Past events

Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density, the so-called Dean-Kawasaki equation.

Our aim is to derive and study continuum models for the mesoscopic behavior of particle systems. In particular, we are interested in finite size effects. We will introduce nonlinear and non-Gaussian models that approximate the Dean-Kawasaki equation, in the special case of non-interacting particles. We want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation.  This is the joint work with H. Kremp (TU Wien) and N. Perkowski (FU Berlin).

We discuss $C^1$-regularity and developability of isometric immersions of flat domains into $\mathbb{R}^3$ enjoying a local fractional Sobolev $W^{1+s;2/s}$-regularity for $2/3 \leq s < 1$, generalising the known results on Sobolev (by Pakzad) and H\"{o}lder (by De Lellis--Pakzad) regimes. Ingredients of the proof include analysis of the weak Codazzi equations of isometric immersions, the study of $W^{1+s;2/s}$-gradient deformations with symmetric derivative and vanishing distributional Jacobian determinant, and the theory of compensated compactness. Joint work with M. Reza Pakzad and Armin Schikorra.

Floer theory, which mimics an infinite dimensional Morse approach to the study of critical points of smooth functions, appeared as an attempt to prove Arnold conjecture. The theory is more or less well understood in some compact cases.

Non-compact symplectic manifolds can sometimes be compactified as singular symplectic manifolds where the symplectic form "blows up" along a hypersurface in a controlled way (b^m-symplectic manifolds). In natural examples in Celestial mechanics such as the 3-body problem, these compactifications are given by regularization transformations à la Moser/Mc Gehee etc.

I will use the theory of b^m-symplectic/b^m-contact manifolds (introduced by Scott, Guillemin-Miranda Weitsman, and Miranda-Oms) as a guinea pig to propose ways to extend the study of Hamiltonian/Reeb Dynamics to singular symplectic/contact manifolds. This, in particular, yields new results for non-compact symplectic manifolds and for special (but, yet, meaningful) classes of Poisson manifolds.

Inspiration comes from several results extending the Weinstein conjecture to the context of b^m-contact manifolds and its connection to the study of escape orbits in Celestial mechanics and Fluid Dynamics. Those examples motivate a model for (singular) Floer homology.

I'll describe the motivating examples/results and some ideas to attack the general questions.

The Hardy Lectureship was founded in 1967 in memory of G.H. Hardy (LMS President 1926-1928 & 1939-1941). The Hardy Lectureship is a lecture tour of the UK by a mathematician with a high reputation in research.

Schauder estimates are a classical tool in linear and nonlinear elliptic and parabolic PDEs. They describe how regularity of coefficients reflects in regularity of solutions. They basically have a perturbative nature. This means that they can be obtained by perturbing the estimates available for problems without coefficients. This paradigm works as long as one deals with uniformly elliptic equations. The nonuniformly elliptic case is a different story and Schauder's theory turns out to be not perturbative any longer, as shown by counterexamples. In my talk, I will present a method allowing to bypass the perburbative schemes and leading to Schauder estimates in the nonuniformly elliptic regime. For this I will concentrate on the case of nonuniformly elliptic functionals with nearly linear growth, also covering a borderline case of so-called double phase energies. From recent, joint work with Cristiana De Filippis (Parma). 

Counting incompressible surfaces in hyperbolic 3-manifolds.

1:30pm

Marc Lackenby (Oxford)

Incompressible embedded surfaces play a central role in 3-manifold theory. It is a natural and interesting question to ask how many such surfaces are contained in a given 3-manifold M, as a function of their genus g. I will present a new theorem that provides a surprisingly small upper bound. For any given g, there a polynomial p_g with the following property. The number of closed incompressible surfaces of genus g in a hyperbolic 3-manifold M is at most p_g(vol(M)). This is joint work with Anastasiia Tsvietkova.

~~~~

The rates of growth in hyperbolic groups.

2:45pm

Koji Fujiwara (Kyoto)

For a finitely generated group of exponential growth, we study the set of exponential growth rates for all possible finite generating sets. Let G be a hyperbolic group. It turns out that the set of growth rates is well-ordered.  Also, given a number, there are only finitely many generating sets that have this number as the growth rate. I also plan to discuss the set of growth for a family of groups.

~~~~

Character varieties of random groups.

