Abstract: We classify all possible configurations of vectors in three-dimensional space with the property that any two of the vectors form a rational angle (measured in degrees).
As a corollary, we find all tetrahedra whose six dihedral angles are all rational. While these questions (and their answers) are of an elementary nature, their resolution will take us on a tour through cyclotomic number fields, computational algebraic geometry, and an amazing fact about the geometry of tetrahedra discovered by physicists in the 1960s.
Joint work with Sasha Kolpakov, Bjorn Poonen, and Michael Rubinstein.
About the speaker: Kiran S. Kedlaya is a Professor of Mathematics at the University of California San Diego. He received his Ph.D. from the Massachusetts Institute of Technology in 2000. Before joining UC San Diego, he held positions at the Institute for Advanced Study, the University of California Berkeley, and the faculty of the Massachusetts Institute of Technology. His research is in number theory and arithmetic geometry, with a focus on p-adic methods. He was an invited speaker at the International Congress of Mathematicians (ICM 2010) in Hyderabad.
More about him: https://kskedlaya.org/
https://en.wikipedia.org/wiki/Kiran_Kedlaya
Abstract: Topological Data Analysis (TDA) offers a geometric and algebraic lens for understanding the shape of complex data. A central tool in this framework is persistent homology, which associates to a dataset a graded family of homology groups that record how topological features—such as connected components, loops, and higher-dimensional cavities—appear and disappear across scales. These invariants can be organized as graded modules over polynomial rings, and many questions in persistence theory reduce to understanding these modules up to isomorphism. This perspective connects data analysis to familiar concepts from algebraic topology (such as Betti numbers) and to the structure theory of finitely generated modules.
In this talk, we will introduce the basic ideas of persistent homology, emphasizing how classical algebraic constructions translate into tools for real-world data exploration. We will describe the algebraic classification of persistence modules, explain the role of barcodes and persistence diagrams, and discuss key theoretical results such as the stability theorem, which ensures robustness under noise. Along the way, we will highlight concrete applications in medicine, illustrating how abstract algebraic techniques can illuminate the geometry hidden in high-dimensional data.
About the speaker: Prof. Thomas Brüstle is a leading expert in Representation Theory with over 1800 citations. He is an active member of the prestigious Centre de Recherches Mathématiques (CRM) in Montréal, and currently holds Maurice Auslander Chair at Bishop’s University. He has successfully led efforts to bring together Topological Data Analysis (TDA) and Representation Theory communities through publications, organization of BIRS, ISM and upcoming Lorentz center workshops on this theme, as well as through collaborations with Quantum Institute (Sherbrooke) on Quantum TDA, and with Optina Diagnostics (Montreal) on TDA and ML methods applied to medical imaging.
Abstract: Given a metric space (X, d), there are several notions of it being negatively curved. In this talk, we single out a consequence of negative curvature in Riemannian manifolds that can be adapted to the complex-analytic setting. This adaptation turns out to be very useful in proving results about holomorphic maps. We shall illustrate what this means by sketching a new proof of a classical result called the Wolff-Denjoy theorem. (This theorem says that given a holomorphic self-map f of the open unit disc, exactly one of the following holds true: either f has a fixed point in the open unit disc or there exists a point p on the unit circle such that ALL orbits under the successive iterates of f approach p.) This talk is intended to be accessible to a broad mathematical audience. In particular, no knowledge of curvature will be assumed -- the notion of negative curvature will only serve as an analogy for the specific concepts that we shall see. Indeed: for most of this talk, we shall focus on metric spaces and basic complex analysis. Towards the end, I shall present a very brief overview of some recent research that relies on the above perspective on negative curvature.
About the speaker: Prof. Gautam Bharali is a mathematician working in the field of Several Complex Variables. He obtained his Ph.D. in Mathematics from the University of Wisconsin–Madison in 2002 and is an alumnus of IIT Kanpur, where he completed his Integrated M.Sc. in Mathematics in 1997.
He was awarded the Swarnajayanti Fellowship in 2015 and is a Fellow of the Indian Academy of Sciences.
Abstract: For a finite group $G$, the number of irreducible complex characters of $G$ is equal to the number of conjugacy classes of $G$. When $G$ is a finite group of Lie type, (say $G={\rm GL}_n(\mathbb{F}_q)$), the conjugacy classes of $G$ are parametrized by Jordan decomposition. The works of J.A. Green or Deligne--Lusztig construct, for a given conjugacy class in $C$ of $G$, a complex irreducible character $\chi_C:G\rightarrow \mathbb{C}$ such that the values of $\chi_C$ are explicitly related to $C$.
A conjugacy class in ${\rm GL}_n(\mathbb{F}_q)$ gives a unique conjugacy class in ${\rm GL}_n(\mathbb{F}_{q^m})$ (a simple fact from linear algebra). The analogue of this map of conjugacy classes on the spectral side, i.e., the corresponding map from the set of irreducible complex representations of ${\rm GL}_n(\mathbb{F}_q)$ to the set of irreducible complex representations of ${\rm GL}_n(\mathbb{F}_{q^m})$ is called Shintani lifting.
