Past

In this talk we will start by introducing the notion of Siegel-Jacobi modular form and explain its close relation to Siegel modular forms through the Fourier-Jacobi expansion. Then we will discuss how one can attach an L-function to an appropriate (i.e. eigenform) Siegel-Jacobi modular form due to Shintani, and report on joint work with Jolanta Marzec on analytic properties of this L-function, extending results of Arakawa and Murase. 

We propose a randomised version of the Heston model--a widely used stochastic volatility model in mathematical finance--assuming that the starting point of the variance process is a random variable. In such a system, we study the small- and large-time behaviours of the implied volatility, and show that the proposed randomisation generates a short-maturity smile much steeper (`with explosion') than in the standard Heston model, thereby palliating the deficiency of classical stochastic volatility models in short time. We precisely quantify the speed of explosion of the smile for short maturities in terms of the right tail of the initial distribution, and in particular show that an explosion rate of $t^\gamma$ (gamma in [0,1/2]) for the squared implied volatility--as observed on market data--can be obtained by a suitable choice of randomisation. The proofs are based on large deviations techniques and the theory of regular variations. Joint work with Fangwei Shi (Imperial College London)

In this talk, I am reflecting on the last 8 extremely enjoyable years I spent in the department (DPhil, OCIAM, 2008-2012, post-doc, WCMB, 2012-2016). My story is a little unusual: coming from an Engineering undergraduate background, spending 8 years in the Maths department, and now moving to a faculty position at the Medical School. However, I think it highlights well the enormous breadth and applicability of mathematics beyond traditional disciplinary boundaries. I will discuss different projects during my time in Oxford, focusing on time-series, signal processing, and statistical machine learning methods, with diverse applications in real-world problems.

I will present a broad family of stochastic algorithms for inverting a matrix, including specialized variants which maintain symmetry or positive definiteness of the iterates. All methods in the family converge globally and linearly, with explicit rates. In special cases, the methods obtained are stochastic block variants of several quasi-Newton updates, including bad Broyden (BB), good Broyden (GB), Powell-symmetric-Broyden (PSB), Davidon-Fletcher-Powell (DFP) and Broyden-Fletcher-Goldfarb-Shanno (BFGS). After a pause for questions, I will then present a block stochastic BFGS method based on the stochastic method for inverting positive definite matrices. In this method, the estimate of the inverse Hessian matrix that is maintained by it, is updated at each iteration using a sketch of the Hessian, i.e., a randomly generated compressed form of the Hessian. I will propose several sketching strategies, present a new quasi-Newton method that uses stochastic block BFGS updates combined with the variance reduction approach SVRG to compute batch stochastic gradients, and prove linear convergence of the resulting method. Numerical tests on large-scale logistic regression problems reveal that our method is more robust and substantially outperforms current state-of-the-art methods.

This talk will introduce the notion of quasi-convex subgroups. As an application, we will prove that the intersection of two finitely generated subgroups of a free group is again finitely generated.
 

Isogenies are algebraic group morphisms of elliptic curves. Let E, E' be two (ordinary) elliptic curves defined over a finite field of characteristic p, and suppose that there exists an isogeny ψ between E and E'. The explicit isogeny problem asks to compute a rational function expression for ψ. Various specializations of this problem appear naturally in point counting and elliptic curve cryptography. There exist essentially two families of algorithms to compute isogenies. Algorithms based on Weierstraß' differential equation are very fast and well suited in the point count setting, but are clumsier in general. Algorithms based on interpolation work more generally, but have exponential complexity in log(p) (the characteristic of the finite field). We propose a new interpolation-based algorithm that solves the explicit isogeny problem in polynomial time in all the involved parameters. Our approach is inspired by a previous algorithm of Couveignes', that performs interpolation on the p-torsion on the curves. We replace the p-torsion in Couveignes' algorithm with the ℓ-torsion for some small prime ℓ; however this adaptation requires some non-trivial work on isogeny graphs in order to yield a satisfying complexity. Joint work with Cyril Hugounenq, Jérôme Plût and Éric Schost.

