Past

We introduce a novel approach to global optimization  via continuous-time dynamic programming and Hamilton-Jacobi-Bellman (HJB) PDEs. For non-convex, non-smooth objective functions,  we reformulate global optimization as an infinite horizon, optimal asymptotic stabilization control problem. The solution to the associated HJB PDE provides a value function which corresponds to a (quasi)convexification of the original objective.  Using the gradient of the value function, we obtain a  feedback law driving any initial guess towards the global optimizer without requiring derivatives of the original objective. We then demonstrate that this HJB control law can be integrated into other global optimization frameworks to improve its performance and robustness. 

In this talk, I will discuss a conjecture of Penrose, which asserts a lower bound on the mass of a spacetime in terms of the area of a suitable horizon. Whilst Penrose presented a physical motivation for this inequality in the 1970s, the only proofs heavily rely upon PDE arguments, and in particular the use of geometric flows. I hope to show in this talk, through this concrete example (and without unpleasant technical details!), how ideas from geometric PDE theory can be helpful in obtaining results in physics.
 

I will talk about the connections between the Siu inequality and existence of the model companion for GVFs. The talk will be partially based on a joint work with Antoine Sedillot.

The hyperbolic plane and its higher-dimensional analogues are well-known
objects. They belong to a larger class of spaces, called rank-one
symmetric spaces, which include not only the hyperbolic spaces but also
their complex and quaternionic counterparts, and the octonionic
hyperbolic plane. By a result of Pansu, two of these families exhibit
strong rigidity properties with respect to their self-quasiisometries:
any self-quasiisometry of a quaternionic hyperbolic space or the
octonionic hyperbolic plane is at uniformly bounded distance from an
isometry. The goal of this talk is to give an overview of the rank-one
symmetric spaces and the tools used to prove Pansu's rigidity theorem,
such as the subRiemannian structure of their visual boundaries and the
analysis of quasiconformal maps.

This is  a notion we defined with Johan de Jong. If a finitely presented group  is the topological fundamental group of a smooth quasi-projective complex variety, then we prove that it is weakly integral. To this aim we use the Langlands program (both arithmetic to produce companions and geometric to use de Jong’s conjecture). On the other hand there are finitely presented groups which are not weakly integral (Breuillard). So this notion is an obstruction.
 

Self-similar groups are groups of automorphisms of infinite rooted trees obeying a simple but powerful rule. Under this rule, groups with exotic properties can be generated from very basic starting data, most famously the Grigorchuk group which was the first example of a group with intermediate growth.

Nekrashevych introduced a groupoid and a C*-algebra for a self-similar group action on a tree as models for some underlying noncommutative space for the system. Our goal is to compute the K-theory of the C*-algebra and the homology of the groupoid. Our main theorem provides long exact sequences which reduce the problems to group theory. I will demonstrate how to apply this theorem to fully compute homology and K-theory through the example of the Grigorchuk group.

This is joint work with Benjamin Steinberg.

Let G be a free group of rank N, let f be an automorphism of G and let Fix(f) be the corresponding subgroup of fixed points. Bestvina and Handel showed that the rank of Fix(f) is at most N, for which they developed the theory of train track maps on free groups. Different arguments were provided later on by Sela, Paulin and Gaboriau-Levitt-Lustig. In this talk, we present a new proof which involves the Linnell division ring of G. We also discuss how our approach relates to previous ones and how it gives new insight into variations of the problem.

While the brain has long been conceptualized as a network of neurons connected by synapses, attempts to describe the connectome using established models in network science have yielded conflicting outcomes, leaving the architecture of neural networks unresolved. Here, we analyze eight experimentally mapped connectomes, finding that the degree and the strength distribution of the underlying networks cannot be described by random nor scale-free models. Rather, the node degrees and strengths are well approximated by lognormal distributions, whose emergence lacks a mechanistic model in the context of networks. Acknowledging the fact that the brain is a physical network, whose architecture is driven by the spatially extended nature of its neurons, we analytically derive the multiplicative process responsible for the lognormal neuron length distribution, arriving to a series of empirically falsifiable predictions and testable relationships that govern the degree and the strength of individual neurons. The lognormal network characterizing the connectome represents a novel architecture for network science, that bridges critical gaps between neural structure and function, with unique implications for brain dynamics, robustness, and synchronization.

Upper bounds on the number of incidences between points and lines, tubes, and other geometric objects, have many applications in combinatorics and analysis. On the other hand, much less is known about lower bounds. We prove a general lower bound for the number of incidences between points and tubes in the plane under a natural spacing condition. In particular, if you take $n$ points in the unit square and draw a line through each point, then there is a non-trivial point-line pair with distance at most $n^{-2/3+o(1)}$. This quickly implies that any $n$ points in the unit square define a triangle of area at most $n^{-7/6+o(1)}$, giving a new upper bound for the Heilbronn's triangle problem.

