Probability Seminar

• 6/10/2025 at 15:00, online at meet.google.com/zqf-wrka-cpq, Natasa Dragovic, University of St. Thomas, US

Title: Political centrism and extremism

Abstract: To win elections, politicians adopt strategies to make themselves appeal to voters. In our two-party system, these strategies are often driven by extensive polling and careful positioning of one candidate relative to their opponent. And yet, sometimes the same strategy that helps a candidate win in one election causes them to lose another. In recent work, we explore a simplified mathematical model of this political process and uncover unexpected phenomena and mechanisms that we compare to real-life political strategies and outcomes. The surprise is that the change in the optimal candidate position can, in certain circumstances, be discontinuous. The underlying mathematical mechanism is a blue-sky bifurcation. I will discuss these different strategies in the talk. I will also present a web-based app that you can use to understand the model better. I will finish the talk with the recent new results, like extending the model to three candidate scenarios. 

• 4/30/2025  at 15:00, FMI room 12, Andrada Fiscutean (FJSC): Science journalism and reporting

Abstract: Science communication is not an easy task, which requires deep understanding of research methodology and its findings, as well as communication skills. Researchers, students, and journalists should balance scientific literacy with writing and presentation skills that get the right message across. This presentation showcases essential tools and tips for the task, either for a research paper, a conference or workshop, or a journalistic article.

• 4/16/2025  at 15:00, FMI room 12, Andrada Fiscutean (FJSC): Science journalism and reporting

Abstract: Science communication is not an easy task, which requires deep understanding of research methodology and its findings, as well as communication skills. Researchers, students, and journalists should balance scientific literacy with writing and presentation skills that get the right message across. This presentation showcases essential tools and tips for the task, either for a research paper, a conference or workshop, or a journalistic article.

• 4/15/2025 NOTICE THE TIME:TUESDAY, 16:00 in ROOM 120 at FMI.  Vlad Mărgărint, University of North Carolina Charlotte: 

Title: A bridge between Random Matrix Theory and Schramm-Loewner Evolutions

Abstract: I will describe a newly introduced toolbox that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh.

• 4/9/2025  at 15:00, FMI room 12, Adrian Manea (UPB): Data processing and visualisations using Jupyter

Abstract: We introduce Jupyter, a literate programming tool based on Python and Markdown, and showcase some of its applications for scientific computations and visual representations. This way, it becomes a tool that is appropriate for teachers, researchers, students, and journalists alike, putting together the writing and communicating part with the source, input, and output of their computations.

• 4/2/2025 at 15:00, FMI room 12,  Title: Convergence rates for the Adam optimizer,  Arnulf Jentzen, 

[1] School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen), China

[2] Applied Mathematics: Institute for Analysis and Numerics, Faculty of Mathematics and Computer Science, University of Münster, Germany

Abstract: Stochastic gradient descent (SGD) optimization methods are nowadays the method of choice for the training of deep neural networks (DNNs) in artificial intelligence systems. In practically relevant training problems, usually not the plain vanilla standard SGD method is the employed optimization scheme but instead suitably accelerated and adaptive SGD optimization methods such as the famous Adam optimizer are applied. In this work we establishing optimal convergence rates for the Adam optimizer for a large class of stochastic optimization problems covering strongly convex stochastic optimization problems. The key ingredient of our convergence analysis is a new vector field function which we propose to refer to as the Adam vector field. This Adam vector field accurately describes the macroscopic behaviour of the Adam optimization process but differs from the negative gradient of the objective function (the function we intend to minimize) of the considered stochastic optimization problem. In particular, our convergence analysis suggests that Adam does typically not converge to critical points of the objective function (zeros of the gradient of the objective function) of the considered optimization problem but converges with rates to zeros of this Adam vector field. Finally, we present acceleration techniques for Adam in the context of deep learning approximations for partial differential equation and optimal control problems. The talk is based on joint works with Steffen Dereich, Thang Do, Robin Graeber, and Adrian Riekert.

References:

[1] S. Dereich & A. Jentzen, Convergence rates for the Adam optimizer, arXiv:2407.21078 (2024), 43 pages.

