Past

We show that given $m$ disjoint chains in the Boolean lattice, we can create $m$ disjoint skipless chains that cover the same elements (where we call a chain skipless if any two consecutive elements differ in size by exactly one). By using this result we are able to answer two conjectures about the asymptotics of induced saturation numbers for the antichain, which are defined as follows. For positive integers $k$ and $n$, a family $\mathcal{F}$ of subsets of $\{1,\dots,n\}$ is $k$-antichain saturated if it does not contain an antichain of size $k$ (as induced subposet), but adding any set to $\mathcal{F}$ creates an antichain of size $k$. We use $\textrm{sat}^{\ast}(n,k)$ to denote the smallest size of such a family. With more work we pinpoint the exact value of $\textrm{sat}^{\ast}(n,k)$, for all $k$ and sufficiently large $n$. Previously, exact values for $\textrm{sat}^{\ast}(n,k)$ were only known for $k$ up to 6. We also show that for any poset $\mathcal{P}$, its induced saturation number (which is defined similar as for the antichain) grows at most polynomially: $\textrm{sat}^{\ast}(n, \mathcal{P})=O(n^c)$, where $c \leq |\mathcal{P}|^2/4+1$. This is based on joint works with Carla Groenland, Maria-Romina Ivan, Hugo Jacob and Tom Johnston.

Let w(x_1,...,x_r) be a word in a free group. For any group G, w induces a word map w:G^r-->G. For example, the commutator word w=xyx^(-1)y^(-1) induces the commutator map. If G is finite, one can ask what is the probability that w(g_1,...,g_r)=e, for a random tuple (g_1,...,g_r) of elements in G.

In the setting of finite simple groups, Larsen and Shalev showed there exists epsilon(w)>0 (depending only on w), such that the probability that w(g_1,...,g_r)=e is smaller than |G|^(-epsilon(w)), whenever G is large enough (depending on w).

In this talk, I will discuss analogous questions for compact groups, with a focus on the family of unitary groups; For example, given r independent Haar-random n by n unitary matrices A_1,...,A_r, what is the probability that w(A_1,...,A_r) is contained in a small ball around the identity matrix?

Based on a joint work with Nir Avni and Michael Larsen.  

I will review some aspects of gravity in asymptotically flat spacetime and mention important challenges to obtain a holographic description in this framework. I will then present two important approaches towards flat space holography, namely Carrollian and celestial holography, and explain how they are related to each other. Similarities and differences between flat and anti-de Sitter spacetimes will be emphasized throughout the talk. 
 

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time. 

Students presenting are:

Sara-Jean Meyer, supervisor Massimiliano Gubinelli

Satoshi Hayakawa, supervisor Harald Oberhauser 

I will present some results from the newly developed theory of wavelets; based on the joint work with the following authors:

P.M. Luthy, H.Šikić, F.Soria, G.L.Weiss, E.N.Wilson.One-DimensionalDyadic Wavelets.Mem. Amer. Math. Soc. 280 (2022), no 1378, ix+152 pp.

About two and a half decades ago and based on the influential book by Fernandez and Weiss, an approach was developed to study wavelets from the point of view of their connections with Fourier analysis. The idea was to study wavelets and other reproducing function systems via the four basic equations that characterized various properties of wavelet systems, like frame and basis properties, completeness, orthogonality, etc. Despite hundreds of research papers and the impressive development of the theory of such systems, some questions remain open even in the basic case of dyadic wavelets on the real line. Among the most thorough treatments that we provide in this lengthy paper is the issue of the understanding of the low-pass filters associated with the MRA structure. In this talk, I will focus on some of these results. As it turned out, a more general and abstract approach to the problem, using the study of dyadic orbits and the newly introduced Tauberian function, reveals several interesting properties and opens an interesting context for some older results

For many spaces of interest to number theorists one can construct cycles which in some ways behave like the coefficients of modular forms. The aim of this talk is to give an introduction to this idea by focusing on examples coming from modular curves and Heegner points and the relevant work of Zagier, Gross-Kohnen-Zagier and Borcherds. If time permits I will discuss generalizations to other spaces.

