Department of Mathematics and Systems Analysis



The Department of Mathematics and Systems Analysis organizes regular colloquia on topics in mathematics and systems analysis for a non-specialist audience. Informal discussion continues after the colloquium in the common room.

Spring semester 2020

  • February 25, 15:15-16:15, hall U5: Prof. Tuomas Hytönen (University of Helsinki) “Commutators and Jacobians”

    The commutator of two objects A and B is the expression AB-BA, a measure of the extent to which A and B fail to commute. Probably the most famous instance arises from the uncertainty principle in Quantum Mechanics, when A and B are the position and momentum operators, or (in a common representation of these operators) multiplication by x and differentiation in x, respectively. The commutators featuring in this talk are distant cousins of the Heisenberg commutator: again, one of the operators is multiplication by a function, while the other one is an (singular) integral operator. Among other things, such commutators have am interesting connection to a distinguished nonlinear partial differential equation, the prescribed Jacobian problem.

  •  January 28, 15:15-16:15, hall U5: Prof. Volker Mehrmann (TU Berlin) "Stability analysis of energy based dynamical system models"

    Dissipative port-Hamiltonian systems are an important class of models that arise in all areas of science and engineering, whenever one uses energy as the major modeling concept. Despite the fact that the model class looks very unstructured at first sight, it has remarkable algebraic and geometric properties. Systems can be coupled in a network fashion in a structure preserving way, Galerkin projection preserves the stucture.  We will illustrate these and further system properties, showing for examples that stability and passivity are automatic. In the linear case Jordan structures for purely imaginary eigenvalues, eigenvalues at infinity, and even singular blocks in the Kronecker canonical form are very restricted and furthermore the structure leads to fast and efficient iterative solution methods for the  associated linear systems. 

    Motivated from an industrial application of studying brake squeal, we study questions like the spectral properties or distance to instability/stability  for this system class. We use this large scale industrial finite element model to illustrate our theoretical findings with numerical computation results.

Fall semester 2019

  • November 26, 15:15-16:15, hall U6: Prof. Kaisa Nyberg (Aalto University) "Cryptographic nonlinearity criteria"

    Abstract: Block ciphers are arguably the most important cryptographic primitives, since they can be used as building blocks of cryptographic protocols that provide various kind of information security services, such as confidentiality, integrity and authentication. To be secure a block cipher must be resistant against statistical attacks that exploit probabilistic properties that allow distinguishing the cipher from a random permutation. Since the publication of the first statistical cryptanalysis methods in early 1990s which exploited differential propagation and linear relations, a number of new more complex properties, e.g. boomerangs and differential-linear relations have been discovered that can be exploited efficiently to distinguish the cipher from random and also to recover part of the secret key. It is quite well understood how to construct differential and linear distinguishers of iterated block ciphers based on the differential and linear properties of the round function captured by DDT tables and LAT tables. These tools have been also used to construct boomerang and differential-linear distinguishers. Recently, Cid et al. (2018) discovered a more efficient method by identifying the boomerang property of the round function. This involves a tool called Boolean Connectivity Table (BCT). Subsequently, Bar-On et al. (2019) found a similar improvement for differential-linear relations using a tool named Differential-Linear Connectivity Table (DLCT). In this talk a brief survey of these tools and relationships between them will be given. Also examples of commonly used building blocks of round functions and properties of their DDT, LAT, BCT and DLCT tables will be discussed.  
  • October 29, 15:15-16:15, hall U6: Prof. Elina Robeva (University of British Columbia) "Maximum Likelihood Estimation of Totally Positive Densities"

    Abstract: Nonparametric density estimation is a challenging problem in theoretical statistics -- in general a maximum likelihood estimate (MLE) does not even exist! Introducing shape constraints allows a path forward. In this talk I will discuss non-parametric density estimation under total positivity (i.e. log-supermodularity) and log-concavity. I will first show that though they possess very special structure, totally positive random variables are quite common in real world data and possess appealing mathematical properties. Given i.i.d. samples from a totally positive distribution, we prove that the maximum likelihood estimator exists with probability one assuming there are at least 3 samples. We characterize the domain of the MLE and show that it is in general larger than the convex hull of the observations. If the observations are 2-dimensional or binary, we show that the logarithm of the MLE is a tent function (i.e. a piecewise linear function) with "poles" at the observations, and we show that a certain convex program can find it. Instead of using a maximum likelihood estimator, we discuss the possibility of using kernel density estimation. This new estimator raises an abundance of theoretical questions. 

