Seminare und Vorträge im WS 2024/2025

am Montag, 07. Oktober :

Oberseminar Zahlentheorie

Qihang Sun
Title: Exact formulas for rank generating functions
Abstract: Dyson's ranks provided a new understanding of the integer partition function, especially of its congruence properties. In 2009, Bringmann used the circle method to prove an asymptotic formula for the Fourier coefficients of rank generating functions. In this talk, we will prove that the asymptotic formula, when summing up to infinity, converges and becomes a Radamacher-type exact formula for the rank of partitions. This exact formula also provides us a new proof of Dyson's rank conjectures.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 04. November :

Oberseminar Zahlentheorie

Jonathan Conrad
Title: Lattices, Gates, and Curves: GKP Codes as a Rosetta Stone
Abstract: The goal of quantum error correction is to encode quantum information into a physical Hilbert space, such that it is robust to errors and logical gates can be implemented in a fault tolerant fashion. In this talk, we will discuss the class of Gottesman-Kitaev-Preskill (GKP) Codes, which are quantum error correcting codes that encode quantum information into the continuous structure underlying the Hilbert space of a collection of quantum harmonic oscillators by endowing it with a symplectic lattice symmetry. The underlying formalism classifies all quantum error correcting codes that are designed based on Heisenberg-Weyl symmetries.
For a single mode quantum system, the moduli space of such GKP codes becomes equivalent to that of complex elliptic curves and I discuss how fault-tolerant quantum gates arise as homotopically non-trivial loops in this space. This perspective introduces an operational interpretation of the modular discriminant function tied to the quantum error correcting properties of the codes along the traversed path. I point to extensions of this dictionary to general multi-mode GKP codes and motivate questions of future research at the intersection of our fields.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 18. November :

Oberseminar Zahlentheorie

Ken Ono
Title: Traces of Partition Eisenstein series
Abstract: Integer partitions are ubiquitous in mathematics, arising in subjects as disparate as algebraic combinatorics, algebraic geometry, number theory, representation theory, to mathematics physics. Many of the deepest results on partitions have their origin in the work of Ramanujan. In this lecture, we will describe a new and unexpected role for partitions that also arises from the mysterious “lost notebook” of Ramanujan. We discover and explain the role of new q-series called “partition Eisenstein series”. These functions magically pop up as the key device for solving a conjecture of Andrews and Berndt, for studying symmetric functions of 2-dimensional lattice sums, for determining the properties of Andrews-Garvan “crank statistic”, and for representing the Taylor coefficients of many interesting Jacobi forms. This talk will tell the story of the recent discovery of these functions, and will offer a brief tour of these applications.

14:00 Online in ZOOM

am Montag, 25. November :

Oberseminar Zahlentheorie

Dmitrii Adler
Title: Jacobi forms and modular differential equations
Abstract: For modular forms, there is a well-known differential operator that increases the weight of modular forms by 2. One can study differential equations with respect to this differential operator. Examples of such equations are the Ramanujan system of differential equations and the Kaneko-Zagier equation. A similar construction takes place in the case of Jacobi forms. In my talk I will discuss differential equations of Jacobi forms and some applications related to the elliptic genus of Calabi-Yau manifolds. This is joint work with Valery Gritsenko.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 02. Dezember :

Oberseminar Zahlentheorie

Shane Chern
Title: A central limit theorem for a card shuffling problem
Abstract: Given a positive integer \(n\), consider a random permutation \(\tau\) of the set \(\{1,2, \ldots, n\}\). In \(\tau\), we look for sequences of consecutive integers that appear in adjacent positions: a maximal such a sequence is called a block. Each block in \(\tau\) is merged, and after all the merges, the elements of this new set are relabeled from \(1\) to the current number of elements. We continue to randomly permute and merge this new set until only one integer is left. In this talk, I will investigate the asymptotic behavior of \(X_n\), the number of permutations needed for this process to end. In particular, I will display an explicit asymptotic expression for each of the mean value \(\mathbf{E}[X_n]\) and the variance \(\mathbf{Var}[X_n]\) as well as for every higher central moment, and show that \(X_n\) satisfies a central limit theorem. This is joint work with Lin Jiu and Italo Simonelli.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 09. Dezember :

