Weekly Bulletin

Matzat's conjecture (now a theorem) predicts the structure of the absolute differential Galois group of one-variable function fields. In this talk, starting from the definition of the differential Galois group of a linear differential equation, I will give an overview of the proof of Matzat's conjecture.

A recent result of Cineli, Ginzburg and Gürel establishes a connection between the topological entropy of a Hamiltonian diffeomorphism and the persistence features of Lagrangian Floer homology. Following Cineli-Ginzburg-Gürel, we will explain the notion of barcode entropy that measures the exponential growth of not-too-short bars in a sequence of barcodes. Specifically, we will examine the barcodes of Lagrangian Floer homology for a pair of weakly exact Hamiltonian isotopic Lagrangian submanifolds in a closed symplectic manifold. By applying iterations of a Hamiltonian diffeomorphism we obtain a sequence of barcodes. We explain a result of Cineli-Ginzburg-Gürel asserting that the barcode entropy of this sequence is bounded above by the topological entropy of the Hamiltonian diffeomorphism. Lastly, we give some reasons why a reverse inequality is not possible in this simple setting.

This will be an analysis talk with motivation coming from number theory. I would like to present some ideas behind a general Hilbert space framework for solving certain Fourier optimization problems that arise when studying the distribution of the low-lying zeros of families of $L$-functions. For instance, in connection to previous work of Iwaniec, Luo, and Sarnak (2000), we will discuss how to use information from one-level density theorems to estimate the proportion of non-vanishing of $L$-functions in a family at a low-lying height on the critical line. We will also discuss the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). This is based on joint work with M. Milinovich and A. Chirre.

Plateau’s problem is a central problem in Geometric Analysis and PDEs. Given an arbitrary closed curve in R^3, it asks for the existence of an area minimizing embedded surface with boundary equal to the given curve. In 1960, Federer and Fleming conceived the theory of currents as a framework for solving this problem. A different approach was proposed in 1990 by Fröhlich and Struwe, through the study of level sets of semilinar elliptic equations. They showed the existence of a minimal surface which was smooth away from the curve. I will talk about joint work with Stephen Lynch, in which we show that the surface is also smooth up to the boundary, thus completing a new solution to Plateau’s problem.

This talk is designed to be a friendly invitation to the wild world of Partial Differential Equations. After introducing the fascinating and two-centuries old Monge-Ampère equation, and looking at its anatomy, we will trace its relation with the theory of optimal transportation. Time permitting, we will see how this applies to prove one of the most elegant equations of geometric analysis: the isoperimetric inequality.

A t-batch code is a method to store a data record in an encoded form on multiple servers in such a way that the bit-values in any batch of t positions from the record can be retrieved by decoding the bit-values in t disjoint groups of positions. More generally, a t-functional batch code can be used to retrieve any batch of t linear combinations of the stored bit-values. In 2017 by Wang et al., it was shown that the well-known binary simplex code of dimension k is a 2^(k-1)-batch code. The proof of that result is somewhat cumbersome, and the algorithm resulting from the proof requires to storing and using a database containing all the solutions for the cases where k <= 7. More recently, in 2020 Zhang et al., the authors conjecture that the k-dimensional simplex code is even a 2^(k-1)-functional batch code. In this talk, we give a simple, algorithmic proof of a result that falls halfway between the known result for the simplex code and the conjecture. Our approach is to relate the required properties of the simplex code to certain additive problems in finite abelian groups. In addition, we formulate a new conjecture for finite abelian groups that would imply the functional batch conjectures if true for groups of the form (Z_2)^k, and we prove that this new conjecture holds for the case of cyclic groups of prime order. This talk is based on a joint work with Henk D.L. Hollmann, Ago-Erik Riet and Vitaly Skachek. <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

