How much do you know actually about the research that is going on across the department? The SIAM Student Chapter brings you a 3 minute thesis competition challenging a group of DPhil students to go head to head to explain their research in just 3 minutes with the aid of a single slide. This is the perfect opportunity to hear about a wide range of topics within applied mathematics, and to gain insight into the impact that mathematical research can have. The winner will be decided by a judging panel comprising Professors Helen Byrne, Jon Chapman, Patrick Farrell, and Christina Goldschmidt.
 

Cristina Palmer-Anghel: Quantum invariants via topological intersection pairings
The world of quantum invariants for knots started in 1984 with the discovery of a strong link invariant, namely the Jones polynomial. Then, Reshetikhin and Turaev developed a conceptual algebraic method that, starting with any quantum group, produces invariants for knots. The question that we have in mind is to find topological models for certain types of quantum invariants. On the topological side, in 2000, Bigelow, building on earlier work of Lawrence,
interpreted the original Jones polynomial in a homological manner- as a graded intersection pairing in a covering of a configuration space of the punctured disc. On the quantum side of the story, the coloured Jones polynomials are a sequence of quantum invariants constructed through the Reshetikhin-Turaev recipe from the quantum group Uq(sl(2)). The first invariant of this sequence is the original Jones polynomial. In this talk we will present how one can use topological intersection pairings in order to describe a topological model for all coloured Jones polynomials.

Francis Woodhouse: Autonomous mechanisms inspired by biology

Unlike the air around us, biological systems are not in equilibrium: cells consume chemical energy to keep growing and moving, forming a clear arrow of time. The recent creation of artificial versions of these ‘active’ materials suggests that these concepts can be harnessed to power new soft robotic systems fuelled by as simple a source as oxygen. After an introduction to the physics of natural and artificial active systems, we will see how endowing a mechanical network with activity can create an intricate self-actuating mechanism.

In this session we discuss various different routes for promoting your research through a panel discussion with Dawn Gordon (Project Manager, Oxford University Innovation), Dyrol Lumbard (External Relations Manager, Mathematical Institute), James Maynard (Academic Faculty, Mathematical Institute) and Ian Griffiths, and chaired by Frances Kirwan. The panel discussion will include the topics of outreach, impact, and strategies for promoting aspects of mathematics that are less amenable to public engagement. 

The goal of the talk will be to introduce a class of functions that answer a question in topology, can be computed via analytic methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan. If time permits, we will discuss connections with vertex algebras and physics of BPS states which play an important role, but will be hidden "under the hood" in much of the talk.

This work deals with exploiting the low-dimensional hierarchical structure of speech signals towards the  goal  of  improving  acoustic  modelling using deep neural networks (DNN).  To this aim the work employ tools from sparsity aware signal processing under novel frameworks to enrich  the  acoustic  information  present  in  DNN posterior features. 

Graphs are a central object of study in various scientific fields, such as discrete mathematics, theoretical computer science and network science. These graphs are typically studied using combinatorial, algebraic or probabilistic methods, each of which highlights the properties of graphs in a unique way. I will discuss a novel approach to study graphs: the simplex geometry (a simplex is a generalized triangle). This perspective, proposed by Miroslav Fiedler, introduces techniques from (simplex) geometry into the field of graph theory and conversely, via an exact correspondence. We introduce the graph-simplex correspondence, identify a number of basic connections between graph characteristics and simplex properties, and suggest some applications as example.

Reference: https://arxiv.org/abs/1807.06475
 

Pseudorandom functions (PRFs) are one of the fundamental building blocks in cryptography. Traditionally, there have been two main approaches for PRF design: the ``practitioner's approach'' of building concretely-efficient constructions based on known heuristics and prior experience, and the ``theoretician's approach'' of proposing constructions and reducing their security to a previously-studied hardness assumption. While both approaches have their merits, the resulting PRF candidates vary greatly in terms of concrete efficiency and design complexity. In this work, we depart from these traditional approaches by exploring a new space of plausible PRF candidates. Our guiding principle is to maximize simplicity while optimizing complexity measures that are relevant to cryptographic applications. Our primary focus is on weak PRFs computable by very simple circuits (depth-2 ACC^0 circuits). Concretely, our main weak PRF candidate is a ``piecewise-linear'' function that first applies a secret mod-2 linear mapping to the input, and then a public mod-3 linear mapping to the result. We also put forward a similar depth-3 strong PRF candidate.  
The advantage of our approach is twofold. On the theoretical side, the simplicity of our candidates enables us to draw many natural connections between their hardness and questions in complexity theory or learning theory (e.g., learnability of depth-2 ACC^0 circuits and width-3 branching programs, interpolation and property testing for sparse polynomials, and natural proof barriers for showing super-linear circuit lower bounds). On the applied side, the piecewise-linear structure of our candidates lends itself nicely to applications in secure multiparty computation (MPC). Using our PRF candidates, we construct protocols for distributed PRF evaluation that achieve better round complexity and/or communication complexity (often both) compared to protocols obtained by combining standard MPC protocols with PRFs like AES, LowMC, or Rasta (the latter two are specialized MPC-friendly PRFs).
Finally, we introduce a new primitive we call an encoded-input PRF, which can be viewed as an interpolation between weak PRFs and standard (strong) PRFs. As we demonstrate, an encoded-input PRF can often be used as a drop-in replacement for a strong PRF, combining the efficiency benefits of weak PRFs and the security benefits of strong PRFs. We conclude by showing that our main weak PRF candidate can plausibly be boosted to an encoded-input PRF by leveraging standard error-correcting codes.
Joint work with Dan Boneh, Yuval Ishai, Amit Sahai, and David J. Wu.

We provide lattice-based protocols allowing to prove relations among committed integers. While the most general zero-knowledge proof techniques can handle arithmetic circuits in the lattice setting, adapting them to prove statements over the integers is non-trivial, at least if we want to handle exponentially large integers while working with a polynomial-size modulus qq. For a polynomial L, we provide zero-knowledge arguments allowing a prover to convince a verifier that committed L-bit bitstrings x, y and z are the binary representations of integers X, Y and Z satisfying Z=X+Y over the integers. The complexity of our arguments is only linear in L. Using them, we construct arguments allowing to prove inequalities X <Z among committed integers, as well as arguments showing that a committed X belongs to a public interval [α,β], where α and β can be arbitrarily large. Our range arguments have logarithmic cost (i.e., linear in L) in the maximal range magnitude. Using these tools, we obtain zero-knowledge arguments showing that a committed element X does not belong to a public set S using soft-O(n⋅log|S|) bits of communication, where n is the security parameter. We finally give a protocol allowing to argue that committed L-bit integers X, Y and Z satisfy multiplicative relations Z=XY over the integers, with communication cost subquadratic in L. To this end, we use our protocol for integer addition to prove the correct recursive execution of Karatsuba's multiplication algorithm. The security of our protocols relies on standard lattice assumptions with polynomial modulus and polynomial approximation factor.

TBA

Convex and in particular semidefinite relaxations (SDP) for non-convex continuous quadratic optimisation can provide tighter bounds than traditional linear relaxations. However, using SDP relaxations directly in Branch&Cut is impeded by lack of warm starting and inefficiency when combined with other cut classes, i.e. the reformulation-linearization technique. We present a general framework based on machine learning for a strong linear outer-approximation that can retain most tightness of such SDP relaxations, in the form of few strong low dimensional linear cuts selected offline. The cut selection complexity is taken offline by using a neural network estimator (trained before installing solver software) as a selection device for the strongest cuts. Lastly, we present results of our method on QP/QCQP problem instances.