1+x)), and let B(x) = Σk≥2 Bk/(k(k-1)) xk-1, where Bk are Bernoulli numbers. De Bruijn showed that Σk≥0 2-k a2k+1 xk/k! = eB(x) by computing two different asymptotic expansions of the gamma function. The question is whether a purely formal power series proof exists.
(2) Combinatorial formulas for strong digraphs: Let sn be the number of strongly connected digraphs on [n], and tn the number of strong tournaments. Then Σn≥1 sn xn/n! = log(1/(1 - T(x))), where T(x) = Σn≥1 2n(n-1)/2 tn xn/n!. Many formulas like this suggest there should be a combinatorial interpretation.
(3) Point-determining graphs and unlabeled counting: A graph is called point-determining (or R-thin/mating-type) if no two vertices have the same neighborhood. Ronald Read counted both labeled and unlabeled point-determining graphs. The labeled exponential generating function P(x) satisfies P(x) = G(log(1+x)), where G(x) = Σn≥1 2n(n-1)/2 xn/n!. Equivalently, G(x) = P(ex - 1), reflecting that every graph reduces to a point-determining graph by identifying vertices with identical neighborhoods. An analogous problem counts graphs where no two vertices have complementary neighborhoods. Using inclusion-exclusion, the exponential generating function is G(0.5 log(2 ex - 1)). However, counting the corresponding unlabeled graphs remains an open problem. | ECA2022_S387.pdf |
| Larry Guth | (1) The Kakeya problem and related questions, which I have spent the most time pondering.
(2) In combinatorics, the inverse problem for the Szemerédi–Trotter theorem, one of my favorite challenges.
(3) An open problem in additive combinatorics concerning the size of the sumset of a convex set. Specifically, if f(x) is a convex function, then the sequence f(1), f(2), …, f(n) forms a convex set of size n. The question is: if A is a convex set of size n, how large must the sumset A + A be? Some striking recent work has been done by Schoen and Shkredov, yet the problem is still far from fully understood. | ECA2021_S3I7.pdf |
| Gil Kalai | (1) Toric g-vector: Is the toric g-vector of a polytope always an M-vector?
(2) Turán (4,3)-problem: What is the maximum number of triples from [n] with no four vertices containing all four possible triples?
(3) Polynomial Hirsch conjecture: Does the diameter of the graph of a polytope admit a polynomial bound in the number of facets?
(4) Extremal and coding problems: I am also interested in improved bounds for Roth’s theorem, sharper asymptotics for binary and spherical codes, and the dying percolation conjecture in three dimensions.
(5) Diversity in mathematics: How can the combinatorial, mathematical, and academic communities be made more diverse? | ECA2022_S3I7.pdf |
| Gyula O. H. Katona | (1) Turán (4,3) Problem: Determine the maximum number of 3-element subsets of an n-element set such that no four subsets are contained within a set of four elements. This classic extremal problem remains open.
(2) Diamond Problem: Find the largest family of subsets of an n-element set that contains no four sets A, B, C, D with A ⊆ B ∩ C and B ∪ C ⊆ D. It is conjectured that the two middle levels of the Boolean lattice are asymptotically optimal.
(3) Generalization of Baranyai’s Theorem: Extend Baranyai’s decomposition theorem for k-uniform set systems to the case k ∤ n, as conjectured by Baranyai and Katona. | ECA2024_S3I7.pdf |
| Ken-ichi Kawarabayashi | Four Color Theorem: My main open problem is to gain a deeper understanding of the Four Color Theorem and its proof. Relatedly, it would be valuable to explore how metric embedding theory, developed in pure mathematics, can help prove significant results in graph algorithms, as attempted by researchers such as Assaf Naor. More broadly, I am interested in seeing how combinatorial tools can be applied to solve major problems in other areas of mathematics, beyond their current impact in theoretical computer science. | ECA2024_S3I2.pdf |
| Don Knuth | (1) To prove the Exponential Time Hypothesis (ETH):
inf{ log2 τ | there exists an algorithm that solves the satisfiability problem with clauses of size 3 in n variables in τn steps } > 0.
