Abstract:

In his paper on free subgroups of linear groups, Tits proved his famous alternative: a linear group is either virtually solvable, or contains a free subgroup. Since then, Tits' work has been generalized and applied in many different ways. One remaining open question in this field was the one asked by de la Harpe: let \(G\) be a semisimple Lie group without compact factors and with trivial center, and let \(\Gamma\) be a Zariski-dense subgroup of \(G\). Given a prescribed finite subset \(F\) of \(G\), is it always possible to find an element \(\gamma \in \Gamma\) such that for any \(h \in F\), the subgroup generated by \(h\) and \(\gamma\) is freely generated (in that case, we say \(h\) and \(\gamma\) are ping-pong partners). In the talk, we will discuss a variant of the question of de la Harpe, where \(F\) is a finite set of finite subgroups \(H_i\) of \(G\). Using careful refinements of the main steps of Tits' proof of the alternative (which we will recall), we give sufficient conditions for the existence of ping-pong partners for the \(H_i\) in any Zariski-dense subgroup \(\Gamma\). We will also show that these conditions are satisfied for products of copies of \(\mathrm{SL}_n\) over division \(\mathbb{R}\)-algebras. The existence of such free products has applications in the theory of integral group rings of finite groups, which will be briefly mentioned. Joint work with G. Janssens and D. Temmerman.

  Abstract:

The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history, from early works of Sylvester, Satge, Selmer etc. and, up to recent work of Alpoge-Bhargava-Shnidman. A conjecture attributed to Sylvester predicts that the primes >2 in the residue classes 2,5 mod 9 are not sums of two rational cubes, while those in the residue classes 4,7 or 8 mod 9 are. Primes which are 1 mod 9 may or may not be sums of two rational cube sums. We rephrase the problem in terms of elliptic curves, and use certain integral binary cubic forms to prove unconditionally that there are infinitely many primes in each of the residue classes 1 mod 9 and 8 mod 9 that are expressible as sums of two rational cubes. More generally, we prove that every non-zero residue class a (mod q), for any prime q, contains infinitely many primes which are sums of two rational cubes. Among other results, we show that corresponding to any positive integer n, there are infinitely many imaginary quadratic fields in which n is a sum of two cubes. These results represent joint work with Somnath Jha and Dipramit Majumdar. The starting point of this work was an accidental encounter in earlier work with Dipramit Majumdar when we classified all cubic cyclic extensions of \(\mathbb{Q}\).

  Abstract:

Polya frequency sequences are ubiquitous objects with a surprising number of connections to many different areas of mathematics. It has long been known that such sequences admit a "duality" operator that mimics the duality of a Koszul algebra and its quadratic dual, but the precise connection between these notions turns out to be quite subtle. In this talk, we will see how the equivariant analogue of Polya frequency is closely related to the problem of constructing Schur functors "with respect to" an algebra. We will moreover see how these ideas come together to understand the problem of extending Boij--Soederberg theory to other classes of rings, with particular attention given to the case of quadric hypersurfaces. This is based on joint work with Steven V. Sam.

  Abstract:

In various cases, a law that holds in a group with high probability, must actually hold for all elements. For instance, a finite group in which the commutator law \([x,y]=1\) holds with probability larger than \(5/8\), must be abelian. In the talk I'll discuss a probabilistic approach to laws on infinite groups, using random walks, and present results, joint with Greenfeld and Olshanskii, answering a few questions of Amir, Blachar, Gerasimova, and Kozma.

  Abstract:

A foundational result in equivariant commutative algebra is Cohen's theorem that the infinite variable polynomial ring \(R=\mathbb{C}[x_1,x_2,\ldots]\) is Noetherian up to the action of the infinite symmetric group. This has been applied to prove uniformity results for finite-dimensional structures in algebraic geometry, statistics, and algebraic topology, and motivates the study of other aspects of the equivariant commutative algebra of \(R\). In this talk, we explain an approach to developing the local theory of \(R\)-modules in this equivariant setting by studying a generalization of FI-modules to a "weighted" setting. We introduce these weighted FI-modules, and discuss how they too can be studied from the perspective of commutative algebra up to the action of parabolic subgroups of the infinite general linear group.

  Abstract:

Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings. On the other hand, enveloping algebras of infinite-dimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will summarize what is known about the noetherianity of enveloping algebras, with a focus on Lie algebras of derivations of associative algebras.

  Abstract:

We describe a new and elementary proof of the fact that many groups of piecewise-projective transformation of the line are non-amenable by constructing an explicit action without fixed points. One the one hand, such groups provide explicit counter-examples to the Day-von Neumann problem. On the other hand, they illustrate that we can distinguish many "layers" of relative non-amenability between nested subgroups.

  Abstract:

We will survey the class of linear groups, and identify three methods of constructing linear groups: algebraic, geometric, and arithmetic. We will propose that every subgroup comes from one of these types of constructions. There are many interesting consequences all across group theory and geometry if this is indeed the case. Focusing on the case of 2-by-2 matrix groups, the algebraic and arithmetic subgroups are very well-understood. We will fill in some understanding of geometric subgroups in this setting, by showing that for any prime p, there is a closed surface group in \({\rm PSL}_2(\mathbb{Z}[1/p])\). These surface groups will have genus proportional to \(p\), and we speculate that the construction is optimal.

  Abstract:

The symbol length problem is a longstanding question concerning the Brauer group of a field. In the case of fields of positive characteristic, every Brauer class is split by a finite extension of height 1. This observation suggests a connection between the symbol length problem and the Galois theory of purely inseparable extensions, where the restricted Lie algebra naturally arise. In this talk, we will explore how various symbol length problems in Brauer groups relate to restricted Lie algebras and introduce a moduli-theoretic description of restricted subspaces in a restricted Lie algebra.

  Abstract:

We discuss an analogue of the Goldbach conjecture for polynomials with coefficients in semidomains (i.e., subsemirings of an integral domain).