[2] S. Dereich, R. Graeber, & A. Jentzen, Non-convergence of Adam and other adaptive stochastic gradient descent optimization methods for non-vanishing learning rates, arXiv:2407.08100 (2024), 54 pages. 

[3] T. Do, A. Jentzen, & A. Riekert, Non-convergence to the optimal risk for Adam and stochastic gradient descent optimization in the training of deep neural networks, arXiv:2503.01660 (2025), 42 pages.

[4] A. Jentzen & A. Riekert, Non-convergence to global minimizers for Adam and stochastic gradient descent optimization and constructions of local minimizers in the training of artificial neural networks, arXiv:2402.05155 (2024), 36 pages, to appear in SIAM/ASA J. Uncertain. Quantif.

• FRIDAY 13/12/2024 at 13:00 (NOTICE THE DAY TIME CHANGE) in sala Google, Alexandru Hening (U. Texas AM), Stochastic Population Dynamics in Discrete Time

Abstract: I will present a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time stochastic difference equations that can include population structure, eco-environmental feedback or other internal or external factors. Using the general theory, I will showcase some interesting examples, including a discrete Lotka-Volterra model.

• 12/04/2024 at 15:00 sala Google, Alexandra Andriciuc (FMI),  Bridging Intuition and Rigor: Insights from "The Art of Statistics" by David Spiegelhalter

In this talk, we explore key ideas from David Spiegelhalter's The Art of Statistics: Learning from Data, examining the balance between popular statistical concepts and their formal mathematical underpinnings. Topics such as the wisdom of crowds, correlation versus causation, and statistical inference are presented with greater analytical rigor while preserving their accessibility. By delving into these ideas through a combination of intuitive explanations and formal proofs, this talk aims to bridge the gap between popularization and mathematical abstraction, encouraging a deeper appreciation of statistics as both an applied and theoretical discipline.

• 11/27/2024 at 15:00 (online), Simone Bombari,  Memorization and Optimization in Deep Neural Networks with Minimum Over-parameterization

The Neural Tangent Kernel (NTK) has emerged as a powerful tool to provide memorization, optimization and generalization guarantees in deep neural networks.  A line of work has studied the NTK spectrum for two-layer and deep networks with at least a layer with Ω(N ) neurons, N being the number of training samples.  Furthermore, there is increasing evidence suggesting that deep networks with sub-linear layer widths are powerful memorizers and optimizers, as long as the number of parameters exceeds the number of samples. Thus, a natural open question is whether the NTK is well conditioned in such a challenging sub-linear setup. In this paper, we answer this question in the affirmative. Our key technical contribution is a lower bound on the smallest NTK eigenvalue for deep networks with the minimum possible over-parameterization: up to logarithmic factors, the number of suaqre root of parameters is Ω(N) and, hence, the number of neurons is as little as Ω(N). To showcase the applicability of our NTK bounds, we provide two results concerning memorization capacity and optimization guarantees for gradient descent training.

• 11/20/2024 at 15:00 sala Google, Ionel Popescu (FMI and IMAR),  Talagrand's work on maximum of Gausian processes II

I will outline the main work of Talagrand on the maximum of Gaussian processes.  This is based on the key idea of generic chaining.  

• 11/13/2024 at 15:00, (online),  Thomas Lehericy (University of Zurich), The maximum displacement and the volume of the critical Branching Random Walk

The Branching Random Walk describe the evolution of a population where, at each generations, particles die giving birth to new particles which move. Such models naturally appear when modelling population under a selection constraint, with the displacement modelling e.g. the fitness of the particle. A lot of effort has been put on studying the supercritical regime, where each particle gives birth to more than one particle on average, since the population survives forever in this case. But less is known about the critical case, where each particle give birth to one particle on average. Motivated by applications to random geometry, I establish a control over the joint distribution of the volume and maximum displacement of the critical Branching Random Walk.

• 11/6/2024 at 15:00, sala Google. Ionel Popescu (FMI and IMAR), Talagrand's work on maximum of Gausian processes I

I will outline the main work of Talagrand on the maximum of Gaussian processes.  This is based on the key idea of generic chaining.    