4:00pm

Emmanuel Breuillard (Oxford)

In joint work with P. Varju and O. Becker we study the representation and character varieties of random finitely presented groups with values in a complex semisimple Lie group.  We compute the dimension and number of irreducible components of the character variety of a random group. In particular we show that random one-relator groups have many rigid Zariski-dense representations. The proofs use a fair amount of number theory and are conditional on GRH. Key to them is the use of expander graphs for finite simple groups of Lie type as well as a new spectral gap result for random walks on linear groups.

I will explain a formalism for anticyclotomic Euler systems for a large class of Galois representations and explain how to prove analogs of Kolyvagins' celebrated "rank one" results. A novelty of this approach lies in the use of primes that split in the CM field. This is joint work with Dimitar Jetchev and Jan Nekovar. I will also describe some higher-dimensional examples of such Euler systems.

1:30 Milena Vuletic

Simulation of Arbitrage-Free Implied Volatility Surfaces

We present a computationally tractable method for simulating arbitrage-free implied volatility surfaces. We illustrate how our method may be combined with a factor model based on historical SPX implied volatility data to generate dynamic scenarios for arbitrage-free implied volatility surfaces. Our approach conciliates static arbitrage constraints with a realistic representation of statistical properties of implied volatility co-movements.

2:00 Nicola Muca Cirone

Neural Signature Kernels

Motivated by the paradigm of reservoir computing, we consider randomly initialized controlled ResNets defined as Euler-discretizations of neural controlled differential equations (Neural CDEs), a unified architecture which enconpasses both RNNs and ResNets. We show that in the infinite-width-depth limit and under proper scaling, these architectures converge weakly to Gaussian processes indexed on some spaces of continuous paths and with kernels satisfying certain partial differential equations (PDEs) varying according to the choice of activation function, extending the results of Hayou (2022); Hayou & Yang (2023) to the controlled and homogeneous case. In the special, homogeneous, case where the activation is the identity, we show that the equation reduces to a linear PDE and the limiting kernel agrees with the signature kernel of Salvi et al. (2021a). We name this new family of limiting kernels neural signature kernels. Finally, we show that in the infinite-depth regime, finite-width controlled ResNets converge in distribution to Neural CDEs with random vector fields which, depending on whether the weights are shared across layers, are either time-independent and Gaussian or behave like a matrix-valued Brownian motion.

2:30 Break

2:50-3:50 Renyuan Xu, Assistant Professor, University of Southern California

Reversible and Irreversible Decisions under Costly Information Acquisition 

Many real-world analytics problems involve two significant challenges: estimation and optimization. Due to the typically complex nature of each challenge, the standard paradigm is estimate-then-optimize. By and large, machine learning or human learning tools are intended to minimize estimation error and do not account for how the estimations will be used in the downstream optimization problem (such as decision-making problems). In contrast, there is a line of literature in economics focusing on exploring the optimal way to acquire information and learn dynamically to facilitate decision-making. However, most of the decision-making problems considered in this line of work are static (i.e., one-shot) problems which over-simplify the structures of many real-world problems that require dynamic or sequential decisions.

As a preliminary attempt to introduce more complex downstream decision-making problems after learning and to investigate how downstream tasks affect the learning behavior, we consider a simple example where a decision maker (DM) chooses between two products, an established product A with known return and a newly introduced product B with an unknown return. The DM will make an initial choice between A and B after learning about product B for some time. Importantly, our framework allows the DM to switch to Product A later on at a cost if Product B is selected as the initial choice. We establish the general theory and investigate the analytical structure of the problem through the lens of the Hamilton—Jacobi—Bellman equation and viscosity solutions. We then discuss how model parameters and the opportunity to reverse affect the learning behavior of the DM.

This is based on joint work with Thaleia Zariphopoulou and Luhao Zhang from UT Austin.
 