(2) A nonconstructive proof that P = NP, by showing that only finitely many essentially different cases can arise, even though we may never know when we have found them all (similar to what Robertson and Seymour did for graph minors).
(3) A study of the curious sign pattern that arises in the asymptotics of the Gould numbers — the number of set partitions whose tail is a singleton — as illustrated in the answer to exercise 7.2.2.1–190 of The Art of Computer Programming, Volume 4 (TAOCP), Fascicle 5 (2019), pages 146 and 277. | ECA2021_S3I9.pdf |
| Christian Krattenthaler | Common Enumeration Phenomenon: Several combinatorial objects are counted by the same formula ∏i=0n−1 (3i+1)! / (n+i)!, including alternating sign matrices (ASM), totally symmetric self-complementary plane partitions (TSSCPP), descending plane partitions (DPP), and alternating sign triangles.
(1) Find a direct bijection between alternating sign matrices and totally symmetric self-complementary plane partitions.
(2) Find a natural bijection between alternating sign matrices and descending plane partitions that avoids involution-principle arguments.
(3) Construct a bijection between alternating sign matrices and alternating sign triangles.
(4) Find a unified combinatorial framework explaining why all these objects are counted by the same product formula.
(5) Extend existing bijections to refined or weighted versions of these combinatorial families.
(6) Find a simple, explicit, and structural bijection connecting all four families simultaneously. | ECA2026_S3I1.pdf |
| Brendan McKay | (1) Growth of Ramsey Numbers: Understanding the growth of Ramsey numbers remains elusive: despite major effort, the known bounds are still exponentially far apart. In particular, I hope to determine R(5,5). With Vigleik Angeltveit, we reduced the bounds to 43–48, and soon to 43–46.
(2) Theoretical Complexity of Graph Isomorphism: My work on graph isomorphism has been largely practical, but its theoretical complexity continues to puzzle me. Recent breakthroughs by Babai, Grohe, and Schweitzer are remarkable—perhaps they point the way to a full solution.
(3) Unified Methods in Asymptotic Enumeration: More broadly, recent work with Misha Isaev has unified several ad-hoc methods in asymptotic enumeration (e.g., via complex martingales). Still, applying these tools remains cumbersome, suggesting that deeper “master theorems” are yet to be found. | ECA2025_S3I1.pdf |
| Peter Paule | A major area of research explores connections between classical q-series (including q-hypergeometric functions and the theory of partitions) and the theory of modular functions and forms. My focus is on developing computer algebra algorithms to study these connections, with inspiration from the work of Felix Klein. | ECA2023_S3I1.pdf |
| Igor Pak | (1) Hilbert’s Third Problem for the 3-sphere: This problem remains open for the sphere S3 (the hyperbolic space H3 is equally difficult). Formally, the conjecture claims that two spherical polyhedra are scissors congruent if and only if they have the same volume and the same Dehn invariant. In particular, it is unknown whether all spherical tetrahedra with rational dihedral angles and the same volume are scissors congruent. Even this special case is far from being resolved.
(2) Generating Primes Problem: The challenge is to find a deterministic polynomial-time algorithm to produce a prime in the interval [n, 2n], effectively derandomizing Bertrand’s postulate, which currently has only a classical probabilistic algorithm. It is plausible that this could be achieved by testing the primality of n + x for all 0 ≤ x ≤ C(log n)2, but this remains unproven.
(3) Skolem’s Problem: This asks whether it is decidable if a sequence {an} defined by a linear recurrence with constant integer coefficients and initial values has at least one zero: an+1 = c0an + ... + ckan-k, for all n ≥ k, where a0, ..., ak, c0, ..., ck ∈ ℤ. This fundamental problem is surprisingly basic, yet no intuition exists as to its decidability.
(4) P vs. NP Problem: Interestingly, I would rather not see this resolved. Any solution could upset the delicate structure of computational complexity, including its beautiful reductions, powerful theorems, and myriad complexity classes. Fortunately, the problem is so difficult that it is unlikely to be resolved in my lifetime. | ECA2021_S3I11.pdf |
| Greta Panova | (1) A major open problem in arithmetic complexity theory is to distinguish the classes VP and VNP, which appears more accessible than the classical P vs. NP problem.