• 10/30/2024 at 15:00, sala Google.  Ionel Popescu (FMI and IMAR), A few ideas on Talagrand's work.  

     I will talk about a few ideas of Michel Talagrand's work.  I encourage everyone to watch this interview:
https://youtu.be/LzRt61xE3Mg?si=zpR83SxSF0hTgSB6.  

• 8/12/2024 and 8/19/2024 at 11 am, room Ciprian Foias (804 the eight foor at IMAR). We will have a joint working seminar with 

Arnulf Jentzen, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen) & University of Münster:

Convergence and Non-Convergence Results for the ADAM optimizer

based on this paper.

and our group (the most representative Iulian Cimpean and Ionel Popescu): 

Monte Carlo methods for PDEs and Neural Networks

based on this paper 

• 7/24/2024 Arnulf Jentzen: Title: Convergence and non-convergence results for accelerated and adaptive stochastic gradient descent optimization methods, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen) & University of Münster,  at 11am,  room Ciprian Foias (804, eight floor), Institute of Mathematics of the Romanian Academy, 

     Abstract:

Deep learning algorithms - typically consisting of a class of deep neural networks trained by a stochastic gradient descent (SGD) optimization method - are nowadays the key ingredients in many artificial intelligence (AI) systems and have revolutionized our ways of working and living in modern societies. In practical relevant learning problems, usually not the plain vanilla standard SGD optimization method is used for the training of ANNs but instead more sophisticated suitably accelerated and adapted SGD optimization methods are employed. As of today, maybe the most popular variant of such accelerated and adaptive SGD optimization methods is the famous Adam optimizer proposed by Kingma & Ba in 2014. Despite the popularity of the Adam optimizer in implementations, it remained an open problem of research to provide a convergence analysis for the Adam optimizer even in the situation of simple quadratic stochastic optimization problems where the objective function (the function one intends to minimize) is strongly convex. In this talk we present optimal convergence rates for the Adam optimizer for a large class of stochastic optimization problems, in particular, covering simple quadratic stochastic optimization problems. Moreover, in ANN training scenarios we prove that the Adam optimizer and a large class of other SGD optimization methods do with high probability not converge to global minimizers in the optimization landscape. The talk is based on joint works with Steffen Dereich (University of Münster) and Adrian Riekert (University of Münster).

• 7/19/2024 Arnulf Jentzen:  Stochastic gradient descent optimization methods with adaptive learning rates, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen) & University of Münster, Sala Google (214) FMI, 3pm.  

Abstract: Deep learning algorithms - typically consisting of a class of deep neural networks trained by a stochastic gradient descent (SGD) optimization method - are nowadays the key ingredients in many artificial intelligence (AI) systems and have revolutionized our ways of working and living in modern societies. For example, SGD methods are used to train powerful large language models (LLMs) such as versions of ChatGPT and Gemini, SGD methods are employed to create successful generative AI based text-to-image creation models such as Midjourney, DALL-E, and Stable Diffusion, but SGD methods are also used to train DNNs to approximately solve scientific models such as partial differential equation (PDE) models from physics and biology and optimal control and stopping problems from engineering. It is known that the plain vanilla standard SGD method fails to converge even in the situation of several convex optimization problems if the learning rates are bounded away from zero. However, in many practical relevant training scenarios, often not the plain vanilla standard SGD method but instead adaptive SGD methods such as the RMSprop and the Adam optimizers, in which the learning rates are modified adaptively during the training process, are employed. This naturally rises the question whether such adaptive optimizers, in which the learning rates are modified adaptively during the training process, do converge in the situation of non-vanishing learning rates. In this talk we answer this question negatively by proving that adaptive SGD methods such as the popular Adam optimizer fail to converge to any possible random limit point if the learning rates are asymptotically bounded away from zero. Moreover, we propose and study a learning-rate-adaptive approach for SGD optimization methods in which the learning rate is adjusted based on empirical estimates for the values of the objective function of the considered optimization problem (the function that one intends to minimize). In particular, we propose a learning-rate-adaptive variant of the Adam optimizer and implement it in case of several ANN learning problems, particularly, in the context of deep learning approximation methods for PDEs such as deep Kolmogorov methods (DKMs), physics-informed neural networks (PINNs), and deep Ritz methods (DRMs). In each of the presented learning problems the proposed learning-rate-adaptive variant of the Adam optimizer faster reduces the value of the objective function than the Adam optimizer with the default learning rate. For a simple class of quadratic minimization problems we also rigorously prove that a learning-rate-adaptive variant of the SGD optimization method converges to the minimizer of the considered minimization problem.

