Seminare und Vorträge im SS 2023

am Montag, 17. April :

Oberseminar Zahlentheorie

Florian Luca
Title: Recent progress on the Skolem problem
Abstract: The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. One of the main contributions of the talk is to present an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct - a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the \(p\)-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool SKOLEM point to the practical applicability of this method. In the second part of the talk, we present the notion of an Universal Skolem Set, which is a subset of the positive integers on which the Skolem is decidable regardless of the linear recurrence. We give two examples of such sets, one of which is of positive density (that is, contains a positive proportion of all the positive integers).
(Coathors: Y. Bilu, J. Nieuwveld, J. Ouaknine, D. Purser, and J. Worrell)

14:00 Übungsraum 2

am Montag, 15. Mai :

Oberseminar Zahlentheorie

Wing Hong Leung
Title: Short second moment bounds and subconvexity for GL(3) L-functions
Abstract: Subconvexity problem, which asks for an estimate of the L-functions on the central value or line, is one of the major problems in analytic number theory. In this talk, we will discuss our joint work with Keshav Aggarwal and Ritabrata Munshi, in which we obtained a strong t-aspect subconvexity estimate for the GL(3) L-functions via a short second moment approach. We will first briefly review what the subconvexity problem is, focusing on the GL(3) setting. Then we will turn to the statement and the key steps of our proof, highlighting the new ingredients we employ with the delta method. We will also mention some recent follow ups from this work.

14:00 Übungsraum 2

am Montag, 22. Mai :

Oberseminar Zahlentheorie

Joshua Males
Title: Oscillating asymptotics of a Nahm-type sum
Abstract: In his famous '86 paper, Andrews made several conjectures on the function \(\sigma(q)\) of Ramanujan, including that it has coefficients (which count certain partition-theoretic objects) whose sup grows in absolute value, and that it has infinitely many Fourier coefficients that vanish. These conjectures were famously proved by Andrews-Dyson-Hickerson in their '88 Invent. paper, and the function \(\sigma\) has been related to the arithmetic of \(\mathbb{Z}[\sqrt{6}]\) by Cohen (and extensions by Zwegers), and is an important first example of quantum modular forms introduced by Zagier. A closer inspection of Andrews' '86 paper reveals several more functions that have been a little left in the shadow of their sibling \(\sigma\), but which also exhibit extraordinary behaviour. In an upcoming paper with Folsom, Rolen, and Storzer, we study the function \(v_1(q)\), which is given by a Nahm-type sum and whose coefficients count certain differences of partition-theoretic objects. We give explanations of four conjectures made by Andrews on \(v_1\), which require a blend of novel and well-known techniques, and reveal that \(v_1\) should be intimately linked to the arithmetic of the imaginary quadratic field \(\mathbb{Q}[\sqrt{-3}]\).

14:00 Übungsraum 2

am Montag, 05. Juni :

Oberseminar Zahlentheorie

Steven Charlton
Title: Generators of multiple t values, and alternating multiple zeta values
Abstract: Multiple zeta values, and their relatives including the multiple t values, are a prominent but mysterious class of real numbers, which appear in various areas from high energy physics and knot theory, to number theory and the periods of mixed Tate motives. I will review some work by Francis Brown, and some recent work by Takuya Murakami, on how to prove certain elements ζ(2's and 3's), and t(2's and 3's), generate the space of multiple zeta values. I will then extend Murakami's work to show t(1's and 2's) generate the space of multiple t values and alternating multiple zeta values, and explain some progress towards Saha's conjecture that t(1's and 2's, 2 or 3) are a basis for convergent MtV’s.

14:00 Übungsraum 2

am Montag, 19. Juni :

Oberseminar Zahlentheorie

Guo-Niu Han
Title: Hook length formulas from a combinatorist's point of view
Abstract: In the first part, I introduce the hook length expansion technique for integer partitions and for plane trees, and explain how to find old and new hook length formulas. As an application, we derive an expansion formula for the powers of the Euler Product in terms of hook lengths, which is first discovered by Nekrasov-Okounkov and Westburg. We also obtain an extension by adding two more parameters. It appears to be a discrete interpolation between the Macdonald identities and the generating function for t-cores. Surprisingly, our general formula has a link with the long standing Lehmer conjecture about the Ramanujan tau-function. In the second part, I will talk about some developments around the Nekrasov-Okounkov formulas. This is a mix of several topics related to hook lengths. In particular, I introduce the difference operator for functions defined on partitions, and show that the Plancherel average of the even power sum of hook lengths is always a polynomial for certain classes of partitions, such as strict, doubled distinct and self-conjugate partitions.

14:00 Übungsraum 2

am Montag, 10. Juli :

Oberseminar Zahlentheorie

Koustav Banerjee
Title: Error bounds for the asymptotic expansion of the modified Bessel function \(I_\nu (x) \) and its consequences.
Abstract: Study on asymptotics of modified Bessel functions dates back to 18th century. In this talk, first, we will describe how from the Hadamard’s integral representation of the modified Bessel function of first kind (denoted by \(I_\nu (x) \)), one comes up with the asymptotic expansion of \(I_\nu (x) \) and finally derives a bound of the absolute value of the error term of the asymptotic expansion by truncating at a point N for all \(x \geq 1\). In this regard, we will discuss briefly on a result of Bringmann, Kane, Rolen, and Tripp. Then we move on to talk about optimality of the error bound estimation stated before. In the final part of the talk, we will discuss in brevity about applications of such inequalities for \(I_\nu (x) \) in proving log-concavity, higher order Turan inequalities of certain arithmetic sequences arising from Fourier expansion of modular forms and related functions.

14:00 Übungsraum 2

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