Principal Component Analysis (PCA) is a fundamental and ubiquitous tool in statistics and data analysis.
The bare-bones idea is this. Given a data set of n points y_1, ..., y_n, form their sample covariance S. Eigenvectors corresponding to large eigenvalues--namely directions along which the variation within the data set is large--are usually thought of as "important"  or "signal-bearing"; in contrast, weak directions are often interpreted as "noise", and discarded along the proceeding steps of the data analysis pipeline. Principal component (PC) selection is an important methodological question: how large should an eigenvalue be so as to be considered "informative"?
Our main deliverable is ScreeNOT: a novel, mathematically-grounded procedure for PC selection. It is intended as a fully algorithmic replacement for the heuristic and somewhat vaguely-defined procedures that practitioners often use--for example the popular "scree test".
Towards tackling PC selection systematically, we model the data matrix as a low-rank signal plus noise matrix Y = X + Z; accordingly, PC selection is cast as an estimation problem for the unknown low-rank signal matrix X, with the class of permissible estimators being singular value thresholding rules. We consider a formulation of the problem under the spiked model. This asymptotic setting captures some important qualitative features observed across numerous real-world data sets: most of the singular values of Y are arranged neatly in a "bulk", with very few large outlying singular values exceeding the bulk edge. We propose an adaptive algorithm that, given a data matrix, finds the optimal truncation threshold in a data-driven manner under essentially arbitrary noise conditions: we only require that Z has a compactly supported limiting spectral distribution--which may be a priori unknown. Under the spiked model, our algorithm is shown to have rather strong oracle optimality properties: not only does it attain the best error asymptotically, but it also achieves (w.h.p.) the best error--compared to all alternative thresholds--at finite n.

This is joint work with Matan Gavish (Hebrew University of Jerusalem) and David Donoho (Stanford). 

Scattering amplitudes in string theory are written as formal integrals of correlations functions over the moduli space of punctured Riemann surfaces. It's well-known, albeit not often emphasized, that this prescription is only approximately correct because of the ambiguities in defining the integration domain. In this talk, we propose a resolution of this problem for one-loop open-string amplitudes and present their first evaluation at finite energy and scattering angle. Our method involves a deformation of the integration contour over the modular parameter to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop planar and non-planar type-I superstring four-point amplitudes, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants and directly verify long-standing conjectures about their behavior at high energies.

Deep convolutional neural networks such as GoogLeNet and ResNet have become ubiquitous in image classification tasks, whereas
transformer-based language models such as BERT and its variants have found widespread use in natural language processing. In this talk, I
will discuss recent efforts in exploring the topology of artificial neuron activations in deep learning, from images to word embeddings.
First, I will discuss the topology of convolutional neural network activations, which provides semantic insight into how these models
organize hierarchical class knowledge at each layer. Second, I will discuss the topology of word embeddings from transformer-based models.
I will explore the topological changes of word embeddings during the fine-tuning process of various models and discover model confusions in
the embedding spaces. If time permits, I will discuss on-going work in studying the topology of neural activations under adversarial attacks.
 

The formation of surface meltwater has been linked with the disintegration of many ice shelves in the Antarctic Peninsula over the last several decades. Despite the importance of surface meltwater production and transport to ice shelf stability, knowledge of these processes is still lacking. Understanding the surface hydrology of ice shelves is an essential first step to reliably project future sea level rise from ice-sheet melt.

In order to better understand the processes driving meltwater distribution on ice shelves, we present the first comprehensive model of surface hydrology to be developed for Antarctic ice shelves, enabling us to incorporate key processes such as the lateral transport of surface meltwater. Recent observations suggest that surface hydrology processes on ice shelves are more complex than previously thought, and that processes such as lateral routing of meltwater across ice shelves, ice shelf flexure and surface debris all play a role in the location and influence of meltwater. Our model allows us to account for these and is calibrated and validated through both remote sensing and field observations.

We study the positional information conferred by the morphogens Sonic Hedgehog and BMP in neural tube patterning. We use the mathematics of information theory to quantify the information that cells use to decide their fate. We study the encoding, recoding and decoding that take place as the morphogen gradient is formed, triggers a nuclear response and determines cell fates using a gene regulatory network.

Given a perfectoid field, we find an elementary extension and a henselian defectless valuation on it, whose value group is divisible and whose residue field is an elementary extension of the tilt. This specializes to the almost purity theorem over perfectoid valuation rings and Fontaine-Wintenberger. Along the way, we prove an Ax-Kochen/Ershov principle for certain deeply ramified fields, which also uncovers some new model-theoretic phenomena in positive characteristic. Notably, we get that the perfect hull of Fp(t)^h is an elementary substructure of the perfect hull of Fp((t)). Joint work with Franziska Jahnke.

Abstract. I will talk about projects in which we combine heuristics with computational data to develop a theory in problems where it was previously hard to be confident of the guesses that there are in the literature.