References:
[1] Steffen Dereich, Robin Graeber, & Arnulf Jentzen, Non-convergence of Adam and other adaptive stochastic gradient descent optimization methods for non-vanishing learning rates, arXiv:2407.08100 (2024), 54 pages, https://arxiv.org/abs/2407.08100
[2] Steffen Dereich, Arnulf Jentzen, & Adrian Riekert, Learning rate adaptive stochastic gradient descent optimization methods: numerical simulations for deep learning methods for partial differential equations and convergence analyses, arXiv:2406.14340 (2024), 68 pages, https://arxiv.org/abs/2406.14340

• 3/11/2024 (Dan Paraschiv IMAR) Title: Multiply connected Fatou components of rational maps

Historically, the only way to obtain Fatou components of finite connectivity higher than 2 has been by means of quasiconformal surgery. We provide a family of rational maps such that, for specific parameters, the corresponding dynamical planes contain Fatou components of arbitrarily large connectivity. We also introduce a family obtained by using the Chebyshev-Halley family of numerical methods. We show the existence of parameters such that all Fatou components are infinitely connected.

• 12/18 (Martin Fronescu FMI) Magics of the Rubik's cube

Instead of abstract:  Martin is a famous figure among the cubers in Romania for his incredible performance in solving the Rubiks cube.  He will tell us about the various aspects of the Rubiks cube, including fast solvings, algorithms and the conjectures, now theorems on how many moves one needs to solve the cube.   His profile on the the World Cube Association is here: Martin's WCA page.   Here is the presentation of Martin.  

• 12/11/2023 (Viorel Costeanu, JP Morgan) Analytical approximations with Strang Splitting NOTICE THE TIME 15:30 not at the usual time 10:00 AND THIS WILL BE ONLY ONLINE.

Strang splitting is the statement that for 2 non-commuting matrices or operators the approximation exp(t(A+B)) ~ exp(tA/2)exp(tB)exp(tA/2) is second order accurate. This approximation is used in the design of finite difference schemes in order to effectively lower the local dimensionality of a problem. We demonstrate a novel application of Strang splitting where we reduce the calculation of the kernel of a complex stochastic differential equation to the kernel of a simpler one and the solution of an ordinary differential equation. We show that in cases of interest we can arrange so that the simpler kernel is known in closed form as either the regular heat kernel or the heat kernel of a hyperbolic space, and the ODE is either analytically solvable or can be very efficiently solved numerically.

• 11/20/2023 (Ionel Popescu) The fundamental theory of machine learning, the regression case II

I will give an introduction to PAC learning and then introduce the most common version, namely the fundamental theorem of PAC learning for classification.  This is well known.  I will discuss also the case of less known case of regression model.  On the way we will discuss a twist which is given by the McDiarmind inequality and how one can guarantee a good sample size is determined.  

• 11/13/2023 (Ionel Popescu) The fundamental theory of machine learning, the regression case 

I will give an introduction to PAC learning and then introduce the most common version, namely the fundamental theorem of PAC learning for classification.  This is well known.  I will discuss also the case of less known case of regression model.  On the way we will discuss a twist which is given by the McDiarmind inequality and how one can guarantee a good sample size is determined.  

• 11/06/2023 (Maximilian Sebastian Janisch, University of Zurich) Review of generative models from the mathematical and computer science perspective.

