Past

TBA

Abstract: The calibration problem for local stochastic volatility models leads to two-dimensional stochastic differential equations of McKean-Vlasov type. In these equations, the conditional distribution of the second component of the solution given the first enters the equation for the first component of the solution. While such equations enjoy frequent application in the financial industry, their mathematical analysis poses a major challenge. I will explain how to prove the strong existence of stationary solutions for these equations, as well as the strong uniqueness in an important special case. Based on joint work with Daniel Lacker and Jiacheng Zhang.  
 

In this talk I will present results from an ongoing joint research  program with Konrad Waldorf. Its main goal is to understand the  relation between gerbes on a manifold M and open-closed smooth field  theories on M. Gerbes can be viewed as categorified line bundles, and  we will see how gerbes with connections on M and their sections give  rise to smooth open-closed field theories on M. If time permits, we  will see that the field theories arising in this way have several characteristic properties, such as invariance under thin homotopies,  and that they carry positive reflection structures. From a physical  perspective, ourconstruction formalises the WZW amplitude as part of  a smooth bordism-type field theory.

I will describe an ongoing research project which aims to encode the DT invariants of a CY3 triangulated category in a geometric structure on its space of stability conditions. More specifically we expect to find a complex hyperkahler structure on the total space of the tangent bundle. These ideas are closely related to the work of Gaiotto, Moore and Neitzke from a decade ago. The main analytic input is a class of Riemann-Hilbert problems involving maps from the complex plane to an algebraic torus with prescribed discontinuities along a collection of rays.

We will show that minimally supersymmetric SU(N+2) SQCD models in the middle of the conformal window can be engineered by compactifying certain 6d SCFTs on three punctured spheres. The geometric construction of the 4d theories predicts numerous interesting strong coupling effects, such as IR symmetry enhancements and duality. We will discuss this interplay between simple geometric and group theoretic considerations and complicated field theoretic strong coupling phenomena. For example, one of the dualities arising geometrically from different pair-of-pants decompositions of a four punctured sphere  is an $SU(N+2)$ generalization of the Intriligator-Pouliot duality of $SU(2)$ SQCD with $N_f=4$, which is a degenerate, $N=0$, instance of our discussion. 

Paolo Aceto

Knot concordance and homology cobordisms of 3-manifolds 

We introduce the notion of knot concordance for knots in the 3-sphere and discuss some key problems regarding the smooth concordance group. After defining homology cobordisms of 3-manifolds we introduce the integral and rational homology cobordism groups and briefly discuss their relationship with the concordance group. We conclude stating a few recent results and open questions on the structure of these groups.

Contagion maps are a family of maps that map nodes of a network to points in a high-dimensional space, based on the activations times in a threshold contagion on the network. A point cloud that is the image of such a map reflects both the structure underlying the network and the spreading behaviour of the contagion on it. Intuitively, such a point cloud exhibits features of the network's underlying structure if the contagion spreads along that structure, an observation which suggests contagion maps as a viable manifold-learning technique. We test contagion maps as a manifold-learning tool on several different data sets, and compare its performance to that of Isomap, one of the most well-known manifold-learning algorithms. We find that, under certain conditions, contagion maps are able to reliably detect underlying manifold structure in noisy data, when Isomap is prone to noise-induced error. This consolidates contagion maps as a technique for manifold learning. 

Pattern formation emerges during development from the interplay between gene regulatory networks (GRNs) acting at the single cell level and cell movements driving tissue level morphogenetic changes. As a result, the timing of cell specification and the dynamics of morphogenesis must be tightly cross-regulated. In the developing zebrafish, mesoderm progenitors will spend varying amounts of time (from 5 to 10hrs) in the tailbud before entering the pre-somitic mesoderm (PSM) and initiating a stereotypical transcriptional trajectory towards a mesodermal fate. In contrast, when dissociated and placed in vitro, these progenitors differentiate synchronously in around 5 hours. We have used a data-driven mathematical modelling approach to reverse-engineer a GRN that is able to tune the timing of mesodermal differentiation as progenitors leave the tailbud’s signalling environment, which also explains our in vitro observations. This GRN recapitulates pattern formation at the tissue level when modelled on cell tracks obtained from live-imaging a developing PSM. Our “live-modelling” framework also allows us to simulate how perturbations to the GRN affect the emergence of pattern in zebrafish mutants. We are now extending this analysis to cichlid fishes in order to explore the regulation of developmental time in evolution.

Following Grothendieck, periods can be interpreted as numbers arising as coefficients of a comparison isomorphism between two cohomology theories. Due to the influence of the “yoga of motives” these numbers are omnipresent in arithmetic algebraic geometry. The first part of the talk will be a crash course on how to study periods, as well as the action of the motivic Galois group on them, via an elementary category of realizations. In the second part, we will see how one uses this framework to study Feynman integrals -- an interesting family of periods arising in quantum field theory. We will finish with a brief overview of some of the recent work in algebraic geometry inspired by the study of periods arising in physics.

Consider a network of identical phase oscillators with sinusoidal coupling. How likely are the oscillators to globally synchronize, starting from random initial phases? One expects that dense networks have a strong tendency to synchronize and the basin of attraction for the synchronous state to be the whole phase space. But, how dense is dense enough? In this (hopefully) entertaining Zoom talk, we use techniques from numerical linear algebra and computational Algebraic geometry to derive the densest known networks that do not synchronize and the sparsest networks that do. This is joint work with Steven Strogatz and Mike Stillman.

