Florian Richter (EPFL)
Multiplicative Number Theory from a dynamical perspective abstract
Abstract:
Multiplicative number theory is the study of prime numbers, factorizations, divisors, and other basic mathematical concepts related to the multiplicative structure of the integers. The focus of this lecture lies on some of the newly emerging interfaces between multiplicative number theory and ergodic theory, which has led to significant advancements in recent years. We will provide an overview of the latest developments and explore some of the new methods and applications that have emerged.
10:30 • Université de Genève, Conseil Général 7-9, Room 1-05
Hanspeter Schmidli (University of Cologne, Germany)
Stabilizing the surplus process through the control of Drawdowns abstract
Abstract:
The drawdown is the loss of the surplus process compared its historical maximum. In order to stabilize the surplus one tries to keep the surplus close the its maximum; that is, keeping the drawdown small. We use proportional reinsurance to control the drawdown and measure the time the drawdown spends below a predefined barrier. If the time in drawdown is the only criterion, the maximum will never increase under the optimal strategy, see Brinker and Schmidli (2022). To avoid this, we maximise simultaneously the increase of the maximum. Dependent on how we weight the two contradicting criteria, we obtain different strategies, of which some are surprising. The talk is based on joint work with Leonie Brinker.
11:00 • EPF Lausanne, UniL campus, Extranef - 110
Prof. Dr. Evgeny Shinder (University of Sheffield)
Motivic invariants of birational maps abstract
Abstract:
I will speak about invariants of birational maps taking values in the Grothendieck ring of varieties, and about applications of these invariants to the properties of birational maps and to the structure of the Grothendieck ring. This is joint work with Hsueh-Yung Lin.
13:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 25
Dominic Phillips (University of Edinburgh)
Numerics with Coordinate Transforms for Efficient Brownian Dynamics Simulations abstract
Abstract:
Many stochastic processes in the physical and biological sciences can be modelled using Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.
14:00 • Université de Genève, Conseil Général 7-9, Room 1-05
Freya Behrens (EPFL)
The statistical mechanics of synchronous local processes on graphs abstract
Abstract:
The friendly partition problem involves determining whether a given graph allows for a partition of its nodes into two nonempty sets, where each node has at least as many neighbors in its own set as in the other. Notably, not all graphs permit such a friendly partition, and even fewer accommodate a partition where a node requires an additional margin of neighbors in its own set compared to the other. We investigate the existence of such partitions and the algorithmic feasibility of finding them.A natural question is: how does a graph evolve when nodes directly adapt their states to meet these local constraints? When this adaptation occurs synchronously, it models scenarios like majority voting or cellular automata. However, it is not a given that a graph, where each node greedily iteratively applies the local rule, converges to a global solution. Our analysis examines the different types of attractors that emerge in locally constrained problems and the role of initialisation in shaping the outcome.Our tool to answer these questions is the cavity method and the backtracking dynamical cavity method from statistical physics for synchronous update processes on regular graphs. They provide the sharp transitions on the existence of solutions, as well as the dynamical phase transitions of local processes in the large system limit.
14:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Anna Skorobogatova (University of Princeton)
Higher codimension area-minimizers: structure of singularities and uniqueness of tangent cones abstract
Abstract:
The problem of determining the size and structure of the interior singular set of area-minimizing surfaces has been studied thoroughly in a number of different frameworks, with many ground-breaking contributions. In the framework of integral currents, when the codimension of the surface is higher than 1, the presence of singular points with flat tangent cones creates an obstruction to easily understanding the interior singularities. Little progress has been made in full generality since Almgren’s celebrated (m-2)-Hausdorff dimension bound on the singular set for an m-dimensional area-minimizing current, which was since revisited and simplified by De Lellis and Spadaro. In this talk I will discuss recent joint works with Camillo De Lellis and Paul Minter, where we establish (m-2)-rectifiability of the interior singular set of an m-dimensional area-minimizing integral current and show that the tangent cone is unique at \\mathcal{H}^{m-2}-a.e. interior point.
14:15 • EPF Lausanne, MA B2 485
Ben Pineau (UC Berkeley)
Sharp Hadamard well-posedness for the incompressible free boundary Euler equations abstract
Abstract:
I will talk about a recent preprint in which we establish an optimal local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations on a connected fluid domain. Some components of this result include: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: A uniqueness result which holds at the level of the Lipschitz norm of the velocity and the $C^{1,\\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove essentially scale invariant energy estimates for solutions, relying on a newly constructed family of refined elliptic estimates; (v) Continuation criterion: We give the first proof of a continuation criterion at the same scale as the classical Beale-Kato-Majda criterion for the Euler equations on the whole space. Roughly speaking, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\\infty}$ and the free surface is in $L_T^1C^{1,\\frac{1}{2}}$; (vi) A novel proof of the construction of regular solutions.\\\\Our entire approach is in the Eulerian framework and can be adapted to work in relatively general fluid domains. This is based on joint work with Mihaela Ifrim, Daniel Tataru and Mitchell Taylor.
15:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Ivo Sachs (Munich)
Strings from graded superalgebras abstract
Abstract:
The world sheet approach to string theory, through its BRST quantization, gives rise to a graded superalgebra. Conversely, for a given choice of a superalgebra, one may construct a theory of background fields, using a graded Chevalley-Eilenberg construction. I will describe some solutions corresponding to different choices of the graded algebra.
15:30 • Université de Genève, Section de mathématiques, 7-9 rue du Conseil-Général, Room 1-07
Isabelle Chalendar (Université Gustave Eiffel, Champs-sur-Marne)
Composition operators on model spaces. abstract
Abstract:
A bounded holomorphic function on the unit disk $\\D$ is inner if its boundary values are almost everywhere of modulus 1. Motivated by the study of composition operators on model spaces launched by Mashreghi and Shabankha we consider the following problem: for a given inner function $\\phi\\not\\in\\mathsf{Aut}(\\D)$, find a non-constant inner function $\\Psi$ satisfying the functional equation $\\Psi\\circ\\phi=\\tau\\Psi$, where $\\tau$ is a unimodular constant. We prove that this problem has a solution if and only if $\\phi$ is of positive hyperbolic step. Joint work with P. GUMENYUK, and J.E. McCARTHY.
17:00 • Université de Neuchâtel, Institut de Mathématiques, B103
Lai-Sang Young (New York University)
Heinz Hopf lecture I: What happens when oscillators are disturbed?
17:15 • UZH Zentrum, Building RAA, Room G 01