Past

Let (V, ≥) be a finite, partially ordered set. Say a directed forest on V is a set of directed edges [x,y> with x ≤ y such that no vertex has indegree greater than one.

Thus for a finite measure μ on some partially ordered measurable space D we may define a Poisson random forest by choosing a set of vertices V according to a Poisson point process weighted by the number of directed forests on V, and selecting a directed forest uniformly.

We give a necessary and sufficient condition for such a process to exist and show that the process may be expressed as a multi-type branching process with type space D.

We build on this observation, together with a construction of the simple birth death process due to Kurtz and Rodrigues [2011] to develop a coalescent theory for rapidly expanding populations.

There are many recent points of contact of model theory and other 
parts of mathematics: o-minimality and Diophantine geometry, geometric group 
theory, additive combinatorics, rigid geometry,...  I will probably 
emphasize  long-standing themes around stability, Diophantine geometry, and 
analogies between ODE's and bimeromorphic geometry.

Thermomechanical processes observed in deformable solids under intensive dynamic or quasi-static loadings consist of coupled mechanical, thermal and fracturing stages. The fracturing processes involve formation, motion and interaction of defects in crystals, phase transitions, breaking of bonds between atoms, accumulation of micro-structural damages (pores, cracks), etc. Irreversible deformations, zones of adiabatic shear micro-fractures are caused by these processes. Dynamic fracturing is a complicated multistage process, which includes appearance, evolution and confluence of micro-defects and formation of embryonic micro-cracks and pores that can grow and lead to the breaking-up of bodies with formation of free surfaces. This results in a need to use more advanced mathematical and numerical techniques.

This talk presents modelling of irreversible deformation near the tip of a crack in a porous domain containing oil and gas during the hydraulic fracturing process. The governing equations for a porous domain containing oil and gas are based on constructing a mathematical model of thermo-visco-elasto-plastic media with micro-defects (micro-pores) filled with another phase (e.g., oil or/and gas). The micro-pores can change their size during the process of dynamical irreversible deformation. The existing pores can expand or collapse. The model was created by using fundamental thermodynamic principles and, therefore, it is a thermodynamically consistent model. All the processes (i.e., irreversible deformation, fracturing, micro-damaging, heat transfer) within a porous domain are strongly coupled. An explicit normalized-corrected meshless method is used to solve the resulting governing PDEs. The flexibility of the proposed technique allows efficient running using a great number of micro- and macro- fractures. The results are presented, discussed and future studies are outlined.

We wish to discuss the role of Modelling in Health Care. While risk factor prevalences vary and change with time it is difficult to anticipate the change in disease incidence that will result without accurately modelling the epidemiology. When detailed study of the prevalence of obesity, tobacco and salt intake, for example, are studied clear patterns emerge that can be extrapolated into the future. These can give rise to estimated probability distributions of these risk factors across age, sex, ethnicity, social class groups etc into the future. Micro simulation of individuals from defined populations (eg England 2012) can then estimate disease incidence, prevalence, death, costs and quality of life. Thus future health and other needs can be estimated, and interventions on these risk factors can be simulated for their population effect. Health policy can be better determined by a realistic characterisation of public health. The Foresight microsimulation modelling of the National Heart Forum (UK Health Forum) will be described. We will emphasise some of the mathematical and statistical issues associated with so doing.

The study of difference algebraic geometry stems from the efforts of Macintyre and Hrushovski to

count the number of solutions to difference polynomial equations over fields with powers of Frobenius.

We propose a notion of multiplicity in the context of difference algebraic schemes and prove a first principle

of preservation of multiplicity. We shall also discuss how to formulate a suitable intersection theory of difference schemes.

Self-renewal and pluripotency of mouse embryonic stem (ES) cells are controlled by a complex transcriptional regulatory network (TRN) which is rich in positive feedback loops. A number of key components of this TRN, including Nanog, show strong temporal expression fluctuations at the single cell level, although the precise molecular basis for this variability remains unknown. In this talk I will discuss recent work which uses a genetic complementation strategy to investigate genome-wide mRNA expression changes during transient periods of Nanog down-regulation. Nanog removal triggers widespread changes in gene expression in ES cells. However, we found that significant early changes in gene expression were reversible upon re-induction of Nanog, indicating that ES cells initially adopt a flexible “primed” state. Nevertheless, these changes rapidly become consolidated irreversible fate decisions in the continued absence of Nanog. Using high-throughput single cell transcriptional profiling we observed that the early molecular changes are both stochastic and reversible at the single cell level. Since positive feedback commonly gives rise to phenotypic variability, we also sought to determine the role of feedback in regulating ES cell heterogeneity and commitment. Analysis of the structure of the ES cell TRN revealed that Nanog acts as a feedback “linchpin”: in its presence positive feedback loops are active and the extended TRN is self-sustaining; while in its absence feedback loops are weakened, the extended TRN is no longer self-sustaining and pluripotency is gradually lost until a critical “point-of-no-return” is reached. Consequently, fluctuations in Nanog expression levels transiently activate different sub-networks in the ES cell TRN, driving transitions between a (Nanog expressing) feedback-rich, robust, self-perpetuating pluripotent state and a (Nanog-diminished), feedback-depleted, differentiation-sensitive state. Taken together, our results indicate that Nanog- dependent feedback loops play a central role in controlling both early fate decisions at the single cell level and cell-cell variability in ES cell populations.

Counter-examples to the Hasse principle are known for many families of geometrically rational varieties. We discuss how often such failures arise for Chatelet surfaces and certain higher-dimensional hypersurfaces. This is joint work with Regis de la Breteche.