  • October 15 (postponed from September 24), 15:15-16:15, hall M1: Prof. Fabricio Oliveira (Aalto University) "Optimisation under uncertainty for real-world production systems: theoretical aspects and practical challenges"

    In this talk, we introduce the framework of optimisation under uncertainty, which consists of collection of disciplines such as stochastic programming, robust optimisation, scenario generation, decomposition methods, and others related. When properly combined, these allow the development of mathematical programming-based decision support tools that can meaningfully consider the inherent uncertainty associated with input data. We illustrate the capabilities of such framework by means of examples derived from real-world problems in which the combination of two or more of these disciplines allowed the development of enhanced models which, ultimately, led to more efficient decision support tools. We will also discuss some of the technical details behind the development of these applications and present future perspectives in terms of research development.

Spring semester 2019

  • May 15th, 15:15 -16:15, hall M1: Professor Aldo Conca (University of Genova, Italy) "Introduction to Gröbner bases."

    Gröbner bases and related algorithms can be seen as generalizations of  Gaussian elimination for linear systems and  Euclid's algorithm for computing polynomial greatest common divisors of univariate polynomials.  They can be used to solve algorithmically questions related to polynomials as, for example, the following:

    1) deciding whether a system of polynomial equations has solutions, 
    2) deciding whether a polynomial can be written as linear combinations with polynomials coefficients of given polynomials,
    3) deciding whether a polynomial can be written as polynomial function of given polynomials,
    4) find the implicit equations of a locus given by a polynomial parametrization.

    Questions of this type and their variations have several applications in  mathematics, science and engineering. The goal of the talk is to present a gentle introduction to Gröbner bases and related algorithms and their use to answer the questions above.

  • April 23rd, 15:15 -16:15, hall M1: Professor Vitaly Skachek (Tartu University)  "Constructing Asynchronous Batch Codes using Hypergraphs."

    Abstract: Batch codes were first proposed by Ishai et al. as a means for balancing load in distributed storage systems. They are also of potential use in private information retrieval. In this talk, we present a new variant of batch codes termed "asynchronous batch codes", which are designed for parallel recovery of information symbols from the coded database, where different requests take different service time (i.e. the requests are served in an asynchronous manner).

    It turns out that the graph-based batch codes studied by Rawat et al. are asynchronous. Bybuilding on the ideas therein, we show that hypergraphs of Berge girth at least 4 yield graph-basedasynchronous batch codes. We prove the hypergraph-theoretic proposition that the maximumnumber of hyperedges in a hypergraph of a fixed Berge girth equals the quantity in a certaingeneralization of the hypergraph-theoretic (6,3)-problem. We then apply the constructions andbounds by Erdos, Frankl and Rodl to obtain batch code constructions and bounds on the optimalredundancy of the graph-based asynchronous batch codes.

    We show that the optimal redundancy $\rho(k)$ of graph-based asynchronous batch codes of dimension $k$ with the query size $t=3$ is $2\sqrt{k}$. Moreover, for a general fixed value of $t \ge 4$, $\rho(k) = O\left({k}^{1/(2-\epsilon)}\right)$ for any small $\epsilon > 0$. For a general value of $t \ge 4$, $\lim_{k \rightarrow \infty} \rho(k)/\sqrt{k} = \infty$. 

    (Joint work with Ago-Erik Riet and Eldho K. Thomas)

  • March 26th, 16:15-17:15, hall M1: Prof. Paavo Pylkkänen (University of Helsinki): "Explaining consciousness in terms of information."

    onscious experience has become the focus of intensive interdisciplinary study over the past few decades. By now, there are a number of candidate theories of consciousness, many of which make use of notions of information in their attempted explanations of consciousness. This talk provides first a brief overview of consciousness studies and then considers how consciousness can be understood in the light of two approaches that make an appeal to information: David Bohm’s “active information” approach, and Giulio Tononi’s mathematical “integrated information” theory of consciousness.

    References: Bohm, D. and Hiley, B. J. (1987) An Ontological Basis for Quantum Theory: I. Non-relativistic Particle Systems. Phys. Rep. 144 (6): 323-348.

