Umeå DiGResS

Soffía Árnadóttir, DTU

Anurag Bishnoi, DELFT

Altar Ciceksiz, Umeå 

Clive Elphick, Birmingham

Victor Falgas Ravry, Umeå

Alexey Gordeev, Umeå

Jakob Hultgren, Umeå 

Sabrina Lato, Umeå 

Signe Lundqvist, Umeå

Klas Markström, Umeå

Tovohery Randrianarisoa, Umeå

Maryam Sharifzadeh, Umeå 

Klara Stokes, Umeå 

Istvan Tomon, Umeå 

Joannes Vermant, Umeå

Monday

9:30 - 10:00 Registration and coffee

10:00 - 10:30 Open Problem Session (Clive): My conjectures in spectral graph theory

10:30-11:00 Open Problems in Finite Geometry (Anurag)

11:00 - 11:30 Break

11:30-12 Open Problem Session (Soffía)

Lunch

13:30 - 15:00 Group Discussions

Coffee

15:00 - 15:30 Open Problem Session (Jakob)

15:30 - 17:00 Group Discussions

19:00 Workshop Dinner

Tuesday

9:30 - 10:30 Group Discussions

Coffee

11-12 Seminar (Anurag)

Lunch 

13:30 - 17 Excursion

Wednesday

9:30 - 12 Group Work

Open problem sessions

Clive Elphick: My conjectures in spectral graph theory

I will give a brief overview of applications of spectral graph theory and then discuss four of my conjectures which have been proved and four which are unproven. The unproven conjectures include lower bounds for the clique number and the fractional chromatic number. My final unproven conjecture was placed 1st in a survey of 25 unsolved problems in spectral graph theory published last year.

Anurag Bishnoi: Open problems in finite geometry

I will present some open problems related to blocking sets, which are vertex covers in hypergraphs constructed from finite affine or projective spaces. Traditionally, the methods for bounding the size of these objects have involved combinatorial and geometric arguments combined with the polynomial method. More recently, probabilistic and coding-theoretic methods have shown great potential. Additionally, I will present new explicit constructions that we obtained using expander graphs. These have opened new avenues for further development in finite geometry.

Soffía Árnadóttir:  Bi-Cayley graphs, spectra and coherent configurations 

Cayley graphs arise from groups and are therefore important objects in algebraic graph theory. The spectrum of a Cayley graph can be studied using tools such as representation theory and association schemes. In this talk, we will look at a bipartite analogue of Cayley graphs, which we will call bi-Cayley graphs (introduced by Evra et al as Cayley bigraphs). Behind those graphs is an algebraic structure, somewhat analogous to association schemes, called bipartite coherent configurations (introduced recently by Sabrina Lato). The hope is that bipartite coherent configurations will prove useful for studying properties of bi-Cayley graphs (such as their spectrum) however, this work is very recent and the talk might contain many questions and few answers. This is joint work with Sabrina Lato. 

Jakob Hultgren: An asymptotic minimal matching problem on convex lattice polytopes, with applications in mirror symmetry

A reflexive polytope is a convex lattice polytope embedded in a vector space with vertices in a lattice, such that the vertices of its dual are lattice points in the dual lattice. This is a rather restrictive condition: up to the action of GL(Z,d), there are only finitely many such polytopes in R^d for each d, although the actual number grows quickly (R^2 has 16 reflexive polytopes, R^3 has 4319, R^4 has 473,800,776 and R^5 has [unknown]). For a positive integer k, I'm interested in the minimal matching problem that arises when matching the (1/k)-lattice points in the boundary of the original polytope with the (1/k)-lattice points in the boundary of the dual polytope, subject to the cost function given by -<m,n> for the standard pairing/scalar product. In particular, I want to know under what circumstances a minimal matching maps each open face into the star of the corresponding vertex in the dual. Knowing this (asymptotically as k tends to infinity) is related to patching together local torus fibrations in Calabi-Yau manifolds, which is a central problem in the SYZ point of view of mirror symmetry. 

Some notes:

• a (1/k)-lattice point is a point m such that km is a lattice point

• The entry in the Online Encyclopedia of Integer Sequences concerning reflexive polytopes has number a090045

• Here is an image of the 16 reflexive polytopes in R^2: https://www.researchgate.net/figure/All-16-reflexive-2-dimensional-polytopes-This-is-43-Fig-15_fig1_340712259

• The cost function in the minimal matching problem -<m,n> might seem odd at first. By identifying the vector spaces and completing the square, you can show that form the point of view of optimization, it is equivalent to the squared distance function |m-n|^2. It also enforces a monotonicity property of the minimal matching when thought of as a partially defined map T on R^d . Namely, <T(m)-T(m'),m-m'> >= 0 for all (1/k)-lattice points m,m' in the boundary. 

• The carfeul reader may have noticed that the number of lattice points on the boundary of a polytope might not be the same as the number of lattice points on the boundary of its dual. In principle it is ok to throw away a small number of points on the side which has to many, but asymptotically, we need to adjust the rates 1/k above to take into account the fact that the number of (1/k)-lattice points on the boundary may grow significantly quicker or slower (wrt k) than the number of (1/k)-lattice points on the boundary of its dual. 

Seminar

Anurag Bishnoi: Constructing strong blocking sets and minimal codes using expander graphs 

A strong blocking set in a finite projective space is a set of points that intersects each hyperplane in a spanning set of the hyperplane. I will present a new graph theoretic construction of such sets: combining constant-degree expanders with asymptotically good codes, to obtain explicit strong blocking sets in the (k − 1)-dimensional projective space over Fq that have size at most cqk for some universal constant c. Since strong blocking sets have been shown to be equivalent to minimal linear codes, this construction gives the first explicit construction of Fq-linear minimal codes of length n and dimension k, for every prime power q, for which n ≤ cqk. This solves one of the main open problems on minimal codes.

Joint work with Noga Alon, Shagnik Das, and Alessandro Neri. 

Place: The workshop will take place at Umeå University, Campus Umeå, mostly in the vicinity of Vardagsrummet in HumLab. 

Umeå has an airport approximately 15 min taxi or bus ride from the University. Most itineraries, especially international, will involve a connection in Stockholm. 

There are also good train connections from Stokholm, both by day and by night.

Contact Klara Stokes (klara.stokes@umu.se) or Jakob Hultgren (jakob.hultgren@umu.se) for further information and registration.