The study of convergence of random walks to well defined curves is founded in the fields of complex analysis, probability theory, physics and combinatorics. The foundations of this subject were motivated by physicists interested in the properties of one-dimensional models that represented some form of physical phenomenon. By taking physical models and generalizing them into abstract mathematical terms, macroscopic properties about the model could be determined from the microscopic level. By using model specific objects known as observables, the convergence of the random walks on particular lattice structures can be proven to converge to continuous curves such as Brownian Motion or Stochastic Loewner Evolution as the size of the lattice step approaches 0. This seminar will introduce the field of statistical lattice models, the types of observables that can be used to prove convergence as well as a proof for the q-state Potts model showing that local non-commutative matrix observables do not exist. No prior physics knowledge is required for this seminar.

# An introduction to statistical lattice models and observables

# A least-squares monte carlo approach to the calculation of capital requirements

The calculation of capital requirements for financial institutions usually entails a reevaluation of the company’s assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. Relying on a well-known method for pricing non-European derivatives, the current paper proposes and analyzes a novel approach to this computational problem based on least-squares regression and Monte Carlo simulations. We study convergence of the algorithm and analyze the resulting estimate for practically important risk measures. Moreover, we address the problem of how to choose the regressors, and show that an optimal choice is given by the left singular functions of the corresponding valuation operator. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs, particularly when relying on the optimal basis functions.

Joint work with Hongjun Ha (Georgia State University)

# Keeping your distance is hard

We will look at the computational complexity of deciding who wins from a given position in specific two-player games. The players alternately color the vertices of a given graph with red or blue, subject to distance conditions. One example is the game of COL, where adjacent vertices cannot be colored with the same color. In general graph distance games, two sets describe at which distances like or different colors are not allowed.

Using the fact that some members of this family, namely COL, SNORT, and NODEKAYLES, are PSPACE-hard, we can show that a large number of graph distance games are also PSPACE-hard. The proof uses the insertion of a subgraph that creates a bijection between the positions of a game with known computational complexity and a game whose complexity is to be determined. This talk does not require prior knowledge of combinatorial games or computational complexity. Come prepared to play! (Joint work with Kyle Burke, Melissa Huggan, and Svenja Huntemann.)

# Solyanik Estimates in Harmonic Analysis

Let $mathcal{B}$ be a collection of open sets in $mathbb{R}^n$. Associated to $mathcal{B}$ is the geometric maximal operator $M_{mathcal{B}}$ defined by $$M_{mathcal{B}}f(x) = sup_{x in R in mathcal{B}}int_R|f|;.$$ For $0 < alpha < 1$, the associated emph{Tauberian constant} $C_{mathcal{B}}(alpha)$ is given by $$C_{mathcal{B}}(alpha) = sup_{E subset mathbb{R}^n : 0 < |E| < infty} rac{1}{|E|}|{x in mathbb{R}^n : M_{mathcal{B}}chi_E(x) > alpha}|;.$$ A maximal operator $M_mathcal{B}$ such that $lim_{alpha ightarrow 1^-}C_{mathcal{B}}(alpha) = 1$ is said to satisfy a emph{Solyanik estimate}. In this talk we will prove that the uncentered Hardy-Littlewood maximal operator satisfies a Solyanik estimate. Moreover, we will indicate applications of Solyanik estimates to smoothness properties of Tauberian constants and to weighted norm inequalities. We will also discuss several fascinating open problems regarding Solyanik estimates.

This research is joint with Ioannis Parissis.

# Solyanik Estimates in Harmonic Analysis

Let $mathcal{B}$ be a collection of open sets in $mathbb{R}^n$. Associated to $mathcal{B}$ is the geometric maximal operator $M_{mathcal{B}}$ defined by $$M_{mathcal{B}}f(x) = sup_{x in R in mathcal{B}}int_R|f|;.$$ For $0 < alpha < 1$, the associated emph{Tauberian constant} $C_{mathcal{B}}(alpha)$ is given by $$C_{mathcal{B}}(alpha) = sup_{E subset mathbb{R}^n : 0 < |E| < infty} rac{1}{|E|}|{x in mathbb{R}^n : M_{mathcal{B}}chi_E(x) > alpha}|;.$$ A maximal operator $M_mathcal{B}$ such that $lim_{alpha ightarrow 1^-}C_{mathcal{B}}(alpha) = 1$ is said to satisfy a emph{Solyanik estimate}. In this talk we will prove that the uncentered Hardy-Littlewood maximal operator satisfies a Solyanik estimate. Moreover, we will indicate applications of Solyanik estimates to smoothness properties of Tauberian constants and to weighted norm inequalities. We will also discuss several fascinating open problems regarding Solyanik estimates.