Suppose that we have a finite quotient singularity $\mathbb C^n/G$ admitting a crepant resolution $Y$ (i.e. a resolution with $c_1 = 0$). The cohomological McKay correspondence says that the cohomology of $Y$ has a basis given by irreducible representations of $G$ (or conjugacy classes of $G$). Such a result was proven by Batyrev when the coefficient field $\mathbb F$ of the cohomology group is $\mathbb Q$. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology+ of $Y$ in two different ways. This proof also extends the result to all fields $\mathbb F$ whose characteristic does not divide $|G|$ and it gives us the corresponding basis of conjugacy classes in $H^*(Y)$. We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.

It follows from the ellipsoid method and results of Grotschel, Lovasz and Schrijver that one can find an optimal colouring of a perfect graph in polynomial time. But no ''combinatorial'' algorithm to do this is known.

Here we give a combinatorial algorithm to do this in an n-vertex perfect graph in time O(n^{k+1}^2) where k is the clique number; so polynomial-time for fixed k. The algorithm depends on another result, a polynomial-time algorithm to find a ''balanced skew partition'' in a perfect graph if there is one.

Joint work with Maria Chudnovsky, Aurelie Lagoutte, and Sophie Spirkl.

The Feynman diagram expansion of scattering amplitudes in perturbative superstring theory can be written (for closed strings) as a series of integrals over compactified moduli spaces of Riemann surfaces with marked points, indexed by the genus. Therefore in genus 0 it is reasonable to find, as it often happens in QFT computations, periods of M_{0,N}, which are known to be multiple zeta values. In this talk I want to report on recent advances in the genus 1 amplitude, which are related to the development of 2 different generalizations of classical multiple zeta values, namely elliptic multiple zeta values and conical sums.

Configuration spaces of points in a manifold are well studied. Giving the points thickness has obvious physical meaning: the configuration space of non-overlapping particles is equivalent to the phase space, or energy landscape, of a hard spheres gas. But despite their intrinsic appeal, relatively little is known so far about the topology of such spaces. I will overview some recent work in this area, including a theorem with Yuliy Baryshnikov and Peter Bubenik that related the topology of these spaces to mechanically balanced, or jammed, configurations. I will also discuss work in progress with Robert MacPherson on hard disks in an infinite strip, where we understand the asymptotics of the Betti numbers as the number of disks tends to infinity. In the end, we see a kind of topological analogue of a liquid-gas phase transition.

We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function in a very directional way. Using an interpretation of this energy as a large deviation rate function for SLE_k as k goes to 0, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal i.e. of the same curve but viewed as going from B to A in D. In particular it measures how far does the chord differ from the hyperbolic geodesic. I will also discuss the relation between the energy of the curve with its regularity, some questions are still open. If time allows, I will present the Loewner energy for loops on the Riemann sphere, and open questions related to it as well.


 

Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small  sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal
discrepancy. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that the standard contact
5 dimensional sphere has a unique Milnor filling up to normalization. This generalizes a Theorem by Mumford.

The class of log-concave densities on $\mathbb{R}^d$ is a very natural infinite-dimensional generalisation of the class of Gaussian densities.  I will show that it also allows the statistician to have the best of both the parametric and nonparametric worlds, in that one can obtain a fully automatic density estimator in the class (via maximum likelihood), with no tuning parameters to choose.  I'll discuss its computation, methodological consequences and theoretical properties, and in particular very recent results on minimax rates of convergence and adaptation.

In this talk I will introduce methods to use 2d gauge theories as a means to describe Calabi-Yau varieties and their moduli spaces. As I review, this description furnishes a natural framework to predict derived equivalences between pairs of (sometimes even non-birational) Calabi-Yau varieties. A prominent example of this kind is realized by the Rødland non-birational pair of Calabi-Yau threefolds.
Using the 2d gauge theory description, I will propose further examples of derived equivalences among non-birational Calabi-Yau varieties.

