Let \(\Gamma\) be a discrete group. A subgroup \(N\) is called confined if there is a finite set \(F\) in \(\Gamma\) which intersects every conjugate of \(N\) outside the trivial element. For example, a nontrivial normal subgroup is confined. 

A discrete subgroup of a semisimple Lie group is confined if the corresponding locally symmetric orbifold has bounded injectivity radius. We proved a generalization of the celebrated NST:  Let \(\Gamma\) be an irreducible lattice in a higher rank semisimple Lie group G. Let \(N<\Gamma\) be a confined subgroup. Then \(N\) is of finite index. 
                        
The case where \(G\) has Kazhdan's property (T) was established in my joint work with Mikolaj Frakzyc. As in the original NST, without property (T) the problem is considerably harder. The main part is to prove a spectral gap for \(L_2(G/N)\). 
                        
This is a joint work with Uri Bader and Arie Levit.
 

Quantum graphs and their characteristics are intriguing generalizations of notions and tools known from discrete mathematics into the quantum world. Their non-trivial relations with quantum information theory provide a bridge between this branch of mathematics and quantum mechanics. In classical graph theory, there are several characteristics that one can associate with given graphs, e.g., chromatic or clique numbers. The famous problem, solved by Mycielski, was to construct a graph that contains a given graph as a subgraph and can have an arbitrarily large chromatic number, but no larger clique is produced. We propose an analog of the Mycielski transformation and its generalizations in the quantum setting and study how they affect the (quantum) characteristics of quantum graphs. Moreover we study relations between quantum automorphism groups of a quantum graph and its Mycielskian. Based on joint work with A. Bochniak (arXiv:2306.09994) and work in progress with A. Bochniak, P.M. Sołtan, and I. Chełstowski.

This talk will explore the connection between Diophantine approximation and the theory of homogeneous dynamics. The first part of the talk will be used to define and study the minimal denominator function (MDF). We compute the limiting distribution of the MDF as one of its parameters tends to zero. We do this by relating the function to the space of unimodular lattices on the plane.

The second part of the talk will be devoted to describing equivariant processes. This will give a general framework to generalize the main theorem in two directions:

1. Higher dimensional Diophantine approximation

2. Statistics of short saddle connections of Veech surfaces

If time allows, we will compute formulas for the statistics of short holonomy vectors of translation surfaces.

In this talk, we will discuss recent work on the extreme eigenvalues of the sample covariance matrix with a spiked population covariance. Extending previous random matrix theory, we will characterize the spiked eigenvalues outside the bulk distribution and their corresponding eigenvectors for a nonlinear version of the spiked covariance model. Our result shows the universality of the spiked covariance model with the same quantitative spectral properties as a linear spiked covariance model. In the proof, we will present a deterministic equivalent for the Stieltjes transform for any spectral argument separated from the support of the limit spectral measure. Then, we will apply this new result to deep neural network models. We will describe how spiked eigenstructure in the input data propagates through the hidden layers of a neural network with random weights. As a second application, we can study a simple regime where the weight matrix has a rank-one signal component over gradient descent training and characterize the alignment of the target function. This is a joint work with Denny Wu and Zhou Fan.

The classical Coates-Sinnott Conjecture and its refinements predict the dee relationship between the special values of L-functions and the structure of the étale cohomology groups attached to number fields. In this talk, we aim to delve deeper along this direction to propose what we call the “Higher Coates-Sinnott Conjectures" which reveal more information about these two types of important arithmetic objects. We introduce the conjectures we formulate and our work towards them. This is joint work with C. Popescu.

I will discuss a new conceptual framework called sequential commutation that has applications to von Neumann algebra theory. These focus on joint works by the speaker and others including Patchell, Gao and Tan. 

Nature-inspired metaheuristics is widely used in computer science and engineering but seems greatly underused in pharmaceutical research, clinical science research, and somewhat in statistical science as well. This class of algorithms is appealing because they are essentially assumptions free, fast and have been shown that they are capable of tackling all sorts of high dimensional complex optimization problems. We first review optimal design theory, some exemplary nature-inspired metaheuristic algorithms and show how they can be applied to (i) find efficient designs for estimating the Biologically Optimal Dose (BOD), (ii) extend Simon’s 2 stage designs for a Phase II trial with a single alternative hypothesis to one with multiple alternative hypotheses to capture the uncertainty of the efficacy of the drug more accurately, and if time permits  (iii) find a D-optimal designs for  estimating parameters in  10 interacting factors. We also indicate how metaheuristics can be applied to develop more realistic and flexible adaptive designs for early phase clinical trials.