Past

abstract:

In my talk I shall try to explain the following speculation and present some
evidence in the form of "correlations" between categoricity conjectures in
model theory and motivic conjectures in algebraic geometry.

Transfinite induction constructions developed in model theory are by now
sufficiently developed to be used to build analogues of objects in algebraic
geometry constructed with a choice of topology, such as a singular cohomology theory,
the Hodge decomposition, and fundamental groups of complex algebraic varieties.
Moreover, these algebraic geometric objects are often conjectured to satisfy
homogeneity or freeness properties which are true for objects constructed by
transfinite induction.


An example of this is Hrushovski fusion used to build Zilber pseudoexponentiation,
i.e. a group homomorphism  $ex:C^+ \to C^*$ which satisfies Schanuel conjecture,
a transcendence property analogous to Grothendieck conjecture on periods.


I shall also present a precise conjecture on "uniqueness" of Q-forms (comparison isomorphisms)
of complex etale cohomology, and will try to explain its relation to conjectures on l-adic
Galois representations coming from the theory of motivic Galois group.
 

Combining work of Galkin, Christopherson-Ilten, and Coates-Corti-Galkin-Golyshev-Kasprzyk we see that all smooth Fano threefolds admit a toric degeneration. We can use this fact to uniformly construct all Fano threefolds: given a choice of a fan we classify reflexive polytopes which break into unimodular pieces along this fan. We can then construct closed torus invariant embeddings of the corresponding toric variety using a technique - Laurent inversion - developed with Coates and Kaspzryk. The corresponding binomial ideal is controlled by the chosen fan, and in low enough codimension we can explicitly test deformations of this toric ideal. We relate the constructions we obtain to known constructions. We study the simplest case of the above construction, closely related to work of Abouzaid-Auroux-Katzarkov, in arbitrary dimension and use it to produce a tropical interpretation of the mirror superpotential via broken lines. We expect the computation to be the tropical analogue of a Floer theory calculation.

Matrix completion is an area of great mathematical interest and has numerous applications, including recommender systems for e-commerce. The recommender problem can be viewed as follows: given a database where rows are users and and columns are products, with entries indicating user preferences, fill in the entries so as to be able to recommend new products based on the preferences of other users. Viewing the interactions between user and product instead as interactions between potential drug chemicals and disease-causing target proteins, the problem is that faced within the realm of drug discovery. We propose a divide and conquer algorithm inspired by the work of [1], who use recursive rank-1 approximation. We make the case for using an LP rank-1 approximation, similar to that of [2] by a showing that it guarantees a 2-approximation to the optimal, even in the case of missing data. We explore our algorithm's performance for different test cases.

[1]  Shen, B.H., Ji, S. and Ye, J., 2009, June. Mining discrete patterns via binary matrix factorization. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 757-766). ACM.

[2] Koyutürk, M. and Grama, A., 2003, August. PROXIMUS: a framework for analyzing very high dimensional discrete-attributed datasets. In Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining (pp. 147-156). ACM.

The classical theory of Erdős–Rényi random graphs is concerned primarily with random subgraphs of $K_n$ or $K_{n,n}$. Lately, there has been much interest in understanding random subgraphs of other graph families, such as regular graphs.

We study the following problem: Let $G$ be a $k$-regular bipartite graph with $2n$ vertices. Consider the random process where, beginning with $2n$ isolated vertices, $G$ is reconstructed by adding its edges one by one in a uniformly random order. An early result in the theory of random graphs states that if $G=K_{n,n}$, then with high probability a perfect matching appears at the same moment that the last isolated vertex disappears. We show that if $k = Ω(n)$, then this holds for any $k$-regular bipartite graph $G$. This improves on a result of Goel, Kapralov, and Khanna, who showed that with high probability a perfect matching appears after $O(n \log(n))$ edges have been added to the graph. On the other hand, if $k = o(n / (\log(n) \log (\log(n)))$, we construct a family of $k$-regular bipartite graphs in which isolated vertices disappear long before the appearance of perfect matchings.

Joint work with Roman Glebov and Zur Luria.
 

There is no more classical problem of numerical PDE than the Laplace equation in a polygon, but Abi Gopal and I think we are on to a big step forward. The traditional approaches would be finite elements, giving a 2D representation of the solution, or integral equations, giving a 1D representation. The new approach, inspired by an approximation theory result of Donald Newman in 1964, leads to a "0D representation" -- the solution is the real part of a rational function with poles clustered exponentially near the corners of the polygon. The speed and accuracy of this approach are remarkable. For typical polygons of up to 8 vertices, we can solve the problem in less than a second on a laptop and evaluate the result in a few microseconds per point, with 6-digit accuracy all the way up to the corner singularities. We don't think existing methods come close to such performance. Next step: Helmholtz?
 

There is a mismatch between levels of crime and its fear and often, cities might see an increase or a decrease in crime over time while the fear of crime remains unchanged. A model that considers fear of crime as an opinion shared by simulated individuals on a network will be presented, and the impact that different distributions of crime have on the fear experienced by the population will be explored. Results show that the dynamics of the fear is sensitive to the distribution of crime and that there is a phase transition for high levels of concentration of crime.