(2) Within algebraic combinatorics, a central challenge is to find a combinatorial interpretation of Kronecker coefficients, as well as other structure constants and coefficients that are "mysteriously" nonnegative. It is possible, however, that such interpretations do not exist, in line with an idea of Igor Pak suggesting that Kronecker coefficients might be provably not in #P under reasonable computational assumptions. | ECA2022_S3I3.pdf |
| Tomaž Pisanski | (1) Geometric Configurations (nk): About 25 years ago, I became interested in geometric configurations of points and lines. It is known that (n3) configurations exist for all n ≥ 3. Branko Grünbaum asked for which n there exist (n4) configurations and resolved almost all cases; only n = 23 remains open. Along with Leah Berman and Gábor Géváy, we explored (nk) configurations in general. For larger k, the number of unsolved cases grows, and most gaps require new techniques. The interplay between geometry and graph theory continues to be fascinating.
(2) Polyhedral Self-Assembly: Inspired by the possibility of designing protein chains that fold into tetrahedral shapes, I became interested in polyhedral self-assembly. Initially modeled with Eulerian graphs, the problem can also be represented as one-face embeddings of a graph on a closed surface. Open questions include cases where edges need not be covered exactly twice by protein segments, or when multiple chains participate in the assembly.
(3) Noncommutative Lattices and Quasi-Lattices: A lattice is a classical algebraic structure with a combinatorial description via its Hasse diagram. Noncommutative lattices relax the commutativity axioms and modify absorption laws. Among these, quasi-lattices are particularly intriguing. I am seeking a faithful combinatorial description of quasi-lattices. | ECA2022_S3I11.pdf |
| Andrei Raigorodskii | (1) Borsuk’s Conjecture and Chromatic Numbers: Understanding the chromatic numbers of metric spaces and related questions connected to Borsuk’s conjecture.
(2) Stability of Independence and Chromatic Numbers in Random Graphs: Determining conditions under which the independence numbers and chromatic numbers of random subgraphs in sequences of graphs remain stable.
(3) Sparse Pseudorandom Graphs: Establishing general results and structural properties of sparse pseudorandom graphs. | ECA2023_S3I4.pdf |
| Amitai Regev | (1) Determine the exact asymptotics of cn(Mk(F)). Very recently, in joint work with Berele, we established a fairly tight sandwich theorem for these invariants. Using the notation cn(Mk(F)) ∼ a nb rn, it follows from a result of Giambruno and Zaicev that b is a half-integer and r is a positive integer; however, no information is yet known about the constant a. Thus, it remains an interesting problem to determine the constants a and b.
(2) In joint work with Beckner, we proved that cn(Mk(F)) is non-algebraic when k is odd. Our earlier result with Beckner partially resolves a problem posed by Richard Stanley. The corresponding question for even k remains open, though the natural conjecture is that the statement holds for all k.
(3) Similar investigations have been conducted for cn(Mk(G)) (see footnote 1), where G is the Grassmann algebra. It is also of interest to determine the constants a, b, and r for verbally prime algebras. | ECA2021_S3I6.pdf |
| Victor Reiner | (1) A combinatorial interpretation of the structure constants arising in Schubert calculus for the type A flag manifold.
(2) The Kronecker problem: finding a combinatorial interpretation of the structure constants sfor tensor products of irreducible characters of the symmetric group.
