Date: 26 January 2023
Speaker: Dmitry Kleinbock
Title: A gentle intro to Diophantine approximation via dynamics
Abstract: I will start with basics of simultaneous Diophantine approximation and show how the set-up can be studied via diagonal flows on the space of lattices. Then I will apply this approach to the problem of improvement of Dirichlet's Theorem, which will lead to the proof of a theorem of Davenport-Schmidt (1969) and a new result in my recent paper with Andreas Strombergsson and Shucheng Yu. No background is required to follow the talk.
Date: 2 February 2023
Speaker: Vasiliy Neckrasov
Title: Completely irrational subspaces and their Diophantine properties
Abstract: In this talk I will introduce the notion of completely irrational subspaces and some recent results about their basic diophantine properties. We will talk about bad approximability and discuss Schmidt's games (and their generalizations) as a useful tool for proofs. If we have enough time, I will also give some new bounds for Diophantine exponents and explain how they can be improved for vectors from a completely irrational subspace.
Date: 16 February 2023
Speaker: Felipe Ramirez
Title: The Duffin-Schaeffer conjecture for systems of linear forms
Abstract: The Duffin-Schaeffer conjecture, posed in 1941 and proved by Koukoulopoulos and Maynard in 2020, gives a complete characterization of when almost every real number can be approximated by infinitely many reduced rationals at a given rate of approximation. Essentially, the characterization is given by the divergence or convergence of a measure sum associated to the desired rate of approximation. Between 1941 and 2020, many related results were proved, including a higher dimensional version of the conjecture by Pollington and Vaughan in 1990. I will discuss these results, as well as a new one showing that the Duffin-Schaeffer conjecture also holds in the setting of systems of linear forms.
Date: 2 March 2023
Speaker: Tariq Osman
Title: Limit Theorems for Theta Sums
Abstract: Theta sums are exponential sums of the form
Date: 9 March 2023
Speaker: Dubi Kelmer
Title: Intrinsic Diophantine approximations on the sphere with and without dynamics
Abstract: Intrinsic Diophantine approximations is concerned with the question of how well can one approximate a real point on an algebraic variety by rational points lying on the same variety. In this talk I will focus on the case of the sphere and describe several problems in intrinsic Diophantine approximations and some results on these problems using methods from dynamics and number theory. I will then describe some refinements of these results, based on recent joint work with Shucheng Yu, relying on a second moment formula of light cone Siegel transform.
Date: 23 March 2023, special time/place: 4:30 PM Volen 106 (Brandeis-Harvard-MIT-Northeastern Joint Colloquium)
Speaker: Barak Weiss
Title: Horocycle flow on the moduli space of translation surfaces
Abstract: I will discuss the dynamics of the horocycle flow on a stratum of translation surfaces (which is an invariant subvariety of the bundle Omega M_g of holomorphic one forms over the moduli space of genus g Riemann surfaces). This flow can be defined as the action of upper triangular matrices with eigenvalue 1, acting linearly on flat charts. Work of Ratner on unipotent flows on homogeneous spaces leads to the question of whether the orbit-closures and invariant measures for this action can be meaningfully classified. I will quickly survey both positive and negative results in this direction. The talk will be based on joint work with Bainbridge, Chaika, Smillie, and Ygouf (in various combinations).
Date: 20 April 2023
Speaker: Manuel Luethi
Title: Almost prime times in the maximal horospherical flow on lattices in three dimensions
Abstract: We give an effective equidistribution result for the standard maximal horospherical subgroup U in the special linear group acting on the space of unimodular lattices in three dimensions. More precisely, following earlier work of Taylor McAdam, we know that orbits of the horospherical group are effectively equidistributed unless there is a Diophantine obstruction. In ongoing work, Taylor McAdam and the speaker provide a finer analysis of the Diophantine obstruction. Namely, we show that for a point generic under the horospherical group but facing the Diophantine obstruction, there exists a sequence of well equidistributed intermediate orbit closures closely tracked by U such that the U-orbits are effectively equidistributed inside the intermediate orbit closures. As an application, we remove the base-point dependence of almost-primes in earlier work by Taylor McAdam. This is joint work with Taylor McAdam.
Date: 27 April 2023
Speaker: Shahriar Mirzadeh
Title: Large deviations and dimension drop in homogeneous spaces
Date: 2 May 2023 (Brandeis Thursday)
Speaker: Akshat Das
Title: The three gap theorem: a survey and an adelic version
Abstract: In order to understand problems in dynamics that are sensitive to arithmetic properties of return times to regions, it is desirable to generalize classical results about rotations on the circle to the setting of rotations on adelic tori.
The classical three gap theorem for rotations on the circle was first proved in the late 1950s and since then it has been proved numerous times using new techniques and generalized in many ways. I will begin the talk with an insight into the classical result and some of these works. I will then give an introduction to the setting of an adelic torus and state a natural generalization of the three gap theorem for rotations on adelic tori, which is joint work with Alan Haynes. Our proof uses an adaptation of a lattice based approach to gaps problems in Diophantine approximation that was originally introduced by Marklof and Strombergsson. I will explain the reformulation of our problem as a problem about bounding a certain function on a space of lattices. I will end by giving an exhaustive list of examples to prove that the bound we get is best possible.
Date: 11 May 2023
Speaker: Demi Allen
Title: Shrinking targets and recurrence
Abstract: In this talk, time permitting(!), I aim to discuss two somewhat disjoint yet related topics. First of all, I aim to discuss Hausdorff measure results for shrinking target sets in self-conformal sets. Secondly, I plan to discuss almost sure recurrence rates of Gibbs measures for Hoelder continuous potentials over subshifts of finite type. The first part of the talk will be based on joint work with Balázs Bárány (Budapest), and the latter part will be based on joint work with Simon Baker (Loughborough) and Balázs Bárány (Budapest).