Past

In this talk, we will introduce two problems of contract theory, in continuous–time, with a multitude of agents. First, we will study a model of optimal contracting in a hierarchy, which generalises the one–period framework of Sung (2015). The hierarchy is modeled by a series of interlinked principal–agent problems, leading to a sequence of Stackelberg equilibria. More precisely, the principal (she) can contract with a manager (he), to incentivise him to act in her best interest, despite only observing the net benefits of the total hierarchy. The manager in turn subcontracts the agents below him. Both agents and the manager each independently control a stochastic process representing their outcome. We will see through a simple example that even if the agents only control the drift of their outcome, the manager controls the volatility of the Agents’ continuation utility. Even this first simple example justifies the use of recent results on optimal contracting for drift and volatility control, and therefore the theory on 2BSDEs. We will also discuss some possible extensions of this model. In particular, one extension consists in the elaboration of more general contracts, indexing the compensation of one worker on the result of the others. This increase in the complexity of contracts is beneficial for the principal, and constitutes a first approach to even more complex contracts, in the case, for example, of a continuum of workers with mean–field interactions. This will lead us to introduce the second problem, namely optimal contracting for demand–response management, which consists in extending the model by Aïd, Possamaï, and Touzi (2019) to a mean–field of consumers. Finally, we will conclude by mentioning that this principal-agent approach with a multitude of agents can be used to address many situations, for example to model incentives for
lockdown in the current epidemic context.
 

Data scientists are often faced with the challenge of understanding a high dimensional data set organized as a table. These tables may have columns of different (sometimes, non-numeric) types, and often have many missing entries. In this talk, we discuss how to use low rank models to analyze these big messy data sets. Low rank models perform well --- indeed, suspiciously well — across a wide range of data science applications, including applications in social science, medicine, and machine learning. In this talk, we introduce the mathematics of low rank models, demonstrate a few surprising applications of low rank models in data science, and present a simple mathematical explanation for their effectiveness.

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We study the evolution of solid surfaces and pattern formation by
surface diffusion. Phase field models with degenerate mobilities are
frequently used to model such phenomena, and are validated by
investigating their sharp interface limits. We demonstrate by a careful
asymptotic analysis involving the matching of exponential terms that a
certain combination of degenerate mobility and a double well potential
leads to a combination of both surface and non-linear bulk diffusion to
leading order. If time permits, we will discuss implications and extensions.

We consider the homogenization of a Stokes system in a domain having many small random holes. This model mainly arises from problems of solid-fluid interaction (e.g. the flow of a viscous and incompressible fluid through a porous medium). We aim at the rigorous derivation of the homogenization limit both in the Brinkmann regime and in the one of Darcy’s law. In particular, we focus on holes that are distributed according to probability measures that allow for overlapping and clustering phenomena.

We characterize which Borel functions are decomposable into
a countable union of functions which are piecewise continuous on
$\Pi^0_n$ domains, assuming projective determinacy. One ingredient of
our proof is a new characterization of what Borel sets are $\Sigma^0_n$
complete. Another important ingredient is a theorem of Harrington that
there is no projective sequence of length $\omega_1$ of distinct Borel
sets of bounded rank, assuming projective determinacy. This is joint
work with Adam Day.

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

A 3-manifold fibers over the circle if it can be identified with the mapping torus of a surface homeomorphism. If the surface is compact with non-empty boundary then the corresponding 3-manifold group is free-by-cyclic, and the action of the cyclic group on the free group is induced by the surface homeomorphism. Although most free-by-cyclic groups do not arise as fundamental groups of 3-manifolds which fiber over the circle, there is a strong analogy between the two families.

In this talk I will discuss how dynamical properties of the monodromy affect the geometry/algebra of the corresponding mapping torus. We will see how the same 3-manifold or group can admit multiple fiberings and what properties of the monodromy are known to be preserved under different fiberings.

In this talk I review the recent discovery of Yangian symmetry for massive Feynman integrals and how it can be used to set up a Yangian Bootstrap. I will provide elementary proofs of the symmetry at one and two loops, whereas at generic loop order I conjecture that all graphs cut from regular tilings of the plane with massive propagators on the boundary enjoy the symmetry. After demonstrating how the symmetry may be used to constrain the functional form of Feynman integrals on explicit examples, I comment on how a subset of the diagrams for which the symmetry is conjectured to hold is naturally embedded in a Massive Fishnet theory that descends from gamma-deformed Coulomb branch N=4 Super-Yang-Mills theory in a particular double scaling limit.

