L3

Mathematical and computational techniques can improve our understanding of diseases. In this talk, I’ll present ways in which data from cancer patients can be combined with mathematical modelling and used to improve cancer treatments.

Given the variability in individual responses to cancer treatments, agent-based modelling has been a useful technique for accurately capturing cellular behaviours that may lead to stochasticity in patient outcomes. Using a hybrid agent-based model and partial differential equation system, we developed a model for brain cancer (glioblastoma) growth informed by ex-vivo patient samples. Extending the model to capture patient treatment with an oncolytic virus rQNestin, we used our model to propose reasons for treatment failure, which was later confirmed with further patient samples. More recently, we extended this model to investigate the effectiveness of combination treatments (chemotherapy, virotherapy and immunotherapy) informed by individual patient imaging mass cytometry.

This talk hopes to provide examples of ways mathematical and computational modelling can be used to run “virtual” clinical trials with the goal of obtaining more effective treatments for diseases.  

Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, cells in a developing tissue, or locusts in a swarm. Here I will focus on two examples of such emergent behavior in biology, specifically cell interactions during pattern formation in zebrafish skin and gametophyte development in ferns. Different modeling approaches provide complementary insights into these systems and face different challenges. For example, vertex-based models describe cell shape, while more efficient agent-based models treat cells as particles. Continuum models, which track the evolution of cell densities, are more amenable to analysis, but it is often difficult to relate their few parameters to specific cell interactions. In this talk, I will overview our models of cell behavior in biological patterns and discuss our ongoing work on quantitatively relating different types of models using topological data analysis and data-driven techniques.

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.

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.

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.
 

The Zilber-Pink conjecture predicts that if V is a proper subvariety of an arithmetic variety S (e.g. abelian variety, Shimura variety, others) not contained in a proper special subvariety of V, then the “unlikely intersections” of V with the proper special subvarieties of S are not Zariski dense in V. In this talk I will present a strong counterpart to the Zilber-Pink conjecture, namely that under some natural conditions, likely intersections are in fact Euclidean dense in V.  This is joint work with Tom Scanlon.

This will be a continuation of the talk from last week (9 May). 

The speakers are Maya Stein (University of Chile), Mathias Schacht (Hamburg), János Pach (Rényi Institute, Hungary and IST Austria), Marthe Bonamy (Bordeaux), Mehtaab Sawhney (Cambridge/MIT), and Julian Sahasrabudhe (Cambridge). Please see the event website for further details including titles, abstracts, and timings. Anyone interested is welcome to attend, and no registration is required.

A conjecture due to Zilber predicts that the complex exponential field is quasiminimal: that is, that all subsets of the complex numbers that are definable in the language of rings expanded by a symbol for the complex exponential function are countable or cocountable.
Zilber showed that this conjecture would follow from Schanuel's Conjecture and an existential closedness type property asserting that certain systems of exponential-polynomial equations can be solved in the complex numbers; later on, Bays and Kirby were able to remove the dependence on Schanuel's Conjecture, shifting all the focus to the existence of solutions. In this talk, I will discuss recent work about the quasiminimality of a reduct of the complex exponential field, that is, the complex numbers expanded by multivalued power functions. This is joint work with Jonathan Kirby.