Beilinson has given a motivic construction of the Eisenstein cohomology on modular curves. This makes it possible to define Eisenstein classes in Deligne-Beilinson, syntomic, and ´etale cohomology. These Eisenstein classes can be computed in terms of real analytic and p-adic Eisenstein series or modular units. The resulting explicit expressions allow to prove results on special values of classical and p-adic L-functions and lead to explicit reciprocity laws. Harder has more generally defined and studied the Eisenstein cohomology for Hilbert modular varieties by analytic methods. In this talk we will explain a motivic and in particular algebraic construction of Harder’s Eisenstein cohomology classes, which generalizes Beilinson’s result. This opens the way to applications, similar as for modular curves, in the case of Hilbert modular varieties.

Approximate prime numbers -- James Maynard

I will talk about the idea of an 'almost prime' number, and how this can be used to make progress on some famous problems about the primes themselves.

Mathematical biology: An early career retrospective -- Thomas Woolley

Since 2008 Thomas has focused his attention to the application of mathematical techniques to biological problems. Through numerous fruitful collaborations he has been extremely fortunate to work alongside some amazing researchers. But what has he done in the last 8 years? What lessons has he learnt? What knowledge has he produced?

This talk will encompass a brief overview of a range of applications, from animal skin patterns to cellular mechanics, via zombies and Godzilla.

The spreading of a viscous fluid in between a rigid, horizontal substrate and an overlying elastic sheet is presented as a simplified model of the hydraulic fracturing process. In particular, the talk will focus on the case of a permeable substrate for which leak-off arrests the propagation of the fluid and permits the development of a steady state. The different regimes of  gravitationally-driven and elastically-driven flow will be explored, as will the cases of a stiff and flexible sheet, before a discussion of the influence that particles included in the fluid have on the fracture propagation. 

Zhenru Wang
Title: Multi-Index Monte Carlo Estimators for a Class of Zakai SPDEs
Abstract:   
We first propose a space-time Multi-Index Monte Carlo (MIMC) estimator for a one-dimensional parabolic SPDE of Zakai type. We compare the computational cost required for a prescribed accuracy with the Multilevel Monte Carlo (MLMC) method of Giles and Reisinger (2012). Then we extend the estimator to a two-dimensional variant of SPDE. The theoretical analysis shows the benefit of using MIMC in high dimensional problems over MLMC methods. Numerical tests confirm these finding empirically.

Vadim Kaushansky
Title: An extended structural default model with jump risk
Abstact:
We consider a structural default model in an interconnected banking network as in Itkin and Lipton (2015), where there are mutual obligations between each pair of banks. We analyse the model numerically for the case of two banks with jumps in their asset value processes. Specifically, we develop a finite difference method for the resulting two-dimensional partial integro-differential equation, and study its stability and consistency. By applying this method, we compute joint and marginal survival probabilities, as well as prices of credit default swaps (CDS) and first-to-default swaps (FTD), Credit and Debt Value Adjustments (CVA and DVA).

 

One of many overlaps between logic and topology is duality: Stone duality links Boolean algebras with zero-dimensional compact Hausdorff spaces, and gives a useful topological way of describing certain phenomena in first order logic; and there are generalisations that allow one to study infinitary logics also. We will look at a couple of ways in which this duality theory is useful.'

