Note: In-person seminars will be held at `Übungsraum 2' 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 SS 2022
am Montag, 04. April :
Oberseminar Zahlentheorie
Petru Constantinescu
Title: Dissipation of Correlations of Automorphic Forms
Abstract: Mass equidistribution of eigenfunctions is a central topic in quantum
chaos and number theory. In this talk we highlight a generalisation of
the Quantum Unique Ergodicity for holomorphic cusp forms in the weight
aspect. We show that correlations of masses coming from off-diagonal
terms dissipate as the weight tends to infinity. This corresponds to
classifying the possible quantum limits along any sequence of Hecke
eigenforms of increasing weight.
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14:00 |
Übungsraum 2 |
am Montag, 25. April :
Oberseminar Zahlentheorie
Ian Wagner
Title: Laguerre-Pólya type functions with applications in combinatorics and number theory
Abstract: We define a new class of functions which puts the recent work of Griffin, Ono, Rolen, and Zagier into a framework analogous to the classical Laguerre-Pólya class. The main result is a classification of functions in this new class involving multiplier sequences, Jensen polynomials, and generalized Laguerre inequalities. Finally, we discuss some applications and open problems connected to functions in this class with a focus on partitions and the Riemann Xi-function.
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14:00 |
Übungsraum 2 |
am Montag, 23. Mai :
Oberseminar Zahlentheorie
Jan Bruinier
Title: Special cycles on toroidal compactifications of orthogonal Shimura
varieties
Abstract: We report on joint work with Shaul Zemel and Markus Schwagenscheidt. We
define special cycles on orthogonal Shimura varieties and prove a
modularity result for the corresponding generating series. It turns out
that the multiplicities of the irreducible components of the boundary
divisors are given by special values of regularized theta lifts of
Lorentzian and positive definite lattices. There are explicit formulas for
these multiplicities, which can be viewed as generalizations of the
Kronecker class number relations.
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14:00 |
Übungsraum 2 |
am Montag, 30. Mai :
Oberseminar Zahlentheorie
Taylor Garnowski
Title: Asymptotic Analysis of Mixed Mock Modular Forms and Related q-products
(PhD Defense)
Abstract: This thesis contains results of three research projects which study asymptotics for the Fourier coefficients of mixed mock modular forms and twisted q-products arising in combinatorics. To begin, we compute an asymptotic distribution for generalizations of unimodal sequences called odd-balanced unimodal sequences which were defined by Kim, Lim, and Lovejoy in 2016. We find the interesting result that the odd-balanced unimodal sequences with certain restrictions on their rank, are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which are asymptotically related to the partition function. In the second part of this thesis, we compute asymptotic estimates for the Fourier coefficients of two mock theta functions originating from Bailey pairs derived by Lovejoy and Osburn in 2012. We encounter cancellation in our estimates for one of the functions, which requires a careful study of secondary asymptotic terms. We deal with this by using higher order asymptotic expansions for the Jacobi theta functions. In our final result, we find asymptotic estimates for the complex Fourier coefficients of the product \((\zeta q;q)^{-1}_\infty\), with \(\zeta\) a root of unity. This result has interesting applications in analysis and combinatorics. For large \(n\), we are able to predict sign changes of arbitrary linear combinations of the function \(p(a,b;n)\) for fixed \(b\), where \(p(a,b;n)\) counts the number of partitions of \(n\) where the number of parts is congruent to \(a\) modulo \(b\). We see that simple differences of the type \(p(a_1,b;n)-p(a_2,b;n)\) have sign change patterns that oscillate like a cosine.
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14:00 |
Übungsraum 2 |
am Montag, 13. Juni :
Oberseminar Zahlentheorie
Jeremy Lovejoy
Title: Bailey pairs and indefinite quadratic forms
Abstract: The first part of this talk will be a leisurely review of
Bailey pairs and their classical applications to Rogers-Ramanujan type
identities. Next, I will continue in a historical vein, discussing
some old work of Andrews and Andrews-Hickerson on Bailey pairs with
special indefinite quadratic forms and their application to
Ramanujan's mock theta functions. Then I will move to more recent
work on Bailey pairs and generic indefinite quadratic forms and their
applications to mock theta functions and colored Jones polynomials.
Finally, I will sketch some very recent results on Bailey pairs and
indefinite quadratic forms that lead to "false" indefinite theta
series instead of the usual type.
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14:00 |
Übungsraum 2 |
am Montag, 20. Juni :
Oberseminar Zahlentheorie
Ioana Coman
Title: Insights from the quantum modularity of 3-manifold invariants
Abstract: A recently proposed class of topological 3-manifold invariants \(\hat{Z}[M]\) which admit series expansions with integer coefficients has been a focal point of much research over the past few years, proving themselves ubiquitous in a wide range of contexts.
They were originally defined physically as an index which computes the BPS spectra of certain supersymmetric quantum field theories in three dimensions, and associated to 3-manifolds \(M\) through the 3d-3d correspondence.
Mathematically, they have also been shown to possess curious number-theoretic features, giving examples of quantum modular forms.
After reviewing some of these new developments, here I explore certain higher rank extensions and highlight emerging features of the corresponding \(\hat{Z}\) invariants, such as nested relations with respect to their rank and a hidden modular structure.
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14:00 |
Übungsraum 2 |
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