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In the talk, we study the Navier–Stokes-like problems for the flows of homogeneous incompressible fluids. We introduce a new type of boundary condition for the shear stress tensor, which includes an auxiliary stress function and the time derivative of the velocity. The auxiliary stress function serves to relate the normal stress to the slip velocity via rather general maximal monotone graph. In such way, we are able to capture the dynamic response of the fluid on the boundary. Also, the constitutive relation inside the domain is formulated implicitly. The main result is the existence analysis for these problems.

This is a joint work with C.Daw in progress. We study the L_{omega_1,omega}-theory of the modular functions j_n: H -> Y(n). In other words, H is seen here as the universal cover in the class of modular curves. The setting is different from one considered before by Adam Harris and Chris Daw: GL(2,Q) is given here as the sort without naming its individual elements. As usual in the study of 'pseudo-analytic cover structures', the statement of categoricity is equivalent to certain arithmetic conditions, the most challenging of which is to determine the Galois action on CM-points. This turns out to be equivalent to determining the Galois action on SL(2,\hat{Z})/(-1), that is a subgroup of

Out SL(2,\hat{Z})/(-1)   induced by the action of  Gal_Q. We find by direct matrix calculations a subgroup Out_* of the outer automorphisms group which contains the image of Gal_Q. Moreover, we prove that Out_* is the image of Drinfeld's group GT (Grothendieck-Teichmuller group) under a natural homomorphism.

It is a reasonable to conjecture that Out_* is equal to the image of Gal_Q, which would imply the categoricity statement. It follows from the above that our conjecture is a consequence of Drinfeld's conjecture which states that GT is isomorphic to Gal_Q.  

Ideas from physics have predicted a number of important properties of random constraint satisfaction problems such as the satisfiability threshold and the free energy (the exponential growth rate of the number of solutions). Another prediction is the condensation regime where most of the solutions are contained in a small number of clusters and the overlap of two random solutions is concentrated on two points. We establish this phenomena in the random regular NAESAT model. Joint work with Danny Nam and Youngtak Sohn.

We study expectations of powers and correlations for characteristic polynomials of N x N non-Hermitian random matrices. This problem is related to the analysis of planar models (log-gases) where a Gaussian (or other) background measure is perturbed by a finite number of point charges in the plane. I will discuss the critical asymptotics, for example when a point charge collides with the boundary of the support, or when two point charges collide with each other (coalesce) in the bulk. In many of these situations, we are able to express the results in terms of Painlevé transcendents. The application to certain d-fold rotationally invariant models will be discussed. This is joint work with Alfredo Deaño (University of Kent).

What is the connection between phase transitions in statistical physics and the computational tractability of approximate counting and sampling? There are many fascinating answers to this question but many mysteries remain. I will discuss one particular type of a phase transition: the first-order phase in the Potts model on $\mathbb{Z}^d$ for large $q$, and show how tools used to analyze the phase transition can be turned into efficient algorithms at the critical temperature. In the other direction, I'll discuss how the algorithmic perspective can help us understand phase transitions.

Active matter describes ensembles of self-organizing agents, or particles, interacting with their local environments so that their micro-scale behavior determines macro-scale characteristics of the ensemble. While there has been a surge of activity exploring the physics underlying such systems, less attention has been paid to questions of how to program them to achieve desired outcomes. We will present some recent results designing programmable active matter for specific tasks, including aggregation, dispersion, speciation, and locomotion, building on insights from stochastic algorithms and statistical physics.

In the first part of this lecture, I will discuss the proof of convergence of the Lorentz process, in the Boltzmann-Grad limit, to a random process governed by a generalised linear Boltzmann equation. This will hold for general scatterer configurations, including certain types of quasicrystals, and include the previously known cases of periodic and Poisson random scatterer configurations. The second part of the lecture will focus on quantum transport in the periodic Lorentz gas in a combined short-wavelength/Boltzmann-Grad limit, and I will report on some partial progress in this challenging problem. Based on joint work with Andreas Strombergsson (part I) and Jory Griffin (part II).

We study the countable set of rates of growth of a hyperbolic 
group with respect to all its finite generating sets. We prove that the 
set is well-ordered, and that every real number can be the rate of growth 
of at most finitely many generating sets up to automorphism of the group.

We prove that the ordinal of the set of rates of growth is at least $ω^ω$, 
and in case the group is a limit group (e.g., free and surface groups), it 
is $ω^ω$.

We further study the rates of growth of all the finitely generated 
subgroups of a hyperbolic group with respect to all their finite 
generating sets. This set is proved to be well-ordered as well, and every 
real number can be the rate of growth of at most finitely many isomorphism 
classes of finite generating sets of subgroups of a given hyperbolic 
group. Finally, we strengthen our results to include rates of growth of 
all the finite generating sets of all the subsemigroups of a hyperbolic 
group.

Joint work with Koji Fujiwara.

Calibrated geometry, more specifically Calabi-Yau geometry, occupies a modern, rather sophisticated, cross-roads between Riemannian, symplectic and complex geometry. We will show how, stripping this theory down to its fundamental holomorphic backbone and applying ideas from classical complex analysis, one can generate a family of purely holomorphic invariants on any complex manifold. We will then show how to compute them, and describe various situations in which these invariants encode, in an intrinsic fashion, properties not only of the given manifold but also of moduli spaces.

Interest in these topics, if initially lacking, will arise spontaneously during this informal presentation.

I will talk about a novel idea on the Swampland program that uses consistency of what lives on the string probes in gravitational theories. The central charges and the levels of current algebras of 2d CFTs on these strings can be calculated by anomaly inflow mechanism and used to provide constraints on the supergravity theories based on unitarity of the worldsheet CFT. I will show some of the theories with 8 or 16 supersymmetries, which are otherwise consistent looking, belong to the Swampland.