In this talk, we will present results regarding the regularity and

rigidity of solutions to the Monge-Ampere equation, inspired by the role

played by this equation in the context of prestrained elasticity. We will

show how the Nash-Kuiper convex integration can be applied here to achieve

flexibility of Holder solutions, and how other techniques from fluid

dynamics (the commutator estimate, yielding the degree formula in the

present context) find their parallels in proving the rigidity. We will indicate

possible avenues for the future related research.

Let F be a fixed infinite, vertex-transitive graph. We say a graph G is `r-locally F' if for every vertex v of G, the ball of radius r and centre v in G is isometric to the ball of radius r in F. For each positive integer n, let G_n = G_n(F,r) be a graph chosen uniformly at random from the set of all unlabelled, n-vertex graphs that are r-locally F. We investigate the properties that the random graph G_n has with high probability --- i.e., how these properties depend upon the fixed graph F. 
We show that if F is a Cayley graph of a torsion-free group of polynomial growth, then there exists a positive integer r_0 such that for every integer r at least r_0, with high probability the random graph G_n = G_n(F,r) defined above has largest component of size between n^{c_1} and n^{c_2}, where 0 < c_1 < c_2  < 1 are constants depending upon F alone, and moreover that G_n has at least exp(poly(n)) automorphisms. This contrasts sharply with the random d-regular graph G_n(d) (which corresponds to the case where F is replaced by the infinite d-regular tree).
Our proofs use a mixture of results and techniques from group theory, geometry and combinatorics, including a recent and beautiful `rigidity' result of De La Salle and Tessera.
We obtain somewhat more precise results in the case where F is L^d (the standard Cayley graph of Z^d): for example, we obtain quite precise estimates on the number of n-vertex graphs that are r-locally L^d, for r at least linear in d, using classical results of Bieberbach on crystallographic groups.
Many intriguing open problems remain: concerning groups with torsion, groups with faster than polynomial growth, and what happens for more general structures than graphs.
This is joint work with Itai Benjamini (Weizmann Institute).
 

The classical Lorentz gas model introduced by Lorentz in 1905, studied further by Sinai in the 1960s, provides a rich source of examples of chaotic dynamical systems with strong stochastic properties (despite being entirely deterministic).  Central limit theorems and convergence to Brownian motion are well understood, both with standard n^{1/2} and nonstandard (n log n)^{1/2} diffusion rates.

In joint work with Paulo Varandas, we discuss examples with diffusion rate n^{1/a}, 1<a<2, and prove convergence to an a-stable Levy process.  This includes to the best of our knowledge the first natural examples where the M_2 Skorokhod topology is the appropriate one.



 

I will consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. A PDE (free boundary problem) approach is used to address the problem in general dimensions with space-time inhomogeneous costs given by Lagrangian integrals along the paths.  An Eulerian---mass flow---formulation of the problem is introduced. Its dual is given by Hamilton-Jacobi-Bellman type variational inequalities.  Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space-time. This complements and provides a constructive PDE alternative to recent results of Beiglb\"ock, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.

We describe the moduli space of distributions in terms of Grothendieck’s Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one distributions on the projective space is an irreducible, nonsingular quasi-projective variety.

 We study codimension one holomorphic distributions on projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2. We show how the connectedness of the curves in the singular sets of foliations is an integrable phenomenon. This part of the  talk  is work joint with  M. Jardim(Unicamp) and O. Calvo-Andrade(Cimat).

We also study foliations by curves via the investigation  of their  singular schemes and  conormal  sheaves and we provide a classification  of foliations of degree at most 3 with  conormal  sheaves locally free.  Foliations of degrees  1 and 2 are aways given by a global intersection of two codimension one distributions. In the classification of degree 3 appear Legendrian foliations, foliations whose  conormal sheaves are instantons and other ” exceptional”
type examples. This part of the  talk   is  work joint with  M. Jardim(Unicamp) and S. Marchesi(Unicamp).

Gathering for women and non-binary mathematicians

Supersymmetric localization is a powerful technique to evaluate a class of functional integrals in supersymmetric field theories. It reduces the functional integral over field space to ordinary integrals over the space of solutions of the off-shell BPS equations. The application of this technique to supergravity suffers from some problems, both conceptual and practical. I will discuss one of the main conceptual problems, namely how to construct the fermionic symmetry with which to localize. I will show how a deformation of the BRST technique allows us to do this. As an application I will then sketch a computation of the one-loop determinant of the super-graviton that enters the localization formula for BPS black hole entropy.
 

The moments of characteristic polynomials play a central role in Random Matrix Theory.  They appear in many applications, ranging from quantum mechanics to number theory.  The mixed moments of the characteristic polynomials of random unitary matrices, i.e. the joint moments of the polynomials and their derivatives, can be expressed recursively in terms of combinatorial sums involving partitions. However, these combinatorial sums are not easy to compute, and so this does not give an effective method for calculating the mixed moments in general. I shall describe an alternative evaluation of the mixed moments, in terms of solutions of the Painlevé V differential equation, that facilitates their computation and asymptotic analysis.