We develop the asymptotic formulas for correlations  
\[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\]

where $f_1,\dots,f_m$ are bounded ``pretentious" multiplicative functions, under certain natural hypotheses. We then deduce several desirable consequences: first, we characterize all multiplicative functions $f:\mathbb{N}\to\{-1,+1\}$ with bounded partial sums. This answers a question of Erd{\"o}s from $1957$ in the form conjectured by Tao. Second, we show that if the average of the first divided difference of multiplicative function is zero, then either $f(n)=n^s$ for $\operatorname{Re}(s)<1$ or $|f(n)|$ is small on average. This settles an old conjecture of K\'atai. Third, we discuss applications to the study of sign patterns of $(f(n),f(n+1),f(n+2))$ and $(f(n),f(n+1),f(n+2),f(n+3))$ where $f:\mathbb{N}\to \{-1,1\}$ is a given multiplicative function. If time permits, we discuss multidimensional version of some of the results mentioned above.
 

We introduce sufficient conditions for the solution of a multi-dimensional, Markovian BSDE to have a density. We show that a system of BSDEs possesses a density if its corresponding semilinear PDE exhibits certain regularity properties, which we verify in the case of several examples.

In the first part of the talk, I will describe a fluid-dynamical model for a "anti-surfactant" solution (such as salt dissolved in water) whose surface tension is an increasing function of bulk solvent concentration. In particular, I will show that this model is consistent with the standard model for surfactants, and predicts a novel instability for anti-surfactants not present for surfactants. Some further details are given in the recent paper by Conn et al. Phys. Rev. E 93 043121 (2016).

In the second part of the talk, I will formulate and analyse the governing equations for the flow of a thixotropic or antithixotropic fluid in a slowly varying channel. These equations are equivalent to the equations of classical lubrication theory for a Newtonian fluid, but incorporate the evolving microstructure of the fluid, described in terms of a scalar structure parameter. If time permits, I will seek draw some conclusions relevant to thixotropic flow in porous media. Some further details are given in the forthcoming paper by Pritchard et al. to appear in J Non-Newt. Fluid Mech (2016).

Most current methods of Magnetic Resonance Imaging (MRI) reconstruction interpret raw signal values as samples of the Fourier transform of the object. Although this is computationally convenient, it neglects relaxation and off–resonance evolution in phase, both of which can occur to significant extent during a typical MRI signal. A more accurate model, known as Parameter Assessment by Recovery from Signal Encoding (PARSE), takes the time evolution of the signal into consideration. This model uses three parameters that depend on tissue properties: transverse magnetization, signal decay rate, and frequency offset from resonance. Two difficulties in recovering an image using this model are the low SNR for long acquisition times in single-shot MRI, and the nonlinear dependence of the signal on the decay rate and frequency offset. In this talk, we address the latter issue by using a second order approximation of the original PARSE model. The linearized model can be solved using convex optimization augmented with well-stablished regularization techniques such as total variation. The sensitivity of the parameters to noise and computational challenges associated with this approximation will be discussed.

In this talk, we will explain how the counting theorems of Pila and Wilkie lead to a conditional proof of the aforementioned conjecture. In particular, we will explain how to generalise the work of Habegger and Pila on a product of modular curves. 
Habegger and Pila were able to prove that the Zilber-Pink conjecture holds in such a product if the so-called weak complex Ax and large Galois orbits conjectures are true. In fact, around the same time, Pila and Tsimerman proved a stronger statement than the weak complex Ax conjecture, namely, the Ax-Schanuel conjecture for the $j$-function. We will formulate Ax-Schanuel and large Galois orbits conjectures for general Shimura varieties and attempt to imitate the Habegger-Pila strategy. However, we will encounter an additional difficulty in bounding the height of a pre-special subvariety.

This is joint work with Jinbo Ren.
 

Commitment schemes are a fundamental primitive in cryptography. Their security (more precisely the computational binding property) is closely tied to the notion of collision-resistance of hash functions. Classical definitions of binding and collision-resistance turn out too be weaker than expected when used in the quantum setting. We present strengthened notions (collapse-binding commitments and collapsing hash functions), explain why they are "better", and show how they be realized under standard assumptions.

