Events

Discovering new physics is harder than finding a needle in a haystack — it not only requires a huge amount of data but also statistical tools to evaluate these properly and the physics knowledge on what to look for. In my talk, I will be introducing the basic intuition that goes behind the statistics of discovering new physics, what „discovering new physics“ means and also more recent developments on how to do it without knowing what „new physics“ might look like with the use of machine learning.

More information: https://zucmap.ethz.ch/call_made

We present the proof for Energy Identity for stationary harmonic maps. In particular, given a sequence of stationary harmonic maps weakly converging to a limit with a defect measure for the energy, then m-2 almost everywhere on the support of this measure the density is the sum of energy of bubbles. This is equivalent to saying that annular regions (or neck regions) do not contribute to the energy of the limit. This result is obtained via a quantitative analysis of the energy in annular regions for a fixed stationary harmonic map. The proof is technically involved, but it will be presented in simplified cases to try and convey the main ideas behind it.

In this lecture, we will discuss recent results from our group that explore the relationship between control theory and machinevlearning, specifically supervised learning and universal approximation. We will take a novel approach by considering the simultaneous control of systems of Residual Neural Networks (ResNets). Each item to be classified corresponds to a different initial datum for the ResNet's Cauchy problem, resulting in an ensemble of solutions to be guided to their respective targets using the same control. We will introduce a nonlinear and constructive method that demonstrates the attainability of this ambitious goal, while also estimating the complexity of the control strategies. This achievement is uncommon in classical dynamical systems in mechanics, and it is largely due to the highly nonlinear nature of the activation function that governs the ResNet dynamics. This perspective opens up new possibilities for developing hybrid mechanics-data driven modeling methodologies. Throughout the lecture, we will also address some challenging open problems in this area, providing an overview of the exciting potential for further research and development.

I will talk about a new strategy of studying discrete subgroup of a higher rank semisimple Lie group using the high and low entropy methods for actions of higher rank abelian groups. They were developed by Einsiedler-Katok-Lindenstrauss to study A-invariant measures on (typically finite volume) quotients of semisimple Lie groups. The properties of such measures can reveal a lot about the geometry of the quotient, wich in turn makes them a useful tool in studying (a priori) infinite covolume subgroups of semisimple Lie groups. Based on a joint work with Minju Lee.

We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining the main result with Matomaki-Radziwill. (This is joint work with M. Radziwill.) For instance: for lambda the Liouville function (that is, the completely multiplicative function with lambda(p) = -1 for every prime), (1/\log x) \sum_{n\leq x} lambda(n) \lambda(n+1)/n = O(1/sqrt(log log x)), which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that lambda(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Omega(n)=k, for any "popular" value of k (that is, k = log log N + O(sqrt(log log N)) for n<=N). We shall also discuss a recent generalization by C. Pilatte, who has succeeded in proving that a graph with edges that are rough integers, rather than primes, also has a strong local expander property almost everywhere, following the same strategy. As a result, he has obtained a bound with O(1/(log x)^c) instead of O(1/sqrt(log log x)) in the above, as well as other improvements in consequences across the board.

We will discuss a graph that encodes the divisibility properties of integers by primes. We will prove that this graph has a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier, by combining the main result with Matomaki-Radziwill. (This is joint work with M. Radziwill.) For instance: for lambda the Liouville function (that is, the completely multiplicative function with lambda(p) = -1 for every prime), (1/\log x) \sum_{n\leq x} lambda(n) \lambda(n+1)/n = O(1/sqrt(log log x)), which is stronger than well-known results by Tao and Tao-Teravainen. We also manage to prove, for example, that lambda(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Omega(n)=k, for any "popular" value of k (that is, k = log log N + O(sqrt(log log N)) for n<=N). We shall also discuss a recent generalization by C. Pilatte, who has succeeded in proving that a graph with edges that are rough integers, rather than primes, also has a strong local expander property almost everywhere, following the same strategy. As a result, he has obtained a bound with O(1/(log x)^c) instead of O(1/sqrt(log log x)) in the above, as well as other improvements in consequences across the board.