1/ "Speculations about the number of primes in fast growing sequences". Starting from studying the distribution of primes in sequences like $2^n-3$, Jon Grantham and I have been developing a heuristic to guess at the frequency of prime values in arbitrary linear recurrence sequences in the integers, backed by calculations.

If there is enough time I will then talk about:

2/ "The spectrum of the $k$th roots of unity for $k>2$, and beyond".  There are many questions in analytic number theory which revolve around the "spectrum", the possible mean values of multiplicative functions supported on the $k$th roots of unity. Twenty years ago Soundararajan and I determined the spectrum when $k=2$, and gave some weak partial results for $k>2$, the various complex spectra.  Kevin Church and I have been tweaking MATLAB's package on differential delay equations to help us to develop a heuristic theory of these spectra for $k>2$, allowing us to (reasonably?) guess at the answers to some of the central questions.

We propose a novel methodology for modeling and forecasting multivariate realized volatilities using graph neural networks. This approach extends the work of Zhang et al. [2022] (Graph-based methods for forecasting realized covariances) and explicitly incorporates the spillover effects from multi-hop neighbors and nonlinear relationships into the volatility forecasts. Our findings provide strong evidence that the information from multi-hop neighbors does not offer a clear advantage in terms of predictive accuracy. However, modeling the nonlinear spillover effects significantly enhances the forecasting accuracy of realized volatilities over up to one month. Our model is flexible and allows for training with different loss functions, and the results generally suggest that using Quasi-likelihood as the training loss can significantly improve the model performance, compared to the commonly-used mean squared error. A comprehensive series of evaluation tests and alternative model specifications confirm the robustness of our results.

Paper available online: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4375165

I will reflect on my time as Professor of Numerical Analysis.

I’m excited to share with everyone some new, unpublished work that we are just in the process of wrapping up and could use everyone’s reactions. It is a reconciliation of evolutionary game theory and ecological dynamics that I have wrestled with since I moved from an evolution program into an ecology-heavy department. It always seemed like, depending on the problem I was thinking about, I had to change my perspective and approach it as either an evolutionary game theorist, or an ecologist; and yet I had this nagging feeling that, at its core, the problem was often one and the same, and therefore one theoretical framework should suffice. So when should one write down an n-type replicator equation and when should one write down an n-species Lotka-Volterra system; and what does it mean mathematically and biologically when one has made such a choice? In the process of reconciling, I also got a deeper appreciation of what is and is not a proper game, such as a Prisoner’s Dilemma. These findings can help shed light on previously puzzling empirical findings.

Asymptotic dimension was introduced by Gromov as an invariant of finitely generated groups. It can be shown that if two metric spaces are quasi-isometric then they have the same asymptotic dimension. In 1998, the asymptotic dimension achieved particular prominence in geometric group theory after a paper of Guoliang Yu, which proved the Novikov conjecture for groups with finite asymptotic dimension. Unfortunately, not all finitely generated groups have finite asymptotic dimension. 

In this talk, we will introduce some basic tools to compute the asymptotic dimension of groups. We will also find upper bounds for the asymptotic dimension of a few well-known classes of finitely generated groups, such as hyperbolic groups, and if time permits, we will see why one-relator groups have asymptotic dimension at most two.

I will explain how to construct an Euler system for imaginary quadratic fields using Eisenstein series and their cohomology classes. This illustrates a template for a construction that should yield many new Euler systems.

A C*-algebra has the lifting property (LP) if any unital completely positive map into a quotient C*-algebra admits a completely positive lift. The local lifting property (LLP), introduced by Kirchberg in the early 1990s, is a weaker, local version of the LP.  I will present a method, based on non-vanishing of second cohomology groups, for proving the failure of lifting properties for full C*-algebras of countable groups with (relative) property (T). This allows us to derive that the full C*-algebras of the groups $Z^2\rtimes SL_2(Z)$ and $SL_n(Z)$, for n>2, do not have the LLP. The same method allows us to prove that the full C*-algebras of a large class of groups with property (T), including those admitting a probability measure preserving action with non-vanishing second real-valued cohomology, do not have the LP.  In a different direction, we prove that the full C*-algebras of any non-finitely presented groups with property (T) do not have the LP. Time permitting, I will also discuss a connection with the notion of Hilbert-Schmidt stability for countable groups. This is based on a joint work with Pieter Spaas and Matthew Wiersma.