Past

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.

One of the central results in rough path theory is the local Lipschitz continuity of the solution map of a controlled differential equation called Ito-Lyons map. This continuity statement was obtained by T. Lyons in a q-variation resp. 1/q-Hölder type (rough path) metrics for any regularity 1/q>0. We extend this to a new class of Besov-Nikolskii type metrics with arbitrary regularity 1/q and integrability p, which particularly covers the aforementioned results as special cases. This talk is based on a joint work with Peter K. Friz.

 

Title: Integrals and symplectic forms on infinitesimal quotients

Abstract: Lie algebroids are models for "infinitesimal actions" on manifolds: examples include Lie algebra actions, singular foliations, and Poisson brackets.  Typically, the orbit space of such an action is highly singular and non-Hausdorff (a stack), but good algebraic techniques have been developed for studying its geometry.  In particular, the orbit space has a formal tangent complex, so that it makes sense to talk about differential forms.  I will explain how this perspective sheds light on the differential geometry of shifted symplectic structures, and unifies a number of classical cohomological localization theorems.  The talk is
based mostly on joint work with Pavel Safronov.

This paper presents an adaptive version of the Hill estimator based on Lespki’s model selection method. This simple data-driven index selection method is shown to satisfy an oracle inequality and is checked to achieve the lower bound recently derived by Carpentier and Kim. In order to establish the oracle inequality, we derive non-asymptotic variance bounds and concentration inequalities for Hill estimators. These concentration inequalities are derived from Talagrand’s concentration inequality for smooth functions of independent exponentially distributed random variables combined with three tools of Extreme Value Theory: the quantile transform, Karamata’s representation of slowly varying functions, and Rényi’s characterisation for the order statistics of exponential samples. The performance of this computationally and conceptually simple method is illustrated using Monte-Carlo simulations.

http://projecteuclid.org/euclid.ejs/1450456321  (joint work with Maud Thomas)

In 3d gauge theories, monopole operators create and destroy vortices. I will explore this idea in the context of 3d N = 4 supersymmetric gauge theories and explain how it leads to an exact calculation of quantum corrections to the Coulomb branch and a finite version of the AGT correspondence. 

I will report on joint work with Michael Weiss (https://arxiv.org/pdf/1503.00498.pdf):

Let K be a subset of a smooth manifold M. In some cases, functor calculus methods lead to a homotopical formula for M \ K in terms of the spaces M \ S,  where S runs through the finite subsets of K. This is for example the case when K is a smooth compact sub manifold of co-dimension greater or equal to three.

Wondering about how to organise your DPhil? How to make the most of your supervision meetings? How to guarantee success in your studies? Look no further!

In this session we will explore the fundamentals of a successful DPhil with help from faculty members, postdocs and DPhil students.

In the first half of the session Andreas Münch, the Director of Graduate Studies, will give a brief overview of the stages of the DPhil programme in Oxford; after this Marc Lackenby will talk about his experience as a PhD student and supervisor.

The second part of the session will be a panel discussion, with panel members Lucy Hutchinson, Mark Penney, Michal Przykucki, and Thomas Woolley. Senior faculty members will be kindly asked to leave the lecture theatre to ensure that students feel comfortable about discussing their experiences with later year students and postdocs/research fellows.

At 5pm senior and junior faculty members, postdocs and students will reunite in the Common Room for Happy Hour.

About the speakers and panel members:

Andreas Münch received his PhD from the Technical University of Munich under the supervision of Karl-Heinz Hoffmann. He moved to Oxford in 2009, where he is an Associate Professor in Applied Mathematics. As the Director of Graduate Studies he deals with matters related to training and education of graduate students. 

Marc Lackenby received his PhD from Cambridge under the supervision of W. B. Raymond Lickorish. He moved to Oxford in 1999, where he has been a Professor of Mathematics since 2006. 

Lucy Hutchinson is a DPhil student in the Mathematical Biology group studying her final year.

Mark Penney is a fourth-year DPhil student in the Topology group.