The bounded coherent dg-category on (suitable versions of) the Steinberg stack of a reductive group G is a categorification of the affine Hecke algebra in representation theory.  We discuss how to describe the center and universal trace of this monoidal dg-category.  Many of the techniques involved are very general, and the description makes use of the notion of "odd micro-support" of coherent complexes.  This is joint work with Ben-Zvi and Nadler.

Morse theory gives an estimate of the dimensions of the cohomology groups of a manifold in terms of the critical points of a function.
One can do better and compute the cohomology in terms of this function using the so-called Witten complex.
Already implicit in work of Smale in the fifties, it was rediscovered by Witten in the eighties using techniques from (supersymmetric) quantum field theories.
I will explain Witten's (heuristic) arguments and describe the Witten complex.

The symmetric group S_{mn} acts naturally on the collection of set partitions of a set of size mn into n sets each of size m, and the resulting permutation character is the Foulkes character. These characters are the subject of the longstanding Foulkes Conjecture. In this talk, we define a deflation map which sends a character of the symmetric group S_{mn} to a character of S_n. The values of the images of the irreducible characters under this map are described combinatorially in a rule which generalises two well-known combinatorial rules in the representation theory of symmetric groups, the Murnaghan-Nakayama formula and Young's rule. We use this in a new algorithm for computing irreducible constituents of Foulkes characters and verify Foulkes’ Conjecture in some new cases. This is joint work with Anton Evseev (Birmingham) and Mark Wildon (Royal Holloway).

Many methods for solving high-dimensional statistical inverse problems are based on convex optimization problems formed by the weighted sum of a loss function with a norm-based regularizer.

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Particular examples include $\ell_1$-based methods for sparse vectors and matrices, nuclear norm for low-rank matrices, and various combinations thereof for matrix decomposition and robust PCA. In this talk, we describe an interesting connection between computational and statistical efficiency, in particular showing that the same conditions that guarantee that an estimator has good statistical error can also be used to certify fast convergence of first-order optimization methods up to statistical precision.

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Joint work with Alekh Agarwahl and Sahand Negahban Pre-print (to appear in Annals of Statistics)

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http://www.eecs.berkeley.edu/~wainwrig/Papers/AgaNegWai12b_SparseOptFul…

We consider the problem of optimizing a portfolio of medium to low frequency

quant strategies under heavy tailed distributions. Approaching this problem by modelling

returns through mixture distributions, we derive robust and relative robust methodologies

and discuss conic optimization approaches to solving these models.

The Kolmogorov 1941 theory (K41) is, in a way, the starting point for all

models of turbulence. In particular, K41 and corrections to it provide

estimates of small-scale quantities such as increments and energy spectrum

for a 3D turbulent flow. However, because of the well-known difficulties

involved in studying 3D turbulent flow, there are no rigorous results

confirming or infirming those predictions. Here, we consider a well-known

simplified model for 3D turbulence: Burgulence, or turbulence for the 1D

Burgers equation. In the space-periodic case with a stochastic white in

time and smooth in space forcing term, we give sharp estimates for

small-scale quantities such as increments and energy spectrum.

A topological space is called rigid if its only autohomeomorphism is the identity map. Using the Axiom of Choice it is easy to construct rigid subsets of the real line R, but sets constructed in this way always have size continuum. I will explore the question of whether it is possible to have rigid subsets of R that are small, meaning that their cardinality is smaller than that of the continuum. On the one hand, we will see that forcing can be used to produce models of ZFC in which such small rigid sets abound. On the other hand, I will introduce a combinatorial axiom that can be used to show the consistency with ZFC of the statement "CH fails but every rigid subset of R has size continuum". Only a working knowledge of basic set theory (roughly what one might remember from C1.2b) and topology will be assumed.

Self-similarity is a fundamental idea in many areas of mathematics. In this talk I will explain how it has entered group theory and the links between self-similar groups and other areas of research. There will also be pretty pictures.

I will give an introduction to The McKay Correspondence, relating the irreducible representations of a finite subgroup Γ ≤ SL2 (C), minimal resolutions of the orbit space C2 /Γ, and affine Dynkin diagrams.

Complex systems emerging from many biochemical applications often exhibit multiscale and multiphysics (MSMP) features: The systems incorporate a variety of physical processes or subsystems across a broad range of scales. A typical MSMP system may come across scales with macroscopic, mesoscopic and microscopic kinetics,
deterministic and stochastic dynamics, continuous and discrete state space, fastscale and slow-scale reactions, and species of both large and small populations. These complex features present great challenges in the modeling and simulation practice. The goal of our research is to develop innovative computational methods and rigorous fundamental theories to answer these challenges. In this talk we will start with introduction of basic stochastic simulation algorithms for biochemical systems and multiscale
features in the stochastic cell cycle model of budding yeast. With detailed analysis of these multiscale features, we will introduce recent progress on simulation algorithms and computational theories for multiscale stochastic systems, including tau-leaping methods, slow-scale SSA, and the hybrid method. 

While usual percolation concerns the study of the connected components of

random subgraphs of an infinite graph, bootstrap percolation is a type of

cellular automaton, acting on the vertices of a graph which are in one of

two states: `healthy' or `infected'. For any positive integer $r$, the

$r$-neighbour bootstrap process is the following update rule for the

states of vertices: infected vertices remain infected forever and each

healthy vertex with at least $r$ infected neighbours becomes itself

infected. These updates occur simultaneously and are repeated at discrete

time intervals. Percolation is said to occur if all vertices are

eventually infected.

As it is often difficult to determine precisely which configurations of

initially infected vertices percolate, one often considers a random case,

with each vertex infected independently with a fixed probability $p$. For

an infinite graph, of interest are the values of $p$ for which the

probability of percolation is positive. I will give some of the history

of this problem for regular trees and present some new results for

bootstrap percolation on certain classes of randomly generated trees:

Galton--Watson trees.