(3) The Charney–Davis–Gal conjecture concerning the nonnegativity of γ-vectors for flag simplicial spheres. | ECA2022_S3I2.pdf |
| Bruce Sagan | (1) The (3 + 1)-free Conjecture of Richard Stanley and John Stembridge. Stanley defined a symmetric function X(G) associated with any finite graph G, which reduces to the chromatic polynomial under specialization of the variables. A natural question is what can be said about X(G) when expanded in the various standard bases for the algebra of symmetric functions. Associated with any finite poset P is a graph GP whose vertices are the elements of P, with an edge connecting x and y if these elements are incomparable in P. A poset is (3 + 1)-free if it contains no induced subposet isomorphic to a disjoint union of a 3-element chain and a 1-element chain. The conjecture states that for a (3 + 1)-free poset P, the expansion of X(GP) in the elementary symmetric function basis has all coefficients nonnegative. Vesselin Gasharov proved the weaker statement that the expansion in the Schur basis has nonnegative coefficients. Recently, Mathieu Guay-Paquet showed that it suffices to consider posets which are both (3 + 1)-free and (2 + 2)-free. With my then graduate student, David Gebhard, we verified some special cases using the symmetric functions in noncommuting variables introduced with Mercedes Rosas. This work has been continued by Samantha Dahlberg and Stephanie van Willigenburg.
(2) Conjecture of interest is the one described by Morier-Genoud and Ovsienko. We have a method that appears to generate nested chain decompositions of these lattices, which would immediately imply rank unimodality and the strong Sperner property. However, we have been unable to prove that this algorithm always works, despite extensive computer testing.
(3) Even though Huh’s work shows that the coefficients of the chromatic polynomial are log-concave, it remains an open problem to find a purely combinatorial proof. For instance, it might be possible to apply the powerful directed path method of Bernt Lindström, Ira Gessel, and Gérard (now Xavier) Viennot. | ECA2021_S3I8.pdf |
| Carla D. Savage | (1) Symmetric Chain Decomposition of L(m, n): The lattice L(m, n) consists of integer partitions whose Ferrers diagrams fit inside an m × n box, ordered by inclusion. In 1980, Richard Stanley proved that L(m, n) is rank-symmetric and unimodal and conjectured that it admits a symmetric chain decomposition. This remains open for m ≥ 5, although Kathy O’Hara proved unimodality in 1990.
(2) Real-Rootedness of the Durfee Polynomial: The Durfee polynomial Dn(x) = ∑d ≥ 0 p(n, d)xd counts partitions of n by Durfee square size. In 1998, we showed its coefficients are asymptotically normal, unimodal, and log-concave. Empirical evidence suggests Dn(x) has only real roots for all n.
(3) Simple Symmetric n-Venn Diagrams: Rotationally symmetric Venn diagrams exist for all prime n, but may have multiple curves intersecting at a point. The open question is whether such diagrams can always be made simple. This is known for n = 1, 2, 3, 5, 7, 11, 13. | ECA2022_S3I9.pdf |
| Anne Schilling | (1) Combinatorial Interpretations of Structure Coefficients: Find combinatorial models or interpretations for various structure coefficients, including Kronecker coefficients, plethysm coefficients, and the structure coefficients of Schubert and k-Schur functions.
(2) Crystal Structures on Tableaux of Tableaux: Construct or characterize crystal structures on tableaux of tableaux, extending the theory of crystal bases in algebraic combinatorics.
(3) Symmetric Chain Decompositions of Young Subposets: Determine symmetric chain decompositions for the Young subposet of partitions contained in a box. | ECA2023_S3I5.pdf |
| Jeffrey Shallit | (1) Pierce expansion: Start with a real number x in (0,1) and define xn+1 = 1 mod xn. If x = p/q is rational, how many steps are needed to reach 0? If x is irrational, the expected size of xn is about e−n. The rational case is still poorly understood, and a cash prize is offered for improving current results.
(2) Shuffling a binary string with itself. How many binary strings of length 2n can be formed by interleaving a binary string x of length n with itself, allowing any left-to-right interleaving? No efficient method or good asymptotic bounds are known. This is OEIS sequence A191755.
(3) Distinguishing binary strings with automata. What is the minimum number of states needed, in the worst case, for a deterministic finite automaton to distinguish between two binary strings of length n? Known lower bounds are Ω(log n), while the best upper bound is O(n1/3). The gap remains large. | ECA2022_S3I5.pdf |
| Micha Sharir | (1) Point–Object Incidence Problems: Study incidences between points and geometric objects such as lines, curves, and surfaces. Despite major progress over the past decades, many fundamental problems in this area remain open.