Michal Przykucki received his PhD from Cambridge in 2013 under the supervision of Béla Bollobás; he is a member of the Combinatorics research group, and has been a Drapers Junior Research Fellow at St Anne's College since 2014. 

Thomas Woolley received his DPhil from Oxford in 2012 under the supervision of Ruth Baker, Eamonn Gaffney, and Philip Maini. He is a member of the Mathematical Biology Group and has been a St John’s College Junior Research Fellow in Mathematics since 2013.

Atomistic Molecular Dynamics is a well established biomolecular modelling tool that uses the wealth of information available in the Protein Data Bank (PDB). However, biophysical techniques that provide structural information at the mesoscale, such as cryo-electron microscopy and 3D tomography, are now sufficiently mature that they merit their own online repository called the EMDataBank (EMDB). We have developed a continuum mechanics description of proteins which uses this new experimental data as input to the simulations, and which we are developing into a software tool for use by the biomolecular science community. The model is a Finite Element algorithm which we have generalised to include the thermal fluctuations that drive protein conformational changes, and which is therefore known as Fluctuating Finite Element Analysis (FFEA) [1].

We will explain the physical principles underlying FFEA and provide a practical overview of how a typical FFEA simulation is set up and executed. We will then demonstrate how FFEA can be used to model flexible biomolecular complexes from EM and other structural data using our simulations of the molecular motors and protein self-assembly as illustrative examples. We then speculate how FFEA might be integrated with atomistic models to provide a multi-scale description of biomolecular structure and dynamics.

1. Oliver R., Read D. J., Harlen O. G. & Harris S. A. “A Stochastic finite element model for the dynamics of globular macromolecules”, (2013) J. Comp. Phys. 239, 147-165.

Firstly I will outline Dirac cohomology for graded Hecke algebras and the branching rules for the projective representations of $S_n$. Combining these notions with the Jucys-Murphy elements for $\tilde{S}_n$, that is the double cover of the symmetric group, I will go through a method to completely describe the spectrum data for the Jucys-Murphy elements for $\tilde{S}_n$. If time allows I will also explain how this spectrum data gives rise to a a concrete description for the matrices of the action of $\tilde{S}_n$.

Dstl are interested in removing liquid contaminants from capillary features (cracks in surfaces, screw threads etc.). We speculated that liquid decontaminants with low surface tension would have beneficial properties. The colloid literature, and in particular the oil recovery literature, discusss the properties of multiphase systems in terms of “Winsor types”, typically consisting of “brine” (water + electrolyte), “oil” (non-polar, water-insoluble solvent) and surfactant. Winsor I systems are oil-in-water microemulsions and Winsor II systems are water-in-oil microemulsions. Under certain circumstances, the mixture will separate into three phases. The middle (Winsor III) phase is surfactant-rich, and is reported to exhibit ultra-low surface tension. The glycol ethers (“Cellosolve” type solvents) consist of short (3-4) linked ether groups attached to short (3-4 carbon) alkyl chains. Although these materials would not normally be considered to be surfactants, their polar head, non-polar tail properties allow them to form a “surfactantless” Winsor III middle phase. We have found that small changes in temperature, electrolyte concentration or addition of contaminant can cause these novel colloids to phase separate. In our decontamination experiments, we have observed that contaminant-induced phase separation takes the form of droplets of the separating phase. These droplets are highly mobile, exhibiting behaviour that is visually similar to Brownian motion, which induces somewhat turbulent liquid currents in the vicinity of the contaminant. We tentatively attribute this behaviour to the Marangoni effect. We present our work as an interesting physics/ physical chemistry phenomenon that should be suitable for mathematical analysis.

I will describe joint work with Katrin Tent, in which we consider a profinite group equipped with a uniformly definable family of open subgroups. We show that if the family is `full’ (i.e. includes all open subgroups) then the group has NIP theory if and only if it has NTP_2 theory, if and only if it has an (open) normal subgroup of finite index which is a direct product of finitely many compact p-adic analytic groups (for distinct primes p). Without the `fullness’ assumption, if the group has NIP theory then it  has a prosoluble open normal subgroup of finite index.