A branch of machine learning, generative models, has gained a lot of academic and public attention in the recent past. Notable cases of generative models are transformer-based large language models such as ChatGPT, as well as image-generating diffusion models such as Dall-E. Mathematically, generative models aim to “learn” a probability distribution after observing independent samples from this distribution. This talk will outline various methodologies to do so, notably (in order of recency) Restricted Boltzmann machines, Variational Auto-Encoders, Generative Adversarial Networks, and diffusion models,. I will also discuss applications to medical research as well as mathematical questions on the efficacy of such generative models.

• 10/16/2023 (Gabriel Majeri, Univ. of Bucharest) slides: here

Title: Introduction to Statistical Learning Theory: the Vapnik-Chervonenkis Theorem II

A fundamental problem in machine learning is that of generalization: how can we ensure the models we train on a finite sample  perform well over the entire distribution of the data? One of the earliest theoretical results in this direction was given in 1971 by the Soviet mathematicians Vladimir Vapnik and Alexey Chervonenkis. This talk will cover the formal definition of the learning problem in the Empirical Risk Minimization (ERM) framework, the VC dimension of a hypothesis set and the statement and proof of the fundamental inequality of VC theory. If time permits, we will also discuss how these ideas lead to the implementation of Support Vector Machines (SVMs).

    The only required prerequisites are basic measure and probability theory.

• 10/9/2023 (Gabriel Majeri, Univ. of Bucharest) 

Title: Introduction to Statistical Learning Theory: the Vapnik-Chervonenkis Theorem I

A fundamental problem in machine learning is that of generalization: how can we ensure the models we train on a finite sample  perform well over the entire distribution of the data? One of the earliest theoretical results in this direction was given in 1971 by the Soviet mathematicians Vladimir Vapnik and Alexey Chervonenkis. This talk will cover the formal definition of the learning problem in the Empirical Risk Minimization (ERM) framework, the VC dimension of a hypothesis set and the statement and proof of the fundamental inequality of VC theory. If time permits, we will also discuss how these ideas lead to the implementation of Support Vector Machines (SVMs).

    The only required prerequisites are basic measure and probability theory.

     References:

 - "On The Uniform Convergence of Relative Frequencies of Events to Their Probabilities", V. N. Vapnik and A. Ya. Chervonenkis

- "A Probabilistic Theory of Pattern Recognition", L. Devroye, L. Györfi and G. Lugosi

- "Foundations of Machine Learning", M. Mohri, A. Rostamizadeh and A. Talwalkar

• 9/26/2023 at 12pm (Arnulf Jentzen, the Chinese Univ. of Hong Kong, China & Univ. of Münster, Germany) (Notice the date and time change)  This is also the monthly conference at IMAR and it will take place in Miron Nicolescu Amphitheater at the ground level 

Title: Overcoming the course of dimensionality: from nonlinear Monte Carlo to the training of neural networks 

Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. Nearly all traditional approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In this talk we present an efficient machine learning algorithm to approximate solutions of high-dimensional PDE and we also prove that deep artificial neural network (ANNs) do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs. Moreover, we specify concrete examples of smooth functions which cannot be approximated by shallow ANNs without the curse of dimensionality, but which can be approximated by deep ANNs without the curse of dimensionality. In the final part of the talk we present some recent mathematical results on the training of neural networks.

• 9/18/2023 (Tushar Vaidya Singapore) Pathwise Quantum Lasso Regression 

We study the high dimensional aspect of linear regression with an ℓ1 penalty. While classical, numerical algorithms are available for Lasso, our focus is on developing a hybrid quantum algorithm that offers new insights and speedup. Quadratic speedup is theoretically possible over the classical Homotopy (Least Angle Regression) method. In particular, we provide a general setup for Lasso solutions as the penalty term varies.  Several challenges remain in creating such an algorithm.  The task is fraught with difficulties. Still the pursuit is worthwhile and we will elucidate how to go about this. The talk should be accessible to those without knowledge of quantum computing!

