### Department of Mathematics and Systems Analysis

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* Dates within the next 7 days are marked by a star.

Vasiliki Evdoridou (The Open University, UK)**Singularities of inner functions and entire maps of finite order***** ** Wednesday 17 October 2018, 12:15, M3 (M234)

Let f be a transcendental entire function of finite order and U an unbounded, invariant Fatou component of f. We can associate an inner function, g say, to the restriction of f to U. We will show that for two classes of entire functions whose set of singular values is bounded, the number of singularities of g on the unit circle is at most twice the order of f. This is joint work with N. Fagella, X. Jarque and D. Sixsmith.

Seminar on analysis and geometry

Hans Tylli (University of Helsinki)**Structural rigidity of generalised Volterra operators on Hardy spaces**

Wednesday 24 October 2018, 12:15, M3 (M234)

will describe generalised analytic Volterra operators T_g on the Hardy spaces H^p
over the unit disk D for 1 \le p < \infty, where
T_gf(z) = \int_0^z f(w)g'(w) dw,
for z in D and f in H^p. Above the analytic map g from BMOA is the symbol of T_g.
The systematic study of this class of operators was initiated by Aleman, Cima and
Siskakis around 1995. Earlier certain T_g were used by Pommerenke (1977)
and the class contains e.g. a version of the classical Cesaro averaging operator,
which is obtained with g(z) = - \log(1-z).
I will focus attention on recent work on the structural rigidity of the class of
non-compact operators T_g on H^p for p different from 2.
The main result says that if T_g defines an isomorphism from M to T_g(M)
for the infinite-dimensional closed subspace M of H^p,
then M contains a subspace linearly isomorphic to the sequence space \ell^p.
In particular, this implies that the non-compact Volterra operators T_g have a
quite restricted range of linear qualitative behaviour compared to that of
arbitrary bounded operators on H^p (for p different from 2).
This is joint work with Santeri Miihkinen (Åbo), Pekka Nieminen (Turku) and
Eero Saksman (Helsinki).

Seminar on Analysis and Geometry

Razane Tajeddine and Lukas Holzbaur**On private keyword and stream search**

Wednesday 24 October 2018, 15:15, M3 (M234)

TBA

ANTA seminar

Prof Chris Brzuska (Aalto University)**Proof Theory for Cryptography**

Tuesday 30 October 2018, 15:15, E

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

Department Colloquium

Ratan Kumar Giri**Singular Nonlocal Problem Involving Measure Data**

Wednesday 31 October 2018, 12:15, M3 (M234)

Seminar on analysis and geometry

Vito Buffa (University of Ferrara)**BV Functions in Metric Measure Spaces: Traces and Integration by Parts Formulæ**

Wednesday 07 November 2018, 12:15, M3 (M234)

We adapt the tools from the differential structure developed by N. Gigli in order to give a definition of BV functions on RCD(K,\infty) spaces via suitable vector fields and then establish an extended Gauss-Green formula on a class of "regular" domains, which features the "normal trace" of vector fields with finite divergence measure. Then, we pass to the more classical context of a doubling metric measure space supporting a Poincaré inequality, where we reformulate the theory of "rough traces" of BV functions (after V. Maz'ya) in comparison with the Lebesgue-points characterization, and discuss the conditions under which the respective notions of trace coincide. Based on a joint work with M. Miranda Jr.

Seminar on analysis and geometry

Karl Brustad (NTNU, Trondheim)**The dominative p-Laplacian and sublinear elliptic operators**

Wednesday 14 November 2018, 12:15, M3 (M234)

Seminar on analysis and geometry

Aleksis Koski (University of Jyväskylä)**TBA**

Wednesday 21 November 2018, 12:15, M3 (M234)

Seminar on Analysis and Geometry

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