Since the seminal work of Keating and Snaith, the characteristic polynomial of a random (Haar-distributed) unitary matrix has seen several of its functional studied in relation with the probabilistic study of the Riemann Zeta function. We will recall the history of the topic starting with the Montgommery-Dyson correspondance and will revisit the last twenty years of computations of integer moments of some functionals, with a particular focus on the mid-secular coefficients recently studied by Najnudel-PaquetteSimm. The new method here introduced will be compared with one of the classical ways to deal with such functionals, the Conrey-Farmer-Keating-Rubinstein-Snaith (CFKRS) formula.

For high order finite elements (continuous piecewise polynomials) the conditioning of the basis is important. However, so far there seems no generally accepted concept of "a well-conditioned basis”,  or a general strategy for how to obtain such representations. In this presentation, we use the $L^2$ condition number as a measure of the conditioning, and construct representations by frames such that the associated $L^2$ condition number is bounded independently of the polynomial degree. The main tools include the bubble transform, which is a stable decomposition of functions into local modes, and orthogonal polynomials on simplexes.  We also include a brief discussion on potential applications in preconditioning. This is a joint work with Ragnar Winther. 

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We explain how group analogues of Slodowy slices arise by interpreting certain Weyl group elements as braids. Such slices originate from classical work by Steinberg on regular conjugacy classes, and different generalisations recently appeared in work by Sevostyanov on quantum group analogues of W-algebras and in work by He-Lusztig on Deligne-Lusztig varieties.

Our perspective furnishes a common generalisation, essentially solving the problem. We also give a geometric criterion for Weyl group elements to yield strictly transverse slices.

That neural networks may be pruned to high sparsities and retain high accuracy is well established. Recent research efforts focus on pruning immediately after initialization so as to allow the computational savings afforded by sparsity to extend to the training process. In this work, we introduce a new `DCT plus Sparse' layer architecture, which maintains information propagation and trainability even with as little as 0.01% trainable kernel parameters remaining. We show that standard training of networks built with these layers, and pruned at initialization, achieves state-of-the-art accuracy for extreme sparsities on a variety of benchmark network architectures and datasets. Moreover, these results are achieved using only simple heuristics to determine the locations of the trainable parameters in the network, and thus without having to initially store or compute with the full, unpruned network, as is required by competing prune-at-initialization algorithms. Switching from standard sparse layers to DCT plus Sparse layers does not increase the storage footprint of a network and incurs only a small additional computational overhead.

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The study of motifs in networks can help researchers uncover links between structure and function of networks in biology, the sociology, economics, and many other areas. Empirical studies of networks have identified feedback loops, feedforward loops, and several other small structures as "motifs" that occur frequently in real-world networks and may contribute by various mechanisms to important functions these systems. However, the mechanisms are unknown for many of these motifs. We propose to distinguish between "structure motifs" (i.e., graphlets) in networks and "process motifs" (which we define as structured sets of walks) on networks and consider process motifs as building blocks of processes on networks. Using the covariances and correlations in a multivariate Ornstein--Uhlenbeck process on a network as examples, we demonstrate that the distinction between structure motifs and process motifs makes it possible to gain quantitative insights into mechanisms that contribute to important functions of dynamical systems on networks.

I will connect approaches to classical integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations. In particular, I will consider holomorphic Chern-Simons theory on twistor space, defined using a range of meromorphic (3,0)-forms. On shell these are, in most cases, found to agree with actions for anti-self-dual Yang-Mills theory on space-time. Under symmetry reduction, these space-time actions yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces the holomorphic Chern-Simons action to 4d Chern-Simons theory.

Let C be an elliptic or hyperelliptic curve over a p-adic field K. Then C is equipped with a Galois representation, given by the action of the absolute Galois group of K on the Tate module of C. The behaviour of this representation depends on the reduction type of C. We will focus on the case of C having bad reduction, and acquiring potentially good reduction over a wildly ramified extension of K. We will show that, if C is an elliptic curve, the Galois representation can be completely determined in this case, thus allowing one to fully classify Galois representations attached to elliptic curves. Furthermore, the same can be done for a special family of hyperelliptic curves, obtaining a result which is surprisingly similar to that for the corresponding elliptic curves case.
 

