DIAMANT Spring Symposium at NMC 2018

 Coloring the Voronoi tessellation of lattices

In this talk I will introduce the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi tessellation of a lattice so that no two cells sharing a facet are of the same color. I will introduce two lower bounds for the chromatic number: the sphere packing lower bound and the spectral lower bound. Using them I will show how to compute, sometimes using polynomial optimization, the chromatic number of several important lattices. Based on joint work with David Madore and Mathieu Dutour Sikiric.

Spaces of matrices of constant rank, vector bundles, and truncated graded modules

A space of matrices of constant rank is a vector subspace V, say of dimension n+1, of the set of matrices of size axb over a field k, such that any nonzero element of V has fixed rank r. It is a classical problem to look for examples of such spaces of matrices, and to give relations among the possible values of the parameters a,b,r,n. In this talk I will report on several joint projects with D. Faenzi, P. Lella, and E. Mezzetti, introducing new methods to classify and produce examples of such spaces. The techniques that I will explain involve vector bundles on projective spaces, and in particular globally generated bundles and instanton bundles, as well as finitely generated graded modules over the ring of polynomials k[x0,…,xn].

Some facets of marginal polytopes

In this talk, I would like to introduce a class of convex polytopes which appear under different names in different fields of mathematics. For instance, they go by the name of “marginal polytopes” in statistics, they are called “partial constraint satisfaction polytopes” in combinatorial optimization. They occur as representation polytopes of abelian permutation groups, and they are used in tensor decomposition analysis. In the general setting, a polytope in the family is specified by a simplicial complex \Delta with vertices labeled by positive integers. If all labels are equal to 2 (binary case), and if \Delta is a graph, the polytope is a cut polytope. In spite of their many uses, the facet structure of marginal polytopes remains mysterious — particularly so in the non-binary case. In joint work with Benjamin Nill and Andreas Paffenholz, we describe a generalization of the cycle inequalities for cut polytopes to non-binary labels together with a polynomial time separation algorithm if the size of the vertex labels is bounded.

Spins of ideals and arithmetic applications to one-prime-parameter families

We will define three similar but different notions of “spin” of an ideal in a number field, and we will show how a number-field version of Vinogradov’s method (a sieve involving “sums of type I” and “sums of type II”) can be used to prove that spins of prime ideals oscillate. Such equidistribution results have applications to the distribution of 2-parts of class groups of quadratic number fields and 2-Selmer groups of quadratic twists of elliptic curves in families parametrized by prime numbers. Parts of this talk are joint work with Peter Koymans.

Practical combinatorial optimization

Combinatorial optimization is a field that provides results and insights relevant in different domains. In this talk we intend to illustrate this claim by focusing on a number of specific results. As a first example, we consider the Circle Method (which dates back to 1851); this method is widely used to generate schedules for round-robin tournaments. An indicator for the fairness of a schedule is the so-called carry-over effect value. We prove that, for an even number of teams, the Circle Method generates a schedule with maximum carry-over effect value, answering an open question. As a second example, we consider the allocation of kidneys in live kidney exchange. Given a number of pairs, each pair consisting of a patient and a donor, incompatibility of the donor with the patient often prohibits a direct transplantation from the donor of a pair to the patient of that pair. Still, by using the fact that the donor of one pair may be compatible with the patient of another pair, and vice versa, transplantations may be possible. We review some of the existing models that are used for solving this problem. Finally, we consider the operation of a set of locks located along a waterway. Ships that travel along a waterway may have to wait in front of a waterway, and the amount of waiting time depends on the precise operation of these locks. We will investigate the computational complexity of the resulting problems, and discuss solution strategies.

Noetherianity of dualized adjoint representations up to locally diagonal groups

Finite-dimensional vector spaces are Noetherian, i.e. every descending chain of Zariski-closed subsets stabilizes. For infinite-dimensional spaces this is not true. However it can be true for some group G acting on the space that every descending chain of G-stable closed subsets stablizes. We call spaces for which this holds G-Noetherian. Recently, Draisma proved that polynomial functors of finite degree are Noetherian, which gives many representations of the infinite-dimensional general linear group that are Noetherian up to the action of the group. Eggermont and Snowden improved this result to encompass algebraic representations of infinite-dimensional classical groups.

In this talk we will go in the opposite direction. We restrict ourselves to a specific representation, the dual of the adjoint representation, but we consider a wider class of groups, those that are the limit of a sequence of diagonal embeddings between classical groups. We prove that such representations are Noetherian up to the group action.

On linearization for the quadratic shortest path problem

Given an instance of the quadratic shortest path problem (QSPP) on a digraph G, the linearization problem for the QSPP asks whether there exists an instance of the linear shortest path problem on G such that the associated costs for both problems are equal for every s-t path in G. We prove here that the linearization problem for the QSPP on directed acyclic graphs can be solved in O(nm3) time, where n is the number of vertices and m is the number of arcs in G. By exploiting this linearization result, we introduce a family of lower bounds for the QSPP on acyclic digraphs. The strongest lower bound from this family of bounds is the optimal solution of a linear programming problem. To the best of our knowledge, this is the first study in which the linearization problem is exploited to compute bounds for the corresponding optimization problem. Numerical results show that our approach provides the best known linear programming bound for the QSPP.

Harmonic shifted symmetric functions and the Bloch-Okounkov theorem

By the Bloch-Okounkov theorem, certain functions defined by sums over partitions are quasimodular. The space of quasimodular forms is closed under differentiating, which induces a sl_2-action on quasimodular forms. This mapping preserves this sl_2-action. A linear subspace of these functions on partitions – those in the kernel of a certain operator, called the space of harmonic shifted symmetric functions – are actually modular. We find an explicit basis for this kernel using an analogue of the Kelvin transform.

Vinogradov’s Theorem with Fouvry-Iwaniec primes

Vinogradov showed in 1937 that every sufficiently large odd integer can be written as the sum of three primes. We consider this problem restricted to primes p, such that p=k^2+l^2 for some integer k and prime l. Fouvry and Iwaniec showed in 1997 that there are infinitely many of such primes. We extend their work to arithmetic progressions and combine it with B. Green’s transference principle and several sieve related ideas to show that every sufficently large integer x with x = 3 mod 4 can be written as the sum of three such primes.

Knot Invariants using the Quantum Double Construction on sl_3

In this talk I will briefly explain the concept of quantum enveloping algebras. I will construct them using the quantum double construction. I will go into the connection between quantum groups and knot invariants, and explain how we created knot invariants with this construction.