I will try to give a brief description of the use of group theory and character theory in chemistry, specifically vibrational spectroscopy. Defining the group associated to a molecule, how one would construct a representation corresponding to such a molecule and the character table associated to this. Then, time permitting, I will go in to the deconstruction of the data from spectroscopy; finding such a group and hence molecule structure. 

I'll discuss recent joint work with Arend Bayer and Ziyu Zhang in which we define a nef divisor class on moduli spaces of Bridgeland-stable objects in the derived category of coherent sheaves with compact support, generalising earlier work of Bayer and Macri for smooth projective varieties. This work forms part of a programme to study the birational geometry of moduli spaces of Bridgeland-stable objects in the derived category of varieties that need not be smooth and projective.

Given vectors $V = (v_i: i \in [n]) \in R^D$, we define the $V$-intersection of $A,B \subset [n]$ to be the vector $\sum_{i \in A \cap B} v_i$. In this talk, I will discuss a new, essentially optimal, supersaturation theorem for $V$-intersections, which can be roughly stated as saying that any large family of sets contains many pairs $(A,B)$ with $V$-intersection $w$, for a wide range of $V$ and $w$. A famous theorem of Frankl and Rödl corresponds to the case $D=1$ and all $v_i=1$ of our theorem. The case $D=2$ and $v_i=(1,i)$ solves a conjecture of Kalai.

Joint work with Peter Keevash.

Consider the problem of identifying an unknown subspace S from data with erasures and with limited memory available. To estimate S, suppose we group the measurements into blocks and iteratively update our estimate of S with each new block.

In the first part of this talk, we will discuss why estimating S by computing the "running average" of span of these blocks fails in general. Based on the lessons learned, we then propose SNIPE for memory-limited PCA with incomplete data, useful also for streaming data applications. SNIPE provably converges (linearly) to the true subspace, in the absence of noise and given sufficient measurements, and shows excellent performance in simulations. This is joint work with Laura Balzano and Mike Wakin.
 

 For a group algebra over a self-injective ring
there are two stable categories: the usual one modulo projectives
and a relative one where one works modulo representations
which are free over the coefficient ring.
I'll describe the connection between these two stable categories,
which are "birational" in an appropriate sense.
I'll then make some comments on the specific case
where the coefficient ring is Z/nZ and give a more
precise description of the relative stable category.

 In the first meeting of the seminar we, and all participants who wish to do so, will each briefly introduce ourselves and our research interests. We will decide future talks and papers to read during this meeting.

The talk will review the origins
of ambitwistor strings, and  recent progress in extending them to a
wider variety of theories and loop amplitudes.

The (Danish-born) German mathematician Olaus Henrici (1840–1918) studied in Karlsruhe, Heidelberg and Berlin before making his career in London, first at University College and then, from 1884, at the newly formed Central Technical College where he established a Laboratory of Mechanics.  Although Henrici’s original training was as an engineer, he became known as a promoter of projective geometry and as an advocate for the use of mathematical models.  In my talk, I shall discuss the different aspects of Henrici's work and explore connections between them.

We find exact solutions of Maxwell's equations that are the precise analog of plane waves, but in the case that the translation group is replaced by the Abelian helical group. These waves display constructive/destructive interference with helical atomic structures, in the same way that plane waves interact with crystals. We show how the resulting far-field pattern can be used for structure determination. We test the method by doing theoretical structure determination on the Pf1 virus from the Protein Data Bank. The underlying mathematical idea is that the structure is the orbit of a group, and this group is a subgroup of the invariance group of the differential equations. Joint work with Dominik Juestel and Gero Friesecke. (Acta Crystallographica A72 and SIAM J. Appl Math).

(Joint work with Marco Golla and József Bodnár)
We will give a general overview of the plethora of concordance invariants which can be extracted from Ozsváth-Szabó-Rasmussen's knot Floer homology. 
We will then focus on the $\nu^+$ invariant and prove some of its useful properties. 
Furthermore we will show how it can be used to obstruct the existence of cobordisms between algebraic knots.