(2) Substructures in Arrangements: Understand combinatorial properties of substructures in arrangements of geometric objects, including cells, zones, levels, lower envelopes, and unions of objects.
(3) Polynomial Partitioning Techniques: Further develop and apply polynomial partitioning methods, both combinatorially and algorithmically, to resolve open problems in geometry and related areas. | ECA2024_S3I6.pdf |
| Joel Spencer | (1) Komlós Conjecture: Given vectors vi in the Euclidean unit ball, find signs εi ∈ {−1,+1} so that ∑ εi vi has L∞ norm bounded by an absolute constant independent of dimension.
(2) Van der Waerden and Szemerédi Functions: Determine close bounds on functions arising from these theorems: for large K, any 2-coloring of {1,…,Kn} must contain a monochromatic arithmetic progression of length n. Gowers made breakthroughs, but significant gaps remain.
(3) Erdős Unit Distance Problem: Determine the maximal number of unit distances among n points in the plane. Conjecturally, this should grow as n1+o(1). | ECA2023_S3I7.pdf |
| Richard Stanley | (1) Find a combinatorial interpretation of the coefficients in the product of Schubert polynomials.
(2) Prove the Stanley–Stembridge conjecture on the e-positivity of the chromatic symmetric function of certain graphs.
(3) Show that Young’s lattice is the smallest differential poset.
(4) Enumerate solid (three-dimensional) partitions, as a generalization of MacMahon’s celebrated enumeration of plane partitions. | ECA2021_S3I1.pdf |
| Volker Strehl | (1) P vs NP Problem: Whether every problem whose solution can be verified efficiently can also be solved efficiently.
(2) >Hamiltonian Cycle Problem: Deciding whether a graph has a Hamiltonian cycle versus verifying that a given set of edges forms one.
(3) #P-Completeness: Understanding the complexity of counting solutions, even when deciding existence is easy.
(4) Counting Perfect Matchings: Determining the number of perfect matchings in a bipartite graph, a problem that is computationally hard despite easy decision. | ECA2022_S3I10.pdf |
| Einar Steingrímsson | Pattern-Avoiding Permutations: A major open problem is to enumerate permutations avoiding the pattern 1324. While the exact numbers may not be dramatic on their own, solving this requires novel combinatorial techniques that could have broader applications. A related conjecture, made independently by Natasha Blitvić and myself, posits that the enumeration sequence of permutations avoiding any classical pattern forms a moment sequence. This generalizes earlier conjectures for the pattern 1324 (Andrew Elvey Price) and patterns of length 5 (Clisby, Conway, Guttmann, Inoue), and suggests a new framework for studying pattern avoidance using ideas from noncommutative probability.
Other enticing problems include finding a combinatorial proof that the statistics (area, bounce) and (bounce, area) on Dyck paths have the same distribution, and proving the “meta theorem” mentioned previously. Each of these challenges is deceptively simple to state yet surprisingly difficult. | ECA2023_S3I2.pdf |
| Benny Sudakov | (1) Ramsey Numbers of 3-Uniform Hypergraphs: Determine the growth rate of the Ramsey number r(3)(n), the smallest N such that every 2-coloring of the edges of the complete 3-uniform hypergraph on N vertices contains a monochromatic set of size n. The best known bounds are 2cn² ≤ r(3)(n) ≤ 22cn, and it remains open whether the true growth is exponential or doubly exponential.
(2) Turán Numbers of Bipartite Graphs: Understand the order of magnitude of ex(n,H) for bipartite graphs H. A conjecture of Erdős predicts that if H is s-degenerate, then ex(n,H) = O(n2−1/s). Determining which graph parameters govern these extremal numbers remains largely open.