This research is joint with Ioannis Parissis.Let $mathcal{B}$ be a collection of open sets in $mathbb{R}^n$. Associated to $mathcal{B}$ is the geometric maximal operator $M_{mathcal{B}}$ defined by $$M_{mathcal{B}}f(x) = sup_{x in R in mathcal{B}}int_R|f|;.$$ For $0 < alpha < 1$, the associated emph{Tauberian constant} $C_{mathcal{B}}(alpha)$ is given by $$C_{mathcal{B}}(alpha) = sup_{E subset mathbb{R}^n : 0 < |E| < infty} rac{1}{|E|}|{x in mathbb{R}^n : M_{mathcal{B}}chi_E(x) > alpha}|;.$$ A maximal operator $M_mathcal{B}$ such that $lim_{alpha ightarrow 1^-}C_{mathcal{B}}(alpha) = 1$ is said to satisfy a emph{Solyanik estimate}. In this talk we will prove that the uncentered Hardy-Littlewood maximal operator satisfies a Solyanik estimate. Moreover, we will indicate applications of Solyanik estimates to smoothness properties of Tauberian constants and to weighted norm inequalities. We will also discuss several fascinating open problems regarding Solyanik estimates.

This research is joint with Ioannis Parissis.

# Box counting

There are both classical and modern theorems expressing certain sums of 2- and 3-dimensional partitions as beautiful infinite products. I will explain these formulas and ideas behind their proof.

# Curvature Flow, Allen-Cahn and the Spatial Lambda-Fleming-Viot

Hybrid zones are interfaces between populations which occur when two species interbreed, but the hybrids have a lower evolutionary fitness. We can model this situation using the spatial Lambda-Fleming-Viot process (SLFV), and study the behaviour using a dual process of branching and coalescing random walks. We use a duality relation with a Branching Brownian motion to give a probabilistic proof of a PDE result (originally proved by Chen) that in solutions to an Allen-Cahn equation, an interface forms which moves approximately according to curvature flow. Our proof of Chen's result is flexible enough that we can also apply it to the SLFV dual to prove that the hybrid zone evolves approximately according to curvature flow. Joint work with Alison Etheridge and Nic Freeman.

Joint seminar with Mathematical Physics. Followed at 16h00 by a lecture of Andrei Okounkov, in the same location.

# 3-body quantum Coulomb problem: where we are

*Current status of 3-body Coulomb problem (negative hydrogen ion H-, helium atom, lithium ion, H2+ ion etc) will be described.Major emphasis will be given to a question of stability vs Coulomb charge and to analytic structure of the ground state energy. The existence of two critical charges with associated square-root and essential singularities, respectively, is predicted. The 2nd excited, weakly-bound state of H- is predicted as a result.**Current status of 3-body Coulomb problem (negative hydrogen ion H-, helium atom, lithium ion, H2+ ion etc) will be described.Major emphasis will be given to a question of stability vs Coulomb charge and to analytic structure of the ground state energy. The existence of two critical charges with associated square-root and essential singularities, respectively, is predicted. The 2nd excited, weakly-bound state of H- is predicted as a result.**Current status of 3-body Coulomb problem (negative hydrogen ion H-, helium atom, lithium ion, H2+ ion etc) will be described.Major emphasis will be given to a question of stability vs Coulomb charge and to analytic structure of the ground state energy. The existence of two critical charges with associated square-root and essential singularities, respectively, is predicted. The 2nd excited, weakly-bound state of H- is predicted as a result.*

# Interactions of forward- and backward-time isochrons

*In the 1970s Winfree introduced the concept of an isochron as the set of all points in the basin of an attracting periodic orbit that converge to the periodic orbit in forward time with the same asymptotic phase. It has been observed that in slow-fast systems, such as the FitzHugh-Nagumo model, the isochrons of such systems can have complicated geometric features; in particular, regions with high curvature that are related to sensitivity in the system. In order to understand where these features come from, we introduce backward-time isochrons that exist in the basin of a repelling periodic orbit, and we consider their interactions with the forward-time isochrons. We show that a cubic tangency between the two sets of isochrons is responsible for creating high curvature features. This study makes use of a boundary value problem formulation to compute isochrons accurately as parametrised curves.*

# Cluster analysis of genetic sequence data via the Gap Procedure

Phylogenetic clustering typically involves estimating a phylogenetic tree and identifying groups of sequences having small genetic pairwise distances and sufficiently high clade support (either bootstrap or posterior probabilities). In this talk, we explore a simple distance-based clustering algorithm, called the Gap Procedure, which uses gaps in sorted pairwise distances to suggest a natural divide between group members and non-members. We show that the clusters found using the Gap Procedure agree closely with computationally expensive gold standard techniques on well separated groups of HIV DNA sequence data. Simulation studies are also presented to illustrate the scenarios in which this fast and easy to implement algorithm may be employed, and more importantly, when more sophisticated methods are required