Interacting particle systems have found diverse applications in mathematics and several related fields, including statistical physics, population dynamics, and machine learning.  We will focus, in particular, on the zero range process and the symmetric simple exclusion process.  The large-scale behavior of these systems is essentially deterministic, and is described in terms of a hydrodynamic limit.  However, the particle process does exhibit large fluctuations away from its mean.  Such deviations, though rare, can have significant consequences---such as a concentration of energy or the appearance of a vacuum---which make them important to understand and simulate.

In this talk, which is based on joint work with Benjamin Gess, I will introduce a continuum model for simulating rare events in the zero range and symmetric simple exclusion process.  The model is based on an approximating sequence of stochastic partial differential equations with nonlinear, conservative noise.  The solutions capture to first-order the central limit fluctuations of the particle system, and they correctly simulate rare events in terms of a large deviations principle.

The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.

We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.

I will give an overview of the interactions between chromatic homotopy theory and the algebraic K-theory of ring spectra, especially around the subject of Ausoni-Rognes's principle of "chromatic redshift," and some of the recent advances in this field.

Unlike for closed manifolds, the existence of positive scalar curvature (psc) metrics on connected manifolds with
nonempty boundary is unobstructed. We study and compare the spaces of psc metrics on such manifolds with various
conditions along the boundary: H ≥ 0, H = 0, H > 0, II = 0, doubling, product structure. Here H stands for the
mean curvature of the boundary and II for its second fundamental form. "Doubling" means that the doubled metric
on the doubled manifold (along the boundary) is smooth and "product structure" means that near the boundary the
metric has product form. We show that many, but not all of the obvious inclusions are weak homotopy equivalences.
In particular, we will see that if the manifold carries a psc metric with H ≥ 0, then it also carries one which is
doubling but not necessarily one which has product structure. This is joint work with Bernhard Hanke.

Chiral fermion anomalies in any spacetime dimension are computed by evaluating an eta-invariant on a closed manifold in one higher dimension. The APS index theorem then implies that both local and global gauge anomalies are detected by bordism invariants, each being classified by certain abelian groups that I will identify. Mathematically, these groups are connected via a short exact sequence that splits non-canonically. This enables one to relate global anomalies in one gauge theory to local anomalies in another, by which we revive (from the bordism perspective) an old idea of Elitzur and Nair for deriving global anomalies. As an example, I will show how the SU(2) anomaly in 4d can be derived from a local anomaly by embedding SU(2) in U(2).

An important open question in black hole thermodynamics is about the existence of a "mass gap" between an extremal black hole and the lightest near-extremal state within a sector of fixed charge. In this talk, I will discuss how to reliably compute the partition function of 4d Reissner-Nordstrom near-extremal black holes at temperature scales comparable to the conjectured gap. I will show that the density of states at fixed charge does not exhibit a gap in the simplest gravitational non-supersymmetric theories; rather, at the expected gap energy scale, we see a continuum of states whose meaning we will extensively discuss. Finally, I will present a similar computation for nearly-BPS black holes in 4d N=2 supergravity. As opposed to their non-supersymmetric counterparts, such black holes do in fact exhibit a gap consistent with various string theory predictions.

The talk will discuss the use of mathematical models for understanding targeted cancer therapies. One area of focus is the treatment of chronic lymphocytic leukemia with tyrosine kinase inhibitors. I will explore how mathematical approaches have helped elucidate the mechanism of action of the targeted drug ibrutinib, and will discuss how evolutionary models, based on patient-specific parameters, can make individualized predictions about treatment outcomes. Another focus of the talk is the use of oncolytic viruses to kill cancer cells and drive cancers into remission. These are viruses that specifically infect cancer cells and spread throughout tumors. I will discuss mathematical models applied to experimental data that analyze virus spread in a spatially structured setting, concentrating on the interactions of the virus with innate immune mechanisms that determine the outcome of virus spread.  

Mathematicians need to talk and write about their mathematics.  This includes undergraduates and MSc students, who might be writing a dissertation or project report, preparing a presentation on a summer research project, or preparing for a job interview.  It can be helpful to think of this as a form of storytelling, as this can lead to more effective communication.  For a story to be engaging you also need to know your audience.  In this interactive session, we'll discuss what we mean by telling a mathematical story, give you some top tips from our experience, and give you a chance to think about how you might put this into practice.