Department of Mathematics and Systems Analysis

# Colloquium

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 2019

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

Abstract:
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."

Abstract:
C
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, 2Oizumi, 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. https://philpapers.org/rec/PYLCBQTononi, 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":

Abstract:
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.

• February 27th, 15-16, hall D : Acadamy Research Fellow Kaisa Matomäki (University of Turku) :  "Correlations of arithmetic functions" (Link to abstract)

• March 27th, 15-16, hall D :  (Lappeenranta University of Technology) : "Statistical invariance in chaos and random patterns" :

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"

#### 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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