    Hiley, B.J. and Pylkkänen, P. (2005) Can mind affect matter via active information? Mind and Matter, 3, 2, 7-26., M., L. Albantakis and G. Tononi (2014). "From the phenomenology to the mechanisms of consciousness: integrated information theory 3.0." PLoS Comput Biol 10 (5): e1003588.
    Pylkkänen, P. (2016), Can Bohmian quantum information help us to understand consciousness?, in Atmanspacher, H., Filk, T. and Pothos, E. eds., Quantum Interaction 2015: 9th International Conference, QI 2015, selected papers. Heidelberg: Springer, forthcoming., G. and Koch, C (2014) Consciousness: Here, There but not Everywhere. arXiv:1405.7089v1 [q-bio.NC]

  • February 26th, 15:15-16:15, hall M1: Prof. Marcus Greferath (Aalto University): "Spectral Methods for Coding Theory in a non-commutative Setup."

    Abstract: In many areas of Electrical Engineering, Computer Science, and Applicable mathematics, and particularly in Algebraic Coding Theory, there are impressive examples, showing how a discrete Fourier calculus can be used in order to construct codes of prescribed minimum distance. Here, this spectral technique is basically restricted to cyclic codes over finite fields so far, however there are no strict reasons to keep it restrained to this case. This talk is particularly interested in the scenario, where a non-commutative finite group acts on the co-ordinate domain, while the alphabet of the desired code may be a finite ring. We will sketch the successful development of a Fourier Calculus for this setting and observe a few remarkable facts. This is work in progress!

  • January 29th, 15:15 - 16:15, hall A2: Prof. Jaakko Lehtinen (Aalto University): "Simulation + machine learning = interpretable, less data hungry AI?"

    Abstract: How do we make computers perceive the everyday world and deeply understand it just by looking at it? How do we build virtual agents and real robots that build on this perception and are able to move and interact with the world, including us humans, in a natural manner? In this talk, I’ll talk about the currently accelerating congruence of physically-based simulation and machine learning in solving very hard problems in artificial intelligence.  I’ll argue that the classic approach of “merely” learning from human-labeled examples is doomed – there is simply no way for us to cover all the variability in the real world with annotated examples – and that making use of interpretable models (simulators!) in the learning process is the way forward. I’ll give examples of my own work, as well as that of my close colleagues and collaborators, and other highlights from around the world.

Fall semester 2018

  • November 27th, 15-16, hall E: Prof. Tuomo Kuusi (University of Helsinki): "Quantitative Stochastic Homogenization and Large-Scale Regularity":

    Abstract: One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In our recent book, jointly with S. Armstrong and J.-C. Mourrat, we have addressed this problem from a new perspective. Essentially, we use recently developed regularity theory for stochastic homogenization to accelerate the weak convergence of the energy density, flux and gradient of the solutions as we pass to larger and larger length scales, until it saturates at the CLT scaling. I will discuss our approach and give, at the same time, an informal introduction to our book..

  • October 30th, 15-16, hall E: Prof. Chris Brzuska (Aalto University): "Proof Theory for Cryptography":

    Most of our cryptography is not perfectly unbreakable. Given enough time, one could, theoretically, perform an exhaustive search over the key space and e.g., decrypt messages intended for another receiver. Modern cryptography, thus, relies on computationally hard problems that (are conjectured to) require an exorbitant amount of computation to solve. Complex systems such as TLS, the backbone of secure communication on the internet, rely on quite a number of such hard problems, and the protocol itself has a specification of over 100 pages. The relation between the protocol security (in a model) and the underlying assumptions needs to be established via a rigorous reduction proof. Due to the complexity of the protocols, the reduction proofs for modern protocols escape what a human can grasp. Therefore, in recent years, the proofs have been partially delegated to computers which, to be fair, also struggle with the tremendous complexity. We propose a new level of abstraction that allows to recover human understanding of security reductions for complex protocols and show how to apply it to the new TLS 1.3 standard (ongoing work).

    Joint work with Ben Dowling, Antoine Délignat-Lavaud, Cédric Fournet, Konrad Kohbrok & Markulf Kohlweiss

  • September 25th, 15-16, hall EProf. Matthieu Jonckheere (University of Buenos Aires): "Distance learning using Euclidean percolation: Following Fermat's principle":

    Abstract: In unsupervised statistical learning tasks such as clustering, recommendation, or dimension reduction, a notion of distance or similarity between points is crucial but usually not directly available as an input. We discuss recent techniques to infer a metric from observed data. Then we propose a new density-based estimator for weighted geodesic distances that takes into account the underlying density of the data, and that is suitable for nonuniform data lying on a manifold of lower dimension than the ambient space. The consistency of the estimator is proven using tools from first passage percolation. After discussing its properties and implementation, we evaluate its performance for clustering tasks.