In this talk, we will consider a stabilized version of the fundamental existence problem of symplectic structures (McDuff--Salamon, Problem 1). Given a formal symplectic manifold, i.e. a closed manifold M with a non-degenerate 2-form and a non-degenerate second cohomology class, we investigate when its natural stabilization to M x T^2 can be realized by a symplectic form. We show that this can be done whenever the formal symplectic manifold admits a symplectic divisor. It follows that the product with T^2 of an almost symplectic blow up admits a symplectic form. Another corollary is that if a formal symplectic 4-manifold M, which either satisfies that its positive/negative second betti numbers are both at least 2, or that is simply connected, then M x T^2 is symplectic. For instance (CP^2#CP^2#CP^2)xT^2 is symplectic, even though CP^2#CP^2#CP^2 is not, by work of Taubes. These results follow from a stabilized version of Eliashberg--Murphy's h-principle for symplectic cobordisms, which makes no assumptions on overtwistedness at the boundary. This is joint work with Fabio Gironella, Fran Presas, Lauran Toussaint.

Computer graphics has been developing since the advent of first displays in the 1950s. Nowadays we're enjoying the results of this research daily through medical imaging, computer-generated imagery in movies and computer games, through interacting with objects designed with computer-aided design, etc. The aim of this talk is to demonstrate applications of geometry in different areas of computer graphics: mesh acquisition, geometry processing, and rendering.

We consider one dimensional nonlinear Schrodinger equations around a traveling wave. We prove its asymptotic stability for general nonlinearities, under the hypotheses that the orbital stability condition of Grillakis-Shatah-Strauss is satisfied and that the linearized operator does not have a resonance and only has 0 as an eigenvalue. As a by-product of our approach, we show long-range scattering for the radiation remainder. Our proof combines for the first time modulation techniques and the study of space-time resonances. We rely on the use of the distorted Fourier transform, akin to the work of Buslaev and Perelman and, and of Krieger and Schlag, and on precise renormalizations, computations and estimates of space-time resonances to handle its interaction with the soliton. This is joint work with Pierre Germain.

We consider the Kyle model in continuous time, where the informed traders face additional frictions. These frictions may arise due to difficulties in executing large portfolio, or legal penalties in case the informed trader is trading illegally on inside information. The equilibrium is characterised via the solution of a backward stochastic differential equation (BSDE) whose terminal condition is determined as the fixed point of a non-linear operator in equilibrium. A curious connection between the terminal condition of this BSDE and an entropic optimal transport problem appears in equilibrium. We find that informed traders consistently trade a constant multiple of the difference between the fundamental value and their anticipated market price just before their private information is disclosed to the public, reminiscent of the behaviour of a large trader in an Almgren-Chris model. If time permits, as an application to insider trading regulation, we also consider a regulator’s challenge of balancing price informativeness with minimising losses for uninformed traders, given a budget constraint. High legal penalties deter illegal trading and protect uninformed traders but make prices less efficient. An optimal policy suggests that if the cost of investigation exceeds its benefits, it’s best not to investigate.

The quadratic Reciprocity Law for the Legendre or Jacobi-Symbol forms the starting point of all Reciprocity Laws as well as of class field theory. It is closely related to the product formula of the quadratic Hilbert-Symbol over local fields. Various mathematicians have established higher explicit formulae to compute higher Hilbert-Symbols. Analogs were found for formal (Lubin-Tate) groups. Eventually Perrin-Riou has formulated a Reciprocity Law, which allows the explicit computation of local cup product pairings by means of Iwasawa- and p-adic Hodge Theory. In this talk I shall try to give an overview of these topics, at the end I will explain recent developments in this regard.

In the talk I will explain that thanks to methods from A^1-homotopy theory, more precisely recent work of Kass-Levine-Solomon-Wickelgren, it now makes sense to perform weighted counts of plane rational curves of degree d through a point configuration of 3d-1 points over an arbitrary field k, and these curve counts are valued in the Grothendieck-Witt ring GW(k) of quadratic forms over the field k, generalizing Welschinger's invariants for k the real numbers. I will present tropical correspondence theorems for these GW(k)-valued counts, identifying these counts with the count of tropical curves counted with some multiplicity. This is based on joint work with Jaramillo Puentes and forthcomi ng work by Markwig-Jaramillo Puentes-Röhrle.

The moduli space of dimension g principally polarized abelian varieties has a system of nice -- toroidal -- compactifications. I will explain how semistable degenerations of abelian varieties endow each of these compactifications with a tautological ring, and I will discuss methods to obtain relations in these rings.