• 6/12/2023 (Ionel Popescu)  Accelerated methods for optimization

I will present some results of accelerated methods for optimization of a convex function using some accelerated methods.  There are some classical, recent and very recent results which deserve attention.  The papers I am going to base my talk: 

Accelerated variational methods and the interesting paper by Convergence for Heavy Ball methody and also this FISTA for strongly convex functions 

• 5/24/2023 (Max von Renesse, Leipzig University) NOTICE THE TIME: 4pm,  Entropic Regularization for General Linear Programs

We revisit the problem of unbalanced optimal transport for which we introduce an analogue of the the Schr"odinger problem leading to its entropic regularization which then can be solved via by iterative scaling similar to the Sinkhorn algorithm from the standard balanced case. It turns out that entropic regularization and iterative scaling as computational tool can be applied to a much larger class of linear programs. 

• 5/22/2023 (Heinrich Matzinger, Georgia Tech) NOTICE THE TIME CHANGE 11-12:  Problems with deep neural networks for image recognition 

We show how the Convolutional Neural Networks do not use global shape for recognizing objects but use micro-features,     which can lead to errors in more involved industry related problems.

• 5/15/2023 (Evgnosia Kelesidis) FastICA

This is about some classical aspects of ICA (independent component analysis) with some by now standard algorithms.  

• 5/08/2023 (Ionel Popescu) About ICA and related conjectures

I will discuss some conjectures related to independent component analysis with k speakers and p microphones.  The interesting phenomena is when the number of microphones is much smaller than the number of speakers and we will see some conjectures around this.  

• 3/27/2023 (Razvan Moraru) Bazele Cuantificării 

  Pe parcursul seminarului vor fi prezentate bazele cuantizării geometrice, cuprinzând definirea spațiului Hilbert precuantic, modul de utilizare al polarizărilor, precum și relevanța acestora în contextul teoriei probabilităților. 

  Odată ce a fost stabilită o modalitatea precisă de a asocia o dinamică unitară unui curent diferențial, vom încerca să înțelegem corespondența dinamicii cuantice cu dinamica stocastică rezultată prin aplicarea transformatei Wigner-Moyal. 

• 3/13/2023 ( Rishabh Bhardwaj, Singapore) Language Models

Language Models (LMs) aim to model a probability distribution over a sequence of words. This simple setting and its variants have shown a huge potential in solving many AI applications dealing with natural language as well as other modalities. In this presentation, we will discuss how LMs evolved from the perspective of model architectures, including classical RRNs and advanced Transformers. We will cover various algorithms to transfer the learning of an LM to solve a given downstream task efficiently. We will also touch upon emerging approaches to prune large models, preserve user-private information, and de-biasing techniques. In the end, we will discuss the basics of training methods adopted by recent and widely famous systems such as ChatGPT with a large LM as its backbone.

• 3/06/2023 (Ionel Popescu, FMI and IMAR) The Fundamental Theorems of Mathematical Finance III

This is the continuation of the previous two lectures on mathematical finance.  I will finally arrive at the continuous case and discuss the Black-Scholes equation appearence and eventually how one can solve it.  

• 20/27/2023 (Andrei Comaneci, Berlin) Tropical Geometry in Data Analysis and Machine Learning

Tropical geometry deals with certain piecewise linear geometric 

objects with rich combinatorial structure. Its algebraic roots can be 

traced from combinatorial optimization and dynamic programming from 

1960s, but the geometric viewpoint arose from algebraic geometry at 

the beginning of 21st century. This led to various connections to 

other subjects, including computational biology and machine learning.

In this talk we will present the basic notions from tropical geometry 

that appear mostly in data science. On one hand, we focus on tropical 

convexity, which are important in phylogenetics. The data consists of 

evolutionary trees which can be seen as points in a certain tropically 

convex set.  On the other hand, we will focus on tropical hypersurfaces and its 

combinatorial properties. They were recently studied in their 

relationship to neural networks with ReLU activation function. This 

led to complexity results in deep learning theory.

• 2/20/2023 (Ionel Popescu, FMI & IMAR) The Fundamental Theorems of Mathematical Finance II

I will present a simple introduction to the fundamental theorem of mathematical finance.  I will do it first in the case of binomial model in the discrete case where the main concept of no arbitrage market plays the central role which in turn yields the main result.  In the continous case, this is more involved, but the fundamental principle is almost the same.  In this second round of the seminar, the goal is to see the how the Black-Sholes ecuation appears.  