Spring semester 2018

A highly desirable property for a mathematical proof is that its correctness is easier to verify than it is to prepare the proof from scratch. One possibility to quantify such "ease of verification" is to view the tasks of preparing and verifying a proof from a computational perspective and in terms of the computational resources employed for a task. Indeed, such proof-system-based characterizations are in many ways fundamental to our current understanding of computational complexity and complexity classes such as P, NP, and beyond. This talk explores classical and recent work on proof systems for computational problems, including some of our own recent work involving proof systems that tolerate adversarial errors during proof preparation.

We apply a recent  statistical algorithm, originally developed for parameter estimation of chaotic dynamical systems, to identify model parameters of reaction-diffusion systems by ensembles of Turing patterns created by unknown random initial values. The method is tested using the Fitzhugh-Nagumo model, a classical model of excitable media. It is shown that the approach is able to  detect small but systematic structural changes of patterns, practically impossible to distinguish by naked eye.

  • April 24th, 15-16, hall D : Prof. Clément Hongler
     (École polytechnique fédérale de Lausanne) 
    : "Statistical Field Theory and the Ising Model" 

The developments of statistical mechanics and of quantum field theory are among the major achievements of 20th century's science. In the second half of the century, these two subjects started to converge, resulting in some of the most remarkable successes of mathematical physics. At the heart of this convergence lies the conjecture that critical lattice models are connected, in the continuous limit, to conformally symmetric field theories. This conjecture has led to much insight into the nature of phase transitions and to beautiful formulae describing lattice models, which have remained unproven for decades.

In this talk, I will focus on the planar Ising model, perhaps the most studied lattice model, whose investigation has initiated much of the research in statistical mechanics. I will explain how, in the last ten years, we have developed tools to understand mathematically the emerging conformal symmetry of the model, and the connections with quantum field theory. This has led one to the proof of celebrated conjectures for the Ising correlations and for the description of the emerging random geometry. I will then explain how these tools have then yielded a rigorous formulation of the field theory describing this model, allowing one to make mathematical sense of the seminal ideas at the root of the subject of conformal field theory.

Titles and abstracts of past colloquia

Fall semester 2017

  • September 26th, 15-16, hall M1 : Prof. Daniele Boffi ( Università di Pavia, Aalto University ) : "Finite element approximation of resonant modes for the Maxwell cavity problem"
  • October 31st, 15-16, hall U1 : Prof. Lothar Nannen (TU Wien) : "Numerical methods for resonance problems in open systems"
  • November 28th, hall U1 : Prof. Kari Astala (Aalto University) : "Random tilings, variational problems and the Beltrami equation"

Spring semester 2017

  • January 31st, 15-16, hall U1 : Jarkko Kari (University of Turku) : "An Algebraic Geometric Approach to Multidimensional Symbolic Dynamics"
  • February 28th, 15-16, hall M1 : Eero Saksman (University of Helsinki) : "The Riemann zeta function meets Gaussian  multiplicative chaos"
  • March 28th, 15-16, hall M1 : Christian Haase (Freie Universität Berlin) : "Finiteness Theorems for Lattice Polytopes"
  • April 25th, 15-16, hall M1 : Thomas Britz (UNSW Sydney) : "A Nice Proof of Wei's Duality Theorem"
  • May 2nd, 15-16, hall M1 : David Rios Insua (ICMAT-CSIC and Royal Academy of Sciences, Spain) : "Adversarial Risk Analysis: Concepts, Applications and Challenges"

Fall semester 2016

Spring semester 2016

  • January 26th, 15-16, hall M1:Prof. Davy Paindavei (Université Libre de Bruxelles) : Inference on the mode of weak directional signals: a Le Cam perspective on hypothesis testing near singularities
  • February 23rd, 15-16, hall M1: Prof. Jeffery M. Keisler (Aalto University, University of Massachusetts Boston) : A decision analytic modification to deal with uncertain targets in project management
  • March 31th, 15-16, hall M1 : Prof. René Scoof (Università di Roma “Tor Vergata”) : Lagrange's theorem for finite algebraic groups
  • April 26th, 15-16, hall M1 : Prof. Giuseppe Mingione (Università di Parma) : Some regularity problems in the calculus of variations

Fall semester 2015

  • September 29, 15-16, hall M1: Prof. Raimo P. Hämäläinen (Aalto University) : Behavioural operational research. 
  • October 27, 15-16, hall U1: Prof. David Radnell (Aalto University) : Some new developments in quasiconformal Teichmueller theory.
  • November 24th, 15-16, hall U1:  Ph.D Jukka Keränen (Aalto Univerisity) : Group Representations in Number Theory: An Introduction to the Langlands Program

Spring semester 2015

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