(3) Brown–Erdős–Sós Problem: Let f(n,v,e) be the maximum number of edges in a 3-uniform hypergraph on n vertices with no set of v vertices spanning e edges. Brown, Erdős, and Sós conjectured that f(n,e+3,e) = o(n²). While the case (v=6,e=3) is settled, the next case (v=7,e=4) remains completely open. | ECA2024_S3I3.pdf |
| Sheila Sundaram | (1) Plethysm Coefficients: Understanding the structure and computation of plethysm coefficients.
(2) Kronecker Coefficients: Clarifying the behavior and combinatorial meaning of Kronecker coefficients.
(3) Thrall’s Problem: Finding a combinatorial rule for the irreducible multiplicities in higher Lie modules. | ECA2025_S3I2.pdf |
| Xavier Viennot | (1) A combinatorial interpretation of Askey–Wilson polynomials that naturally explains their full four-parameter symmetry.
(2) A combinatorial model for Genocchi numbers in which the ternary symmetry discovered by Dumont and Foata appears naturally.
(3) A unified combinatorial object explaining the connections between Askey–Wilson polynomials, continuous dual Hahn polynomials, and related symmetry phenomena.
(4) A combinatorial explanation of the symmetry in the (q, t)-Catalan polynomials.
(5) A combinatorial explanation of the algebraicity of the density in the hard hexagon gas model. | ECA2022_S3I4.pdf |
| Van Vu | (1) A proof of the Riemann Hypothesis - a goal that continues to inspire mathematicians everywhere.
(2) A fundamentally convincing approach to the Four Color Theorem (and similar problems), based on a core underlying principle rather than extensive case checking.
(3) A solution to the distinct distances problem in all dimensions. (For context, the two-dimensional case was recently solved by Guth and Katz.) | ECA2021_S3I10.pdf |
| Douglas West | (1) Reconstruction Conjecture: Kelly and Ulam's conjecture: determine whether a graph can be uniquely reconstructed from the multiset of its vertex-deleted subgraphs. The extension by Manvel asks the same question when deleting more than one vertex.
(2) Symmetric Chain Decomposition of L(m, n): Determine whether the poset L(m, n) of integer partitions fitting in an m × n box admits a symmetric chain decomposition.
(3) Pancake Problem: Find the worst-case number of prefix reversals required to sort a permutation of [n]. | ECA2023_S3I8.pdf |
| Stuart Whittington | Self-Avoiding Walks and Related Models: A central open problem is to find an exact expression for the number of self-avoiding walks with n edges, cn. While this seems out of reach, determining the exponential growth rate — the connective constant κ = limn→∞ n-1 log cn — is an important step. A further challenge is to understand the rate of convergence to this limit, for instance proving that cn = eκn + O(log n). Similar questions exist for related combinatorial objects such as lattice trees and lattice animals. In a related area of statistical physics, the 3-dimensional Ising model remains a major open problem. | ECA2023_S3I6.pdf |
| Lauren K. Williams | (1) Koornwinder Polynomials: Extend results for Askey-Wilson polynomials to the multivariable Koornwinder (Macdonald) polynomials. Only initial steps using the multispecies ASEP with open boundaries have been completed; full generalization remains open.
(2) Tiling the Amplituhedron for Even m: For the tree amplituhedron Αn,k,m(Z), it is conjectured that the number of top-dimensional strata in a tiling equals the number of plane partitions in a k × (n-k-m) × (m/2) box. This conjecture is still wide open.
(3) Understanding the Amplituhedron–Hypersimplex Correspondence: Tilings of the amplituhedron Αn,k,2(Z) correspond to tilings of the hypersimplex Δk+1,n. The reason for this surprising correspondence is not fully understood, and whether a similar phenomenon occurs for other m remains an open question. | ECA2022_S3I12.pdf |
| Doron Zeilberger | Computer-generated, or at least computer-assisted, proof of:
(1) the Riemann Hypothesis.
(2) P≠NP.
(3) the Collatz conjecture. | ECA2021_S3I3.pdf |
| Yufei Zhao | (1) Maximum sphere packing density in high dimensions.
(2) Sidorenko’s conjecture on graph homomorphism densities.
(3) Shannon capacity of odd cycles.
| ECA2021_S3I5.pdf |
