Note: In-person seminars will be held at `Seminarraum 1' of the Mathematical Institute subject to requirements and guidelines of the University of Cologne on COVID-19. In particular, we have limited room for participants. So please contact `cnazarog@math.uni-koeln.de' in advance to check for room availability if you would like to participate.
Seminare und Vorträge im WS 2021/2022
am Montag, 06. September :
Oberseminar Zahlentheorie
Ken Ono
Frobenius Trace Distributions for Gaussian hypergeometric varieties
Abstract: In the 1980's, J. Greene defined hypergeometric functions over finite fields using Jacobi sums. The framework of his theory establishes that these functions possess many properties that are analogous to those of the classical hypergeometric series studied by Gauss and Kummer. These functions have played important roles in the study of Apéry-style supercongruences, the Eichler-Selberg trace formula, Galois representations, and zeta-functions of arithmetic varieties. We study the value distribution (over large finite fields) of natural families of these functions. For the \(_2F_1\) functions, the limiting distribution is semicircular, whereas the distribution for the \(_3F_2\) functions is the more exotic Batman distribution.
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15:00 |
Online in ZOOM |
am Montag, 25. Oktober :
Oberseminar Zahlentheorie
Wadim Zudilin
Rogers-Ramanujan reflections
Abstract: (Good!) finite (aka polynomial) versions of identities of Rogers-Ramanujan (RR) type often lead to other interesting RR type identities under the q->1/q reflection. I will illustrate the principle on (known and conjectured) examples of finite versions behind the RR (original), Caparelli and Kanade-Russell identities, leaving the audience some room to look for a structure in the chaos. The talk is based on joint project with Ali Uncu.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 08. November :
Oberseminar Zahlentheorie
Walter Bridges
Statistics for partitions and unimodal sequences (Part I)
Abstract: We survey the many techniques that have been used to prove limiting distributions for statistics for integer partitions and unimodal sequences, highlighting the importance of modularity on the occasions it arises. Time permitting, the first lecture will include discussions of elementary recurrences (e.g. the distribution of the largest part), the method of moments (e.g. ranks and cranks), and limit shapes. In the second lecture, we will introduce the so-called "Fristedt conditioning device", powerful probabilistic machinery that can produce distributions for many statistics for partitions at once. We will also discuss the recent adaptation of Fristedt's conditioning device to unimodal sequences by Bringmann and the speaker.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 15. November :
Oberseminar Zahlentheorie
Walter Bridges
Statistics for partitions and unimodal sequences (Part II)
See part I of the talk for the abstract.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 22. November :
Oberseminar Zahlentheorie
Claudia Alfes-Neumann
Harmonic Maass forms and periods
Abstract: We present work in progress on the relation of coefficients of harmonic weak Maass forms of half-integral weight and periods of associated differentials. This generalizes work of Bruinier and Bruinier and Ono who investigated the situation in the case that the harmonic weak Maass forms have weight 1/2.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 29. November :
Oberseminar Zahlentheorie
Amita Malik
Partitions into primes with a Chebotarev condition
Abstract: In this talk, we discuss the asymptotic behaviour of the number of partitions into primes concerning a Chebotarev condition. In special cases, this reduces to the study of partitions into primes in arithmetic progressions. While the study for ordinary partitions goes back to Hardy and Ramanujan, partitions into primes were recently re-visited by Vaughan. Our error term is sharp and improves on previous known estimates in the special case of primes as parts of the partition. As an application, monotonicity of this partition function is established explicitly via an asymptotic formula in connection to a result of Bateman and Erdös.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 06. Dezember :
Oberseminar Zahlentheorie
Michael Mertens
Moonshine Sonata: Theme and Variations
-- Movement 1: Monstrous Moonshine and the Jack Daniels Problem
Abstract: In this series of three lectures we will give a broad overview of some aspects of the theory of Moonshine. The term Moonshine in mathematics is often used to describe an unexpected, sometimes mysterious, connection between two or more seemingly unrelated fields of mathematics, most often finite groups and modular forms.
In the first lecture we consider the classical case of Monstrous Moonshine (concerning the sporadic simple Monster group and the modular j-function), the second will cover Mathieu and Umbral Moonshine, which involves 23 examples of groups which are associated to mock theta functions, and the third will be concerned with Moonshines having some connection to the arithmetic of elliptic curves.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 13. Dezember :
Oberseminar Zahlentheorie
Michael Mertens
Moonshine Sonata: Theme and Variations
-- Movement 2: Mathieu and Umbral Moonshine.
See part I of the talk for the abstract.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 20. Dezember :
Oberseminar Zahlentheorie
Michael Mertens
Moonshine Sonata: Theme and Variations
-- Movement 3 (Finale): Finite groups and Arithmetic.
See part I of the talk for the abstract.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 10. Januar :
Oberseminar Zahlentheorie
Aradhita Chattopadhyaya
Modularity and Scaling Black Hole Solutions
Abstract: Based on arXiv:2110.05504.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 17. Januar :
Oberseminar Zahlentheorie
Christina Roehrig
Siegel Theta Series for Indefinite Quadratic Forms
Abstract: Considering Siegel theta series for positive definite quadratic forms and including harmonic polynomials in the definition of these theta series, we can construct holomorphic Siegel modular forms. In this talk, we discuss a generalization of this construction to quadratic forms of indefinite signature \((r, s)\). In general, we obtain non-holomorphic Siegel modular forms, so we also investigate a specific example for the case \(s = 1\) and give a description of the holomorphic part of this series.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Montag, 24. Januar :
Oberseminar Zahlentheorie
Matthias Storzer
Modularity properties of Nahm sums
Abstract: The modularity properties of so-called Nahm sums are known to be related to certain elements in the Bloch group. Nevertheless, a first conjecture about the characterisation of modular Nahm sums in terms of these elements (sometimes referred to as ’Nahm’s Conjecture’) turned out to be false. In this talk, we will discuss the motivation behind this conjecture and describe a way to correct and refine it.
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14:00 |
Seminarraum 1 des Mathematischen Instituts |
am Freitag, 28. Januar :
Oberseminar Zahlentheorie
Wei-Lun Tsai
Distributions on integers partitions
Abstract: In this talk we examine three types of distributions on integer partitions.
(1) Generalizing a classical theorem of Erdos and Lehner, we determine the distribution of parts in partitions that are multiples of a fixed integer A. These limiting distributions are of Gumbel type. (2) For a fixed positive integer t, we determine the distribution of the number of hook lengths of size t among the partitions of n. As n tends to infinity, the distributions are asymptotically normal. (3) For a fixed integer t > 3, we determine the distribution of the number of hook lengths that are multiples of t among the partitions of n. As n tends to infinity, the distributions are asymptotically shifted Gamma distributions. This is joint work with Michael Griffin, Ken Ono, and Larry Rolen.
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15:00 |
Online in ZOOM |
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