Joint work with Alex Cohen and Cosmin Pohoata.

Let X be a n by n unitary matrix, drawn at random according to the Haar measure on U_n, and let m be a natural number. What can be said about the distribution of X^m and its eigenvalues? 

The density of the distribution \tau_m of X^m can be written as a linear combination of irreducible characters of U_n, where the coefficients are the Fourier coefficients of \tau_m. In their seminal work, Diaconis and Shahshahani have shown that for any fixed m, the sequence (tr(X),tr(X^2),...,tr(X^m)) converges, as n goes to infinity, to m independent complex normal random variables (suitably normalized). This can be seen as a statement about the low-dimensional Fourier coefficients of \tau_m. 

In this talk, I will focus on high-dimensional spectral information about \tau_m. For example: 

(a) Can one give sharp estimates on the rate of decay of its Fourier coefficients?

(b) For which values of p, is the density of \tau_m  L^p-integrable? 

Using works of Rains about the distribution of X^m, we will see how Item (a) is equivalent to a branching problem in the representation theory of certain compact homogeneous spaces, and how (b) is equivalent to a geometric problem about the singularities of certain varieties called (Weyl) hyperplane arrangements.

Based on joint works with Julia Gordon and Yotam Hendel and with Nir Avni and Michael Larsen.

Gauging is a systematic way to construct a model with non-invertible symmetry from a model with ordinary group-like symmetry. In 2+1d dimensions or higher, one can generalize the standard gauging procedure by stacking a symmetry-enriched topological order before gauging the symmetry. This generalized gauging procedure allows us to realize a large class of non-invertible symmetries. In this talk, I will describe the generalized gauging of finite group symmetries in 2+1d lattice models. This talk will be based on my ongoing work with L. Bhardwaj, S.-J. Huang, S. Schäfer-Nameki, and A. Tiwari.

A question we get asked all the time! We'll also be discussing the numerous ways our identities as Mathematicians are shaped by being a minority. Free lunch provided.

The Camassa–Holm equation, which is nonlinear one-dimensional nonlinear PDE which is completely integrable and has  applications in several areas, has received considerable attention. We will discuss recent work regarding the Camassa—Holm equation with transport noise, more precisely, the equation $u_t+uu_x+P_x+\sigma u_x \circ dW=0$ and $P-P_{xx}=u^2+u_x^2/2$. În particular, we will show existence of a weak, global, dissipative solution of the Cauchy initial-value problem on the torus.  This is joint work with L. Galimberti (King’s College), K.H. Karlsen (Oslo), and P.H.C. Pang (NTNU/Oslo).

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of $\mathbb{Q}_p^{ab}$, the maximal extension of the $p$-adic numbers $\mathbb{Q}_p$ with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of $\mathbb{Q}_p$) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

I’ll tell you about some of my favorite algebraic varieties, which are beautiful in their own right, and also have some dramatic applications to algebraic combinatorics.  These include the top-heavy conjecture (one of the results for which June Huh was awarded the Fields Medal), as well as non-negativity of Kazhdan—Lusztig polynomials of matroids.

Cost-sensitive loss functions are crucial in many real-world prediction problems, where different types of errors are penalized differently; for example, in medical diagnosis, a false negative prediction can lead to worse consequences than a false positive prediction. However, traditional learning theory has mostly focused on the symmetric zero-one loss, letting cost-sensitive losses largely unaddressed. In this work, we extend the celebrated theory of boosting to incorporate both cost-sensitive and multi-objective losses. Cost-sensitive losses assign costs to the entries of a confusion matrix, and are used to control the sum of prediction errors accounting for the cost of each error type. Multi-objective losses, on the other hand, simultaneously track multiple cost-sensitive losses, and are useful when the goal is to satisfy several criteria at once (e.g., minimizing false positives while keeping false negatives below a critical threshold). We develop a comprehensive theory of cost-sensitive and multi-objective boosting, providing a taxonomy of weak learning guarantees that distinguishes which guarantees are trivial (i.e., can always be achieved), which ones are boostable (i.e., imply strong learning), and which ones are intermediate, implying non-trivial yet not arbitrarily accurate learning. For binary classification, we establish a dichotomy: a weak learning guarantee is either trivial or boostable. In the multiclass setting, we describe a more intricate landscape of intermediate weak learning guarantees. Our characterization relies on a geometric interpretation of boosting, revealing a surprising equivalence between cost-sensitive and multi-objective losses.

It was recently argued that topological operators (at least those associated with continuous symmetries) need regularization. However, such regularization seems to be ill-defined when the underlying QFT is coupled to gravity. If both of these claims are correct, it means that charges cannot be meaningfully measured in the presence of gravity. I will review the evidence supporting these claims as discussed in [arXiv:2411.08858]. Given the audience's high level of expertise, I hope this will spark discussion about whether this is a promising approach to understanding the fate of global symmetries in quantum gravity.