Since its first usage for solving a coding theory problem in 1994 the Gray maps have been used to make constructions of well-known families of linear and nonlinear codes. The Gray map was first presented as an isometry between Z_4 and Z_2^2 with the Lee and Hamming metrics respectively. Later, generalisation up to finite chain rings of such Gray map were defined. In this talk we will see some properties of this Gray map and how can be used to obtain a compact representation of some families of nonlinear codes. <BR> <BR> (**This eSeminar will take place over Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact zita.fiquelideabreu@math.uzh.ch **)

Quantum topology provides a wealth of highly organized invariants, produced by machinery that operates in very general contexts. For some of them, a more classical homological reformulation is known. This often allows us to better understand the topological content of the resulting invariants, as witnessed by Bigelow’s spectacular proof of the linearity of braid groups. For the mapping class group Mod(Σ) of a surface Σ, we will explain how to recover the family of quantum representations associated with the small quantum group of sl(2) by a classical construction, with Mod(Σ) acting on twisted homology groups of configuration spaces of Σ. This is a joint work with Jules Martel.

In this talk we will discuss recent results of Robert Seiringer and myself, concerning the ground state energy and the energy-momentum relation of the Fröhlich polaron, which is a model describing the interactions between a charged particle and a polarized medium. We especially verify a conjecture by Landau and Pekar from 1948, claiming that the energy-momentum relation asymptotically coincides with the one of a free particle having an increased mass M=alpha^4 m, where m is an explicit constant, in the regime of large couplings alpha between the particle and the medium, and suitably small momenta.

I will discuss a recent result about describing the spectrum of randomly perturbed Berezin-Toeplitz operators, which generalizes a result of Martin Vogel from 2020 about quantizations of the torus. I will briefly discuss similar spectral results regarding randomly perturbed non-self-adjoint operators. Then I will explain how to construct Berezin-Toeplitz operators (which are quantizations of smooth functions on compact Kähler manifolds). Finally, I will discuss the main idea of proving a Weyl-law, which requires constructing an exotic calculus of Berezin-Toeplitz operators.

In the classical collective risk theory it is usually assumed that the capital reserve of a company is placed in the bank account paying zero interest. In the recent two decades the theory was extended to cover a more realistic situation where the reserve is invested, fully or partially, in a risky asset (e.g., in a portfolio evolving as a market index). This natural generalization generates a huge variety of new ruin problems. Roughly speaking, each “classical” ruin problem, e.g., a version of the Cramer-Lundberg model (for the non-life insurance, for the annuity payments etc., with a specific assumption) can be combined a model of price of the risky security (geometric Brownian motion, geometric Lévy process, various models with stochastic volatilities, etc.). In the talk we present new asymptotic results for the ruin probabilities, in particular, for the Sparre Andersen type models with risky investments having the geometric Lévy dynamics and for Cramér-Lundberg type models with investments in a risky asset with a regime switching price.

I will describe recent work giving an asymptotic formula for a count of primitive integral zeros of an isotropic ternary quadratic form in an orbit under integral automorphs of the form. The constant in the asymptotic is given explicitly in terms of local data determined by the orbit. Comparison with the well-known asymptotic for the corresponding count of all primitive zeros yields information on the distribution of the zeros in different orbits, the number of orbits r and the class number h of the genus of the form. For a certain special class of forms the distribution is shown to be uniform and a simple explicit formula is given for hr.

Intensive longitudinal studies (e.g., experience sampling studies) have demonstrated that detecting changes in statistical features across time is crucial to better capture and understand psychological phenomena. For example, it has been uncovered that emotional episodes are characterized by changes in both means and correlations. In psychopathology research, recent evidence revealed that changes in means, variance, autocorrelation and correlation of experience sampling data can serve as early warning signs of an upcoming relapse into depression. In this talk, I will discuss flexible statistical tools for retrospectively and prospectively capturing such changes. First, I will present the KCP-RS framework, a retrospective change point detection framework that can be tailored to capture changes in not only the means but in any statistic that is relevant to the researcher. Second, I will turn to the prospective change detection problem, where I will argue that statistical process control procedures, originally developed for monitoring industrial processes, are promising tools but need tweaking to the problem at hand.