Glacier flow is an example of a gravity driven non-linear viscous flow at low Reynolds numbers. As a glacier flows over an undulating bed, the surface topography is modified in response. Some information about bed conditions is therefore contained in the shape of the surface and the surface velocity field. I will present theoretical and numerical work on how basal conditions on glaciers affect ice flow, and how one can obtain information about basal conditions through surface-to-bed inversion. I’ll give an overview over inverse methodology currently used in glaciology, and how satellite data is now routinely used to invert for bed properties of the Greenland and the Antarctic Ice Sheets.

Computer vision approaches have made huge advances with deep learning research. These algorithms can be employed as a basis for phenotyping of biological traits from imaging modalities. This can be employed, for example, in the context of facial photographs of rare diseases as a means of aiding diagnostic pathways, or as means to large scale phenotyping in histological imaging. With any data set, inherent biases and problems in the data available for training can have a detrimental impact on your models. I will describe some examples of such data set problems and outline how to build models that are not confounded – despite biases in the training data. 

Let F be a totally real or CM number field.  Scholze has constructed Galois representations associated with torsion classes in the cohomology of locally symmetric spaces for GL_n(F).  We show that the nilpotent ideal appearing in Scholze's construction can be removed when F splits completely at the relevant prime.  As a key component of the proof, we show that the compactly supported cohomology of certain unitary and symplectic Shimura varieties with level  Gamma_1(p^{\infty}) vanishes above the middle degree. This is joint work with Ana Caraiani, Chi-Yun Hsu, Christian Johansson, Lucia Mocz, Emanuel Reinecke, and Sheng-Chi Shih. 

J. M. Foster 1 , N. E. Courtier 2 , S. E. J. O’Kane 3 , J. M. Cave 3 , R. Niemann 4 , N. Phung 5 , A. Abate 5 , P. J. Cameron 4 , A. B. Walker 3 & G. Richardson 2 .

1 School of Mathematics & Physics, University of Portsmouth, UK. {@email}

2 School of Mathematics, University of Southampton, UK.

3 School of Physics, University of Bath, UK.

4 School of Chemistry, University of Bath, UK.

5 Helmholtz-Zentrum Berlin, Germany.

Metal halide perovskite has emerged as a highly promising photovoltaic material. Perovskite-based solar cells now exhibit power conversion efficiencies exceeding 22%; higher than that of market-leading multi-crystalline silicon, and comparable to the Shockley-Queisser limit of around 33% (the maximum obtainable efficiency for a single junction solar cell). In addition to fast electronic phenomena, occurring on timescales of nanoseconds, they also exhibit much slower dynamics on the timescales of several seconds and up to a day. One well-documented example of this is the ‘anomalous’ hysteresis observed in current-voltage scans where the applied voltage is varied whilst the output current is measured. There is now a consensus that this is caused by the motion of ions in the perovskite material affecting the internal electric field and in turn the electronic transport.

We will discuss the formulation of a drift-diffusion model for the coupled electronic and ionic transport in a perovskite solar cell as well as its systematic simplification via the method of matched asymptotic expansions. We will use the resulting reduced model to give a cogent explanation for some experimental observations including, (i) the apparent disappearance of current-voltage hysteresis for certain device architectures, and (ii) the slow fading of performance under illumination during the day and subsequent recovery in the dark overnight. Finally, we suggest ways in which materials and geometry can be chosen to reduce charge carrier recombination and improve device performance.

The FEniCS system [1] allows the description of finite element discretisations of partial differential equations using a high-level syntax, and the automated conversion of these representations to working code via automated code generation. In previous work described in [2] the high-level representation is processed automatically to derive discrete tangent-linear and adjoint models. The processing of the model code at a high level eases the technical difficulty associated with management of data in adjoint calculations, allowing the use of optimal data management strategies [3].

This previous methodology is extended to enable the calculation of higher order partial differential equation constrained derivative information. The key additional step is to treat tangent-linear
equations on an equal footing with originating forward equations, and in particular to treat these in a manner which can themselves be further processed to enable the derivation of associated adjoint information, and the derivation of higher order tangent-linear equations, to arbitrary order. This enables the calculation of higher order derivative information -- specifically the contraction of a Kth order derivative against (K - 1) directions -- while still making use of optimal data management strategies. Specific applications making use of Hessian information associated with models written using the FEniCS system are presented.

[1] "Automated solution of differential equations by the finite element method: The FEniCS book", A. Logg, K.-A. Mardal, and  G. N. Wells (editors), Springer, 2012
[2] P. E. Farrell, D. A. Ham, S. W. Funke, and M. E. Rognes, "Automated derivation of the adjoint of high-level transient finite element programs", SIAM Journal on Scientific Computing 35(4), C369--C393, 2013
[3] A. Griewank, and A. Walther, "Algorithm 799: Revolve: An implementation of checkpointing for the reverse or adjoint mode of computational differentiation", ACM Transactions on Mathematical Software 26(1), 19--45, 2000