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Mathematics & Statistics

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Riemann zeta and multiple zeta values

Cezar Lupu

University of Pittsburgh

3:00pm-4:00pm

CMC 130

Razvan Teodorescu

**Abstract**

In this talk, we bring into perspective the infamous Riemann zeta function and its natural generalization, the multiple zeta function. We focus more on the evaluations of such objects at positive integers. The techniques used in these evaluations rely on the properties of some special functions.

Although they look rather simple, it turns out that the Riemann zeta and multiple zeta values play a very important role at the interface of analysis, number theory, geometry and physics with applications ranging from periods of mixed Tate motives to evaluating Feynman integrals in quantum field theory.

The talk should be accessible to non specialists and graduate students.

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Refinement-Based Modeling for Adaptive Networks

Luigia Petre

Åbo Akademi University

Åbo, Finland

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

In this talk we explore computational methodologies for developing adaptive networks with a correct-by-construction methodology, based on abstraction, refinement, and proofs.

When constructing a system by refinement, we start with a very simple model for it, which only encompasses the systemâ€™s basic functionality and satisfies its elementary properties. Given this abstract model of the system, proving that it satisfies the elementary properties is often feasible. Then, we stepwise add the missing details to the model in such a way that the basic functionality and elementary properties continue to be satisfied. In addition, more complex functionality and more sophisticated properties can be proven to hold. The key issue in refinement is that, to prove that a model M1 refines a model M0, some logical conditions (called proof obligations) need to be shown true. The final model is the system itself, and is a correct implementation of its requirements.

The formal method we use is called Event-B, and Event-B models are essentially state transition systems. The state is modelled by variables and the transitions by discrete events. Properties to prove about the model are usually described as invariants, to hold before and after the execution of each event.

We demonstrate this approach through two examples on wireless sensor-actor networks and on a simplified smart electrical grid.

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Network controllability: theory and applications

Ion Petre

Åbo Akademi University and Turku Centre for Computer Science

Åbo, Finland

3:00pm-4:00pm

CMC 130

Nataša Jonoska

**Abstract**

The intrinsic robustness of living systems against perturbations is a key factor that explains why many single-target drugs have been found to provide poor efficacy or to lead to significant side effects. Rather than trying to design selective ligands that target individual receptors only, network polypharmacology aims to modify multiple cellular targets to tackle the compensatory mechanisms and robustness of disease-associated cellular systems, as well as to control unwanted off-target side effects that often limit the clinical utility of many conventional drug treatments. However, the exponentially increasing number of potential drug target combinations makes the pure experimental approach quickly unfeasible, and translates into a need for algorithmic design principles to determine the most promising target combinations to effectively control complex disease systems, without causing drastic toxicity or other side-effects. Building on the increased availability of disease-specific essential genes, we concentrate on the target structural controllability problem, where the aim is to select a minimal set of driver/driven nodes which can control a given target within a network. That is, for every initial configuration of the system and any desired final configuration of the target nodes, there exists a finite sequence of input functions for the driver nodes such that the target nodes can be driven to the desired final configuration. We investigated this approach in some pilot studies linking FDA-approved drugs with cancer cell-line-specific essential genes, with some very promising results.

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Towards Spectral Sparsification of Simplicial Complexes based on Generalized Effective Resistance

Bei Wang

Scientific Computing and Imaging Institute

University of Utah

3:00pm-4:00pm

CMC 130

Masahiko Saito

**Abstract**

As a generalization of the use of graphs to describe pairwise interactions, simplicial complexes can be used to model higher-order interactions between three or more objects in complex systems. There has been a recent surge in activity for the development of data analysis methods applicable to simplicial complexes, including techniques based on computational topology, higher-order random processes, generalized Cheeger inequalities, isoperimetric inequalities, and spectral methods. In particular, spectral learning methods (e.g. label propagation and clustering) that directly operate on simplicial complexes represent a new direction emerging from the confluence of computational topology and machine learning. Similar to the challenges faced by massive graphs, computational methods that operate on simplicial complexes are severely limited by computational costs associated with massive datasets.

To apply spectral methods in learning to massive datasets modeled as simplicial complexes, we work towards the sparsification of simplicial complexes based on preserving the spectrum of the associated Laplacian operators. We show that the theory of Spielman and Srivastava for the sparsification of graphs extends to the generality of simplicial complexes via the up Laplacian. In particular, we introduce a generalized effective resistance for simplexes; provide an algorithm for sparsifying simplicial complexes at a fixed dimension; and gives a specific version of the generalized Cheeger inequalities for weighted simplicial complexes under the sparsified setting. In addition, we demonstrate via experiments the preservation of up Laplacian during sparsification, as well as the utility of sparsification with respect to spectral clustering.

This is a joint work with Braxton Osting and Sourabh Palande.

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On exceptional sets of some operator sequences

Grigori Karagulyan

Institute of Mathematics

National Academy of Sciences of Armenia

3:00pm-4:00pm

CMC 130

Arthur Danielyan

**Abstract**

We will consider general theorems characterizing the divergence sets of the operator sequences satisfying the localization property. Applying those theorems we deduce complete characterizations of the divergence sets of Fourier series and their Cesaro means in classical orthonormal systems.

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Weak Commutativity in Groups

Said N. Sidki

Universidade de Brasília

3:00pm-4:00pm

CMC 130

Dima Savchuk

**Abstract**

The topic of this lecture is a group functor which we introduced in 1980. Precisely, given a group \(G\) and an isomorphic copy of it \(G^\psi\) where \(\psi\) is the isomorphism, we defined $$ \chi(G)=\langle G,G^\psi\mid gg^\psi=g^\psi g\text{ for all }g\in G\rangle. $$ It was shown that the group \(\chi(G)\) has a section isomorphic to the second homology group \(H_2(G,ℤ)\). The functor \(\chi\) preserves a number of classes of groups such as: finite \(\pi\)-groups (\(\pi\) a set of primes); finite nilpotent groups; solvable groups; nilpotent groups; polycyclic-by-finite groups. Recently, Bridson and Kochloukova added to this list the important class of finitely presented groups.