Oberseminar Zahlentheorie

Sven Möller
Title: Symplectic reduction, quasi-lisse vertex operator algebras and quasimodular forms
Abstract: Important properties of a vertex operator algebra V are governed by its associated Poisson variety X. For example, if dim(X)=0 (and if V is rational), then the character of V is a modular form of weight 0 (for some congruence subgroup).
A weaker condition for X is being quasi-lisse, i.e. having finitely many symplectic leaves. In this situation, it is expected that the character of V is quasimodular. I will discuss examples of such vertex operator algebras, which are obtained by symplectic reduction and provide chiral quantisations of quiver varieties. These play an important role in the context of 3d and 4d superconformal field theories.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 16. Dezember :

Oberseminar Zahlentheorie

Larry Rolen
Title: Conjectures of Andrews on partition-theoretic q-series
Abstract: In a famous 1986 paper, Andrews made a number of conjectures on the signs and growth rate of q-series arising from partition theory. Andrews made these based on computer experiments. The first of these functions, the famous function \(\sigma(q):=\sum_{n \geq 0}\frac{q^{\frac{n(n+1)}{2}}}{(-q;q)_n},\) had remarkable growth and vanishing behavior which was finally proven by Andrews-Dyson-Hickerson by tying this series to the arithmetic of the field \(\mathbb{Q}(\sqrt6)\). Cohen further uncovered that the numerical phenomenon was due to the q-series being what we would now call, thanks to work of Lewis-Zagier, a period integral of a Maass waveform. This was also an early example of the new theory of Zwegers mock Maass theta functions, and of a quantum modular form. In the same paper, Andrews also made conjectures on remarkable sign behavior of partition theoretic functions, such as \(v_1(q):=\sum_{n\geq0}\frac{q^{\frac{n(n+1)}{2}}}{(-q^2;q^2)_n}.\) Here, we will discuss recent work, joint with Folsom, Males, and Storzer, establishing some of these. We will also discuss conjectural observations for other questions of Andrews on this function.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 13. Januar :

Oberseminar Zahlentheorie

Claude Duhr
Title: From number theory to particle physics: scattering amplitudes in the 21st century
Abstract: Collider experiments like those carried out at the Large Hadron Collider (LHC) at CERN require very precise theoretical predictions. Over the last 10 years, it was discovered that there is a close connection between precision computations for collider experiments and certain topics in number theory. After giving a brief overview of computations in particle physics, I review this connection and highlight some places where number theory has informed particle physics.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 20. Januar :

Oberseminar Zahlentheorie

Diana Mocanu
Title: Generalized Fermat equations over totally real fields
Abstract: Wiles’ famous proof of Fermat’s Last Theorem pioneered the so-called modular method, in which modularity of elliptic curves is used to show that all integer solutions of the Fermat’s equation are trivial.
In this talk, we briefly sketch a variant of the modular method described by Freitas and Siksek in 2014, proving that for sufficiently large exponents, Fermat’s Last Theorem holds in five-sixths of real quadratic fields. We then extend this method to explore solutions to two broader Fermat-type families of equations. The main ingredients are modularity, level lowering, image of inertia comparisons, and S-unit equations.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 27. Januar :

Oberseminar Zahlentheorie

Winston Heap
Title: Simultaneous extreme values of zeta and L-functions
Abstract: I will discuss a recent joint work with Junxian Li which examines joint distributional properties of L-functions, in particular, their extreme values. Using a modification of the resonance method we demonstrate the simultaneous occurrence of extreme values of L-functions on the critical line. The method extends to other families and can be used to show both simultaneous large and small values.

14:00 Seminarraum 3 des Mathematischen Instituts

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