In this talk, we will present a loop expansion for lattice gauge theories and its application to prove ultraviolet stability in the Abelian Higgs model. We will first describe this loop expansion and how it relates to earlier works of Brydges-Frohlich-Seiler. We will then show how the expansion leads to a quantitative diamagnetic inequality, which in turn implies moment estimates, uniform in the lattice spacing, on the Holder-Besov norm of the gauge field marginal of the Abelian Higgs lattice model. Based on Gauge field marginal of an Abelian Higgs model, which is joint work with Ajay Chandra.

We apply recent ideas in Floer homotopy theory to some questions in symplectic topology. We show that Floer homology can detect smooth structures of certain Lagrangians, as well as using this to find restrictions on symplectic mapping class groups. This is based on joint work-in-progress with Ivan Smith.

TBA

Speaker: Lasse Grimmelt (North Wing)
Title: Modular forms and the twin prime conjecture

Abstract: Modular forms are one of the most fruitful areas in modern number theory. They play a central part in Wiles proof of Fermat's last theorem and in Langland's far reaching vision. Curiously, some of our best approximations to the twin-prime conjecture are also powered by them. In the existing literature this connection is highly technical and difficult to approach. In work in progress on this types of questions, my coauthor and I found a different perspective based on a quite simple idea. In this way we get an easy explanation and good intuition why such a connection should exists. I will explain this in this talk.

Speaker: Yang Liu (South Wing)
Title: Obtaining Pseudo-inverse Solutions With MINRES

Abstract: The celebrated minimum residual method (MINRES) has seen great success and wide-spread use in solving linear least-squared problems involving Hermitian matrices, with further extensions to complex symmetric settings. Unless the system is consistent whereby the right-hand side vector lies in the range of the matrix, MINRES is not guaranteed to obtain the pseudo-inverse solution. We propose a novel and remarkably simple lifting strategy that seamlessly integrates with the final MINRES iteration, enabling us to obtain the minimum norm solution with negligible additional computational costs. We also study our lifting strategy in a diverse range of settings encompassing Hermitian and complex symmetric systems as well as those with semi-definite preconditioners.

Topological data analysis (TDA) is an emerging field of research, which considers the application of topology to data analysis. Recently, these methods have been successfully applied to research problems in the field of geographical information science (GIS). This includes the problems of Point of Interest (PoI), street network and weather analysis. In this talk I will describe how TDA can be used to provide solutions to these problems plus how these solutions compare to those traditionally used by GIS practitioners. I will also describe some of the challenges of performing interdisciplinary research when applying TDA methods to different types of data.

Chronic hepatitis B virus (HBV) infection leads to liver damage that increases the risk of hepatocellular carcinoma and liver cirrhosis. Individuals with chronic HBV infection are often either treated with interferon alpha or nucleoside reverse transcriptase inhibitors (NTRL). While these NTRLs inhibit de novo DNA synthesis, they do not represent a functional cure for chronic HBV infection and so must be taken indefinitely. The resulting life-long treatment leads to an increased risk of selection for treatment resistant strains of HBV. Consequently, there is increased interest in a novel treatment modality, capsid protein allosteric modulators (CPAMs), that blocks a crucial step in the viral life cycle. I'll discuss recent work that identifies HBV serum RNA as an informative biomarker of CPAM treatment efficacy, evaluates CPAMs as a potential functional cure for HBV infection, and illustrates the role of mechanistic modelling in trial design using an age structured, multi-scale mathematical model. 

I will show that the massless irreducible representations of the Poincare group are precisely Corrolian field living on I^+. I will also show that the analogous massless irreducible representation of E11 are just the degrees of freedom of maximal supergravity. Finally I will speculate how spacetime could emerge from an underlying fundamental theory.