• 2/13/2023 (Ionel Popescu, FMI & IMAR) The Fundamental Theorems of Mathematical Finance I

I will present a simple introduction to the fundamental theorem of mathematical finance.  I will do it first in the case of binomial model in the discrete case where the main concept of no arbitrage market plays the central role which in turn yields the main result.  In the continous case, this is more involved, but the fundamental principle is almost the same.    

• 2/06/2023 (Razvan Moraru, IMAR)  Operatorii Lévy. Dinamica stocastică și dinamica Cuantică. III

Abstract:

Conținutul este structurat pe durata a două seminarii în modul următor:

Cu ocazia primului seminar vom pleca de la definiția Proceselor Lévy. Un exemplu de proces Lévy este și procesul de difuzie cu salturi, vom vedea că funcția sa caracteristică are forma prescrisă de Teorema Lévy-Kincin.

 - O consecință a faptului că formula Lévy-Kincin caracterizează exponentul caracteristic al unei distribuții in(de)finit divizabile, iar procesele Lévy au distribuții de acest fel. Mai general, vom vedea cum formula Lévy-Itô de descompunere este inerent legată de Teorema Levy-Kincin

  - De asemenea vom vedea faptul că procesele Lévy sunt procese Markov, având semigrupul de tranziție asociat chiar un semigrup de convoluție, al cărui generator infinitezimal este un Operator Lévy, rezolvă problema Martingalului și constituie un operator pseudodiferențial, al cărui simbol este întocmai exponentul caracteristic prescris de formula Lévy-Kincin.  

O formă aparent identică cu cea a Operatorilor Lévy vom regăsi pe parcursul celui de-al doilea seminar pentru Operatorii Weyl ai Mecanicii Cuantice. 

- În cel de-al doilea seminar vom prezenta legătura inerentă dintre formalismul Hamiltonian al Mecanicii Cuantice, geometria simplectică și operatorii Mecanicii Cuantice. Voi prezenta bazele Cuantificării Geometrice - mai precis, voi prezenta cuantificarea Weyl; astfel încât la sfârșitul celui de-al doilea seminar vom putea avea o imagine unitară a conexiunii dintre:

  - semigrupul de simplectomorfisme (curentul diferențial) asociat unui sistem dinamic în geometria diferențială/mecanica clasică 

   - semigrupul de operatori unitari asociat evoluției în timp a unei observabile în Mecanica Cuantică

   - semigrupul de tranziție asociat unui proces Markov în analiza stocastică.

Ca un pas intermediar, în primul seminar vom discuta și despre aplicarea metodei dilatărilor (P.Halmos) pentru a pune în corespondență dinamica unui lanț Markov și dinamica unui sistem cuantic reprezentat pe o sferă complexă. Aspectele acestei conexiuni nu au doar o valoare pur-teoretică, ele pot fi aplicate în modelarea de qubits - pe sfera Bloch, respectiv în calculul cuantic sau în procedee precum tomografia cuantică. 

 În plus pe această cale este devoalată o legătură subtilă între grafuri (asociate dinamicii unui lanț Markov) și spațiile proiective complexe, respectiv sferele cuantice (asociate dinamicii cuantice unitare).

• 
  

• 1/30/2023 (Razvan Moraru, IMAR)  Operatorii Lévy. Dinamica stocastică și dinamica Cuantică. II

Abstract:

Conținutul este structurat pe durata a două seminarii în modul următor:

Cu ocazia primului seminar vom pleca de la definiția Proceselor Lévy. Un exemplu de proces Lévy este și procesul de difuzie cu salturi, vom vedea că funcția sa caracteristică are forma prescrisă de Teorema Lévy-Kincin.

