The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give an elementary geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geometry. The talk is based on a joint work with Arseniy Akopyan.

This talk is based on joint work with Jens Marklof, and with Roland Roeder. The three distance theorem states that, if \(x\) is any real number and \(N\) is any positive integer, the points \(x, 2x, \ldots , Nx \mod 1\) partition the unit interval into component intervals having at most 3 distinct lengths. We will present two higher dimensional analogues of this problem. In the first we consider points of the form \(mx+ny \mod 1\), where \(x\) and \(y\) are real numbers and \(m\) and \(n\) are integers taken from an expanding set in the plane. This version of the problem was previously studied by Geelen and Simpson, Chevallier, Erdős, and many other people, and it is closely related to the Littlewood conjecture in Diophantine approximation. The second version of the problem is a straightforward generalization to rotations on higher dimensional tori which, surprisingly, has been mostly overlooked in the literature. For the two dimensional torus, we are able to prove a five distance theorem, which is best possible. In higher dimensions we also have bounds, but establishing optimal bounds is an open problem.

I will provide an introductory talk to hypergraph varieties, focusing on the combinatorial aspect. The main themes of the talk are (1) connecting the geometric properties of hypergraphs to their minimal matroids; (2) reducing the geometric invariants of these matroids to grid matroids; and (3) understanding the realizability of these matroids. Finally, I will mention the application to conditional independence models in statistics and will present some geometric questions and computational challenges around this problem. This is based on joint works with Kevin Grace, Oliver Clarke, and Harshit J Motwani.

In this talk, I shall review recent work on reversibility of isometries of Hermitian spaces over the complex numbers and over the quaternions. I shall explain what I mean by reversibility in the talk and how it has been classified in some Lie groups.

Steiner triple systems (STSs) are the simplest and best understood infinite family of block designs. They are equivalent to decompositions of complete graphs into edge disjoint triangles. A sequencing of an STS is a bijection of the points with the integers \([1, \ldots, v]\).

In 2019, Stinson and Kreher introduced L-good sequencings, in which no block is contained in an interval of length L, that is \([d, d+1, \ldots, d+L]\). Stinson and Veitch gave an algorithmic proof that an STS with v points has an L-good sequencing for \(v \geq L^{6}/16 + O(L^{5})\). This was later improved to \(v \geq L^{4}/2 + O(L^{3})\) by Blackburn and Linial with an explicit greedy algorithm.

Using the Lovasz Local Lemma, we prove that every STS with \(v \geq 121L^{2}\) admits an L-good sequencing. The exponent \(2\) is optimal, in the sense that counterexamples are known without L-good sequencings for \(v = \Theta(L^{2-\epsilon})\) for any \(\epsilon > 0\). In fact, we prove a more general result on decompositions of general Steiner systems into sufficiently large (as a function of L) disjoint independent sets. This is joint work with Daniel Horsley of Monash University.

The Berglund–Hübsch–Henningson (BHH–) duality is a particular case of the mirror symmetry. It is described as a duality on the set of pairs \((f, G)\) consisting of an invertible polynomial and a subgroup \(G\) of diagonal symmetries of \(f\). Symmetries of invariants of BHH-dual pairs are related to the mirror symmetry. There is a method to extend the BBH–duality to the set of pairs \((f, G')\), where \(G'\) is the semidirect product of a group \(G\) of diagonal symmetries of \(f\) and a group \(S\) of permutations of the coordinates preserving \(f\). The construction is based on ideas of A. Takahashi and therefore is called the Berglund-Hübsch-Henningson-Takahashi (BHHT–) duality. Invariants of BHHT–dual pairs have symmetries similar to mirror ones only under some restrictions on the group \(S\): the so-called parity condition (PC). Under the PC-condition it is possible to prove symmetries of the orbifold Euler characteristic and of some other orbifold invariants for actions on the Milnor fibers of dual pairs. The talk is based on joint results with W. Ebeling.

Geometric rigidity theory is concerned with the rigidity and flexibility analysis of bar-joint frameworks and related constraint systems of geometric objects. In the beginning of this talk, we will give a brief introduction to this area, which has a rich history that can be traced back to classical work of Euler, Cauchy and Maxwell on the rigidity of polyhedra and skeletal frames. One of the major recent research directions in this field is to study the impact of symmetry on the rigidity of bar-joint frameworks. We show how group representation theory can be used to reveal `hidden' infinitesimal motions and states of self-stress in symmetric frameworks that cannot be detected with Maxwell's basic counting rule from 1864. We then show how this symmetry-adapted counting rule, which was originally discovered by the engineer Simon Guest and the chemist Patrick Fowler, can be used to derive an efficient new method for constructing symmetric frameworks with a large number of `fully-symmetric' or `anti-symmetric' states of self-stress. Maximizing the number of independent states of self-stress of a planar framework, as well as understanding their symmetry properties, has important practical applications, for example in the design and construction of gridshells. We show the usefulness of our method by applying it to some practical examples.

This is joint work with Cameron Millar (SOM), Arek Mazurek (Mazurek Consulting) and William Baker (SOM).

We extend a formula of Carl Ludwig Siegel in the geometry of numbers, allowing the body to contain an arbitrary number of interior lattice points. Our extension involves a lattice sum of the covariogram of any compact set \(K\subset \mathbb{R}^d\), and hinges on a variation of the Poisson summation formula which we derive here. The Fourier methods herein also allow for more general admissible sets. One of the consequences of these results is a new characterization of multi–tilings of Euclidean space by translations, using the lower bound on lattice sums of such covariograms. Some classical results, such as Van der Corput's inequality, also follow from the main result. This is joint work with my graduate student Michel Faleiros Martins.

The celebrated Duffin–Schaeffer conjecture was recently settled in the affirmative by Koukoulopoulos and Maynard. In previous work, Haynes proposed a p–adic version of this conjecture. This latter conjecture was settled in the affirmative in recent, joint work with M. L. Laursen.

In my talk, I will provide the necessary background on the Duffin–Schaeffer conjecture as well as its p–adic counterpart. I will then proceed to sketch a proof of the conjecture. Finally, I will discuss another variant of of the conjecture, which is false but nonetheless raises some interesting questions for further research.

The classical (inhomogeneous) Khintchine–Groshev Theorem tells us that for a monotonic approximating function \(\psi: \mathbb{N} \to [0,\infty)\) the Lebesgue measure of the set of (inhomogeneously) \(\psi\)–well–approximable points in \(\mathbb{R}^{nm}\) is zero or full depending on, respectively, the convergence or divergence of \(\sum_{q=1}^{\infty}{q^{n-1}\psi(q)^m}\). In the homogeneous case, it is now known that the monotonicity condition on \(\psi\) can be removed whenever \(nm>1\), and cannot be removed when \(nm=1\). In this talk I will discuss recent work with Felipe A. Ramírez (Wesleyan, US) in which we show that the inhomogeneous Khintchine–Groshev Theorem is true without the monotonicity assumption on \(\psi\) whenever \(nm>2\). This result brings the inhomogeneous theory almost in line with the completed homogeneous theory. I will survey previous results towards removing monotonicity from the homogeneous and inhomogeneous Khintchine–Groshev Theorem before discussing the main ideas behind the proof our recent result.