The affine and periodic Temperley–Lieb algebras are families of infinite-dimensional algebras with a diagrammatic presentation. They have been studied in the last 30 years, mostly for their physical applications in statistical mechanics, where the diagrammatic presentation encodes the connectivity property of the models. Most of the relevant representations for physics are finite-dimensional. In this work, we define finite-dimensional quotients of these algebras, which we name uncoiled algebras in reference to the diagrammatic interpretation of the quotient, and construct a family of Wenzl–Jones idempotents, each of which projects onto one of the one-dimensional modules these algebras admit. We also prove that the uncoiled algebras are sandwich cellular and sketch some of the applications of the objects we defined. This is joint work with Alexi Morin-Duchesne.

This seminar is part of our Frontiers in Quantitative Finance. Attendance is free of charge but requires prior online registration.

Abstract
Empirical studies consistently find that the price impact of large trades approximately follows a nonlinear power law. Yet, tractable formulas for the portfolios that trade off predictive trading signals, risk, and trading costs in an optimal manner are only available for quadratic costs corresponding to linear price impact. In this paper, we show that the resulting linear strategies allow to achieve virtually optimal performance also for realistic nonlinear price impact, if the “effective” quadratic cost parameter is chosen appropriately. To wit, for a wide range of risk levels, this leads to performance losses below 2% compared to the numerical Viterbi algorithm of Kolm and Ritter (2014) run at very high accuracy. The effective quadratic cost depends on the portfolio risk, but can be computed without any sophisticated numerics by simply maximizing an explicit scalar function.
Read more on this work here.

Habegger showed that a subvariety of a fibre power of the Legendre family of elliptic curves contains a Zariski-dense set of special points if and only if it is special. I'll explain this result, and discuss an effective version that Gal Binyamini, Harry Schmidt, Margaret Thomas and I proved.

We use the term shape optimization when we want to find a minimizer of an objective function that assigns real values to shapes of domains. Solving shape optimization problems can be quite challenging, especially when the objective function is constrained to a PDE, in the sense that evaluating the objective function for a given domain shape requires first solving a boundary value problem stated on that domain. The main challenge here is that shape optimization methods must employ numerical methods capable of solving a boundary value problem on a domain that changes after each iteration of the optimization algorithm.

The first part of this talk will provide a gentle introduction to shape optimization. The second part of this talk will highlight how the finite element framework leads to automated numerical shape optimization methods, as realized in the open-source library fireshape. The talk will conclude with a brief overview of some academic and industrial applications of shape optimization.

In order for artificial neural networks to learn a task, one must solve an inverse design problem. What network will produce the desired output? We have harnessed AI approaches to design physical systems to perform functions inspired by biology, such as protein allostery. But artificial neural networks require a computer in order to learn in top-down fashion by the global process of gradient descent on a cost function. By contrast, the brain learns by local rules on its own, with each neuron adjusting itself and its synapses without knowing what all the other neurons are doing, and without the aid of an external computer. But the brain is not the only biological system that learns by local rules; I will argue that the actin cortex and the amnioserosa during the dorsal closure stage of Drosophila development can also be viewed this way.

The Zilber-Pink Conjecture, which should rule the behaviour of intersections between an algebraic variety and a countable family of "special varieties", does not take into account double intersections; some results related to tangencies with special subvarieties have been obtained by Marché-Maurin in 2014 in the case of powers of the multiplicative group and by Corvaja-Demeio-Masser-Zannier in 2019 in the case of elliptic schemes. We prove that any algebraic curve contained in Y(1)^2 is tangent to finitely many modular curves, which are the one-codimensional special subvarieties. The proof uses the Pila-Zannier strategy: the Pila-Wilkie counting theorem is combined with a degree bound coming from a Weakly Bounded Height estimate. The seminar will be divided into two talks: in the first one, we will explain the general Zilber-Pink Conjecture philosophy, we will describe the main tools used in this context and we will see what the differences in the double intersection case are; in the second one, we will focus on the proofs and we will see how o-minimality plays a main role here. In the case of a curve in Y(1)^2, o-minimality is also used for height estimates (which are then ineffective, which is usually not the case).