Events of the Week

We propose a debiased inference framework to infer the ranking structure in the contextual Bradley-Terry-Luce (BTL) model. We first adopt a nonparametric maximum likelihood estimation method using ReLU neural networks to estimate unknown preference functions in the model. For the inference of pairwise ranking, we introduce a novel random-walk debiased estimator that efficiently aggregates all accessible estimating scores. In particular, under mild conditions, our debiased estimator yields a tractable distribution, and achieves the semiparametric efficiency bound asymptotically. We further extend our method by incorporating multiplier bootstrap techniques for the uniform inference of ranking structures, and adapting it to accommodate the distributional shift of contextual variables. We provide thorough numerical studies to validate the statistical properties of our method, and showcase its applicability in evaluating large language models based on human preferences under different contexts.

Our presentation explores the evolving role of statisticians in the pharmaceutical industry, particularly within drug development. It introduces the various stages of the drug development process, from pre-clinical trials through Phase IV post-market, highlighting statisticians critical roles during those processes. The presentation aims to dispel common myths about the statistical profession in pharma, encouraging more talented graduates devote their career to the mission of bringing innovative treatments to patients.

This talk aims to introduce two fundamental theorems in algebraic geometry: the Riemann-Roch theorem and Serre duality. I will develop the necessary background and present these theorems in the setting of Riemann surfaces, following the approach in Algebraic Curves and Riemann Surfaces by Rick Miranda. I will conclude my talk by given few applications of these theorems.

Every irreducible polynomial f(x) with integer coefficients corresponds uniquely to a field extension of the rational numbers which consists of Q, a root x of f, and all combinations thereof under the standard arithmetic operations. For example, f(x) = x^2-2 produces the field Q(sqrt{2}) = {a + bsqrt{2} : a, b \in Q}. If f is a polynomial in two or more variables, we can produce infinitely many such fields corresponding to solutions to f=0. For f(x,y) = y^2-x^3-x-1, we have solutions (1, \sqrt{3}), (2, \sqrt{11}), (3, \sqrt{31}) and so on, and so the curve defined by f(x,y)=0 ``generates" the fields Q(\sqrt{3}), Q(\sqrt{11}), and Q(\sqrt{31}).
Recently, Mazur and Rubin suggested using this algebraic information as a means to study the geometric properties of a curve. One can easily ask the reverse question: ``If we know something about a curve C, what can we say about the fields that it generates?" We approach this question through the lens of arithmetic statistics by counting the number of such fields with bounded size---under some appropriate notion of size---for an arbitrary fixed plane curve C. This is joint work in progress with Renee Bell, Robert Lemke Oliver, Allechar Serrano L\'{o}pez, and Tian An Wong.

Title and abstract to be announced

Combinatorics has constantly evolved from the mere counting of classes of objects to the study of their underlying algebraic or analytic properties, such as symmetries or deformations. This was fostered by interactions with in particular statistical physics, where the objects in the class form a statistical ensemble, where each element comes with some probability. Integrable systems form a special subclass: that of systems with sufficiently many symmetries to be amenable to exact solutions.
In this talk, we explore various basic combinatorial problems involving discrete surfaces, dimer models of cluster algebra, or two-dimensional vertex models, whose (discrete or continuum) integrability manifests itself in different manners: commuting operators, conservation laws, flat connections, quantum Yang-Baxter equation, etc. All lead to often simple and beautiful exact solutions.

Homeostasis occurs in a biological system when a chosen output variable remains approximately constant despite changes in an input variable. In this work we specifically focus on biological systems which may be represented as chemical reaction networks and consider their infinitesimal homeostasis, where the derivative of the input-output function is zero. The specific challenge of chemical reaction networks is that they often obey various conservation laws complicating the standard input-output analysis. We derive several results that allow to verify the existence of infinitesimal homeostasis points both in the absence of conservation and under conservation laws where conserved quantities serve as input parameters. In particular, we introduce the notion of infinitesimal concentration robustness, where the output variable remains nearly constant despite fluctuations in the conserved quantities. We provide several examples of chemical networks which illustrate our results both in deterministic and stochastic settings.

Phylogenetics is concerned with uncovering the relationships among organisms (the “Tree of Life”), and statistical computational research has made many important contributions to achieving this goal. In this talk I discuss a major overall challenge facing the field as DNA sequencing efforts have become central to this work: many individual genes have tree topologies that do not match the tree describing relationships among species. Such gene-tree discordance poses many new difficulties for inferring the Tree of Life. Here, I present three approaches for dealing with discordance: 1) a deep-learning method for inferring gene-tree topologies; 2) a quartet summary approach that combines many different gene-tree topologies to construct an accurate species tree, even in the presence of duplication and loss; and 3) a probabilistic approach to reconstructing the history of different traits on a species tree in the presence of discordance. These three problems (and their solutions) represent just a fraction of the challenges now facing the field of phylogenetics.

TBA

Title and abstract to be announced

Leibniz algebras were introduced by Blo(k)h and Loday as non-anticommutative analogues of Lie algebras. Many results for Lie algebras and their modules have been proven to hold for Leibniz algebras and Leibniz bimodules, but there are also several results that are not true in this more general context. In this talk we will define three different notions of tensor products for Leibniz bimodules. The "natural" tensor product of Leibniz bimodules is not always a Leibniz bimodule. In order to fix this, we will introduce the notion of a weak Leibniz bimodule and show that the "natural" tensor product of weak bimodules is again a weak bimodule. Moreover, it turns out that weak Leibniz bimodules are modules over a cocommutative Hopf algebra canonically associated to the Leibniz algebra. Therefore, the category of all weak Leibniz bimodules is symmetric monoidal and the full subcategory of finite-dimensional weak Leibniz bimodules in addition is rigid and pivotal. On the other hand, we introduce two truncated tensor products of Leibniz bimodules which are again Leibniz bimodules. These tensor products induce a non-associative multiplication on the Grothendieck group of the category of finite-dimensional Leibniz bimodules. In particular, we prove that in characteristic zero for a finite-dimensional solvable Leibniz algebra over an algebraically closed field this Grothendieck ring is an alternative power-associative commutative Jordan ring, but for a finite-dimensional non-zero semi-simple Leibniz algebra it is neither alternative nor a Jordan ring. We also expect it not to be power-associative in the semi-simple case, but at the moment we are neither able to prove nor to disprove this.
This is joint work with Friedrich Wagemann from Nantes Universit\'e

Over the complex numbers the solutions to enumerative problems are invariant: the number of solutions of a polynomial equation or polynomial system, the number of lines or curves in a surface, etc. Over the real numbers such invariance fails. However, the signed count of solutions may lead to numerical invariants: Descartes' rule of signs, Poincaré-Hopf theorem, real curve-counting invariants. Since many of this problems have a geometric nature, one may ask the same problems for arbitrary fields.
Motivic homotopy theory allows to do enumerative geometry over an arbitrary base, leading to additional arithmetic and geometric information. The goal of this talk is to illustrate a generalized notion of sign that allows us to state a movitic version of classical theorems like the Bézout theorem, the tropical correspondence theorem and a wall-crossing formula for curve counting invariants for points in quadratic field extensions.

Title and abstract to be announced