After reviewing some of these basic results, we will discuss a curious congruence relation between character values of a representation of ${\rm GL}_n(\mathbb{F}_q)$ and the character values of its Shintani lifting to ${\rm GL}_n(\mathbb{F}_{q^\ell})$ mod-$(\ell)$ where $\ell$ is a prime number with $(\ell, q)=1$. I will indicate some connections with local Langlands correspondence and the conjectures of D.Treumann and A. Venkatesh.
About the speaker: Prof. Nadimpalli is a mathematician working in the field of Representation theory. He obtained his Ph.D. in Mathematics from University of Leiden and University of Paris Sud. in 2015.
More about him: https://home.iitk.ac.in/~nsantosh/
Abstract: Mathematicians have studied the interpolation problem for a long time. A well-known version, going back to Euclid himself, asks the following: given a finite set of points in the euclidean plane, is there a polynomial of prescribed degree in two variables having them as roots? A variant which is of interest to us requires the polynomial to vanish at the given points up to a certain “multiplicity”. We will first introduce this problem by looking at some examples and then discuss some techniques to study the problem. Later we will talk about connections with interesting recent work in commutative algebra and algebraic geometry, beginning with the famous “Nagata Conjecture”.
Abstract: A principled approach to cyclicality and intransitivity in paired comparison data is developed.
The proposed methodology enables more precise estimation of the underlying preference profile and facilitates the identification of all cyclic patterns and potential intransitivities.
Consequently, it improves upon existing methods for ranking and prediction, including enhanced performance in betting and wagering systems. Fundamental to our development is a detailed understanding and study of the parameter space that accommodates cyclicality and intransitivity. It is shown that identifying cyclicality and intransitivity reduces to a model selection problem, and a new method for model selection employing geometrical insights, unique to the problem at hand, is proposed. The large sample properties of the estimators and guarantees on the selected model are provided. Thus, it is shown that in large samples all cyclical relations and consequent intransitivities can be identified.
The method is exemplified using simulations and analysis of an illustrative example.
Abstract: We extend the results of Serre on non-lacunarity of prime and integral coefficients of L-functions attached to l-adic Galois representations to linear combinations of such L-functions. One application is to Fourier coefficients of L-functions attached to modular forms.
Another application is to the study of rational points of algebraic varieties defined over number fields: here we recover some of Serre's results proved in his lectures N_X(p). This is ongoing joint work with Rishabh Agnihotri and Mihir Sheth.
Abstract: One method to understand compact finite dimensional complexes is to break it up into simpler pieces. In this context, one of the most useful results is that up to homotopy, these can be built up from disks. Stronger decomposition results can be proved when one considers either the loop space or a suspension. In this talk, I will discuss some of these results and some geometric consequences of them.
Abstract: We consider the family of cyclic coverings of the projective line with fixed ramification data, and degree. The monodromy representation associated to these data is closely related to the "Gassner representation" of the pure braid group, and we characterise the cases when the monodromy group is a product of non-uniform higher rank arithmetic groups.
Abstract: With the advent of modern technologies, it is increasingly common to deal with data of multi-dimensions in various scientific fields of study. In this context Gaussian Process is a very attractive approach of modeling such data. In this presentation, we first develop a generative Gaussian Process and then apply to spatial statistical concept to explain a curvature process. Analyzing the resulting random surface provides deeper insights into the nature of latent dependence within the studied response. We develop Bayesian modeling and inference for rapid changes on the response surface to assess directional curvature along a given trajectory. Such trajectories or curves of rapid change, often referred to as _wombling_ boundaries, occur in geographic space in the form of rivers in a flood plain, roads, mountains or plateaus or other topographic features leading to high gradients on the response surface. We demonstrate fully model based Bayesian inference on directional curvature processes to analyze differential behavior in responses along wombling boundaries. We illustrate our methodology with a number of simulated experiments followed by multiple applications featuring the Boston Housing data; Meuse river data; and temperature data from the Northeastern United States.
Abstract: A partial order on the set of reals is locally countable if the set of predecessors of every real is countable. A long-standing problem of Sacks asks if every such partial order embeds into the Turing degrees. He showed that the answer is yes under the continuum hypothesis but the problem remains open (in ZFC) even for partial orders of finite height. It turns out that to be able to embed such posets, one would need to construct large Turing independent set of reals with some additional features. This has recently motivated developing new methods for constructing Turing independent sets. Many of these involve set-theoretic ideas like forcing. We will discuss these and several related recent results.
Abstract: We consider two examples of statistical inference for two related populations. In one example we characterize two patient populations that are relevant in the construction of a clinical study design, and propose a method to adjust for detected differences.
The second example is about comparative immune profiling under two biologic conditions of interest when we identify shared versus condition-specific homogeneous cell subpopulations.
Bayesian inference in both applications requires prior probability models for two or more related distributions. We build on extensive literature on such models based on Dirichlet process priors. Models are commonly known as dependent Dirichlet processes (DDP), with many variations and extensions beyond the Dirichlet process model.
The special feature in the two motivating applications is the focus on differences in the heterogeneity of the related populations, with one application aiming to adjust for such differences, and the other application aiming to identify and understand immune cell subtypes that are characteristic for one or the other condition.
We briefly review the extensive literature on DDP models and then introduce variations of DDP priors suitable for these inference goals. In both applications the underlying model structures are common atoms mixture models with highly structured priors on the weights.