Related papers: Lower bounds on the essential dimension of reducti…

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.

Log-concave sampling has witnessed remarkable algorithmic advances in recent years, but the corresponding problem of proving lower bounds for this task has remained elusive, with lower bounds previously known only in dimension one. In this…

The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup…

Let E be an elliptic curve over the rationals. Let L be an infinite Galois extension of the rationals with uniformly bounded local degrees at almost all primes. We will consider the infinite extension L(E_tor) of the rationals where we…

We give a purely algebraic treatment of reduction theory for connections over the formal punctured disc. Our proofs apply to arbitrary connected linear algebraic groups over an algebraically closed field of characteristic 0. We also state…

We study orthogonal decompositions of symmetric and ordinary tensors using methods from linear algebra. For the field of real numbers we show that the sets of decomposable tensors can be defined be equations of degree 2. This gives a new…

Originally motivated by questions of P. Etingof related to growth rates of tensor powers in symmetric tensor categories, we obtain general bounds on the order of finite subgroups of ${\rm GL}(n,\mathbb{C})$ with restricted composition…

We introduce the notion of splitting subgroups of quasireducitve supergroups, and explain their significance. For $GL(m|n)$, $Q(n)$, and defect one basic classical supergroups, we give explicit splitting subgroups. We further prove they are…

We prove lower bounds for the minimum distance of algebraic geometry codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds…

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\…

We prove a simple, nearly tight lower bound on the approximate degree of the two-level $\mathsf{AND}$-$\mathsf{OR}$ tree using symmetrization arguments. Specifically, we show that $\widetilde{\mathrm{deg}}(\mathsf{AND}_m \circ…

In the study of extensions of polytopes of combinatorial optimization problems, a notorious open question is that for the size of the smallest extended formulation of the Minimum Spanning Tree problem on a complete graph with $n$ nodes. The…

We show that there is significant cancellation in certain exponential sums over small multiplicative subgroups of finite fields, giving an exposition of the arguments by Bourgain and Chang.

In this paper, we explicitly classify the minimal discriminants of all elliptic curves $E/\mathbb{Q}$ with a non-trivial torsion subgroup. This is done by considering various parameterized families of elliptic curves with the property that…

For a discrete mechanical system on a Lie group $G$ determined by a (reduced) Lagrangian $\ell$ we define a Poisson structure via the pull-back of the Lie-Poisson structure on the dual of the Lie algebra ${\mathfrak g}^*$ by the…

We combine the fundamental results of Breuillard, Green, and Tao on the structure of approximate groups, together with "tame" arithmetic regularity methods based on work of the authors and Terry, to give a structure theorem for finite…

In this paper, we prove super-polynomial lower bounds for the model of \emph{sum of ordered set-multilinear algebraic branching programs}, each with a possibly different ordering ($\sum \mathsf{smABP}$). Specifically, we give an explicit…

We work in the category of locally definable groups in an o-minimal expansion of a field. Eleftheriou and Peterzil conjectured that every definably generated abelian connected group G in this category is a cover of a definable group. We…

Building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work, we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups. Such…

In studying the structure of derived categories of module categories of group algebras or their blocks, it is fundamental to classify support $\tau$-tilting modules. Koshio and Kozakai showed that the structure of support $\tau$-tilting…