 - O consecință a faptului că formula Lévy-Kincin caracterizează exponentul caracteristic al unei distribuții in(de)finit divizabile, iar procesele Lévy au distribuții de acest fel. Mai general, vom vedea cum formula Lévy-Itô de descompunere este inerent legată de Teorema Levy-Kincin

  - De asemenea vom vedea faptul că procesele Lévy sunt procese Markov, având semigrupul de tranziție asociat chiar un semigrup de convoluție, al cărui generator infinitezimal este un Operator Lévy, rezolvă problema Martingalului și constituie un operator pseudodiferențial, al cărui simbol este întocmai exponentul caracteristic prescris de formula Lévy-Kincin.  

O formă aparent identică cu cea a Operatorilor Lévy vom regăsi pe parcursul celui de-al doilea seminar pentru Operatorii Weyl ai Mecanicii Cuantice. 

- În cel de-al doilea seminar vom prezenta legătura inerentă dintre formalismul Hamiltonian al Mecanicii Cuantice, geometria simplectică și operatorii Mecanicii Cuantice. Voi prezenta bazele Cuantificării Geometrice - mai precis, voi prezenta cuantificarea Weyl; astfel încât la sfârșitul celui de-al doilea seminar vom putea avea o imagine unitară a conexiunii dintre:

  - semigrupul de simplectomorfisme (curentul diferențial) asociat unui sistem dinamic în geometria diferențială/mecanica clasică 

   - semigrupul de operatori unitari asociat evoluției în timp a unei observabile în Mecanica Cuantică

   - semigrupul de tranziție asociat unui proces Markov în analiza stocastică.

Ca un pas intermediar, în primul seminar vom discuta și despre aplicarea metodei dilatărilor (P.Halmos) pentru a pune în corespondență dinamica unui lanț Markov și dinamica unui sistem cuantic reprezentat pe o sferă complexă. Aspectele acestei conexiuni nu au doar o valoare pur-teoretică, ele pot fi aplicate în modelarea de qubits - pe sfera Bloch, respectiv în calculul cuantic sau în procedee precum tomografia cuantică. 

 În plus pe această cale este devoalată o legătură subtilă între grafuri (asociate dinamicii unui lanț Markov) și spațiile proiective complexe, respectiv sferele cuantice (asociate dinamicii cuantice unitare).

• 1/16/2023 (Razvan Moraru, IMAR)  Operatorii Lévy. Dinamica stocastică și dinamica Cuantică. I

Abstract:

Conținutul este structurat pe durata a două seminarii în modul următor:

Cu ocazia primului seminar vom pleca de la definiția Proceselor Lévy. Un exemplu de proces Lévy este și procesul de difuzie cu salturi, vom vedea că funcția sa caracteristică are forma prescrisă de Teorema Lévy-Kincin.

 - O consecință a faptului că formula Lévy-Kincin caracterizează exponentul caracteristic al unei distribuții in(de)finit divizabile, iar procesele Lévy au distribuții de acest fel. Mai general, vom vedea cum formula Lévy-Itô de descompunere este inerent legată de Teorema Levy-Kincin

  - De asemenea vom vedea faptul că procesele Lévy sunt procese Markov, având semigrupul de tranziție asociat chiar un semigrup de convoluție, al cărui generator infinitezimal este un Operator Lévy, rezolvă problema Martingalului și constituie un operator pseudodiferențial, al cărui simbol este întocmai exponentul caracteristic prescris de formula Lévy-Kincin.  

O formă aparent identică cu cea a Operatorilor Lévy vom regăsi pe parcursul celui de-al doilea seminar pentru Operatorii Weyl ai Mecanicii Cuantice. 

- În cel de-al doilea seminar vom prezenta legătura inerentă dintre formalismul Hamiltonian al Mecanicii Cuantice, geometria simplectică și operatorii Mecanicii Cuantice. Voi prezenta bazele Cuantificării Geometrice - mai precis, voi prezenta cuantificarea Weyl; astfel încât la sfârșitul celui de-al doilea seminar vom putea avea o imagine unitară a conexiunii dintre:

  - semigrupul de simplectomorfisme (curentul diferențial) asociat unui sistem dinamic în geometria diferențială/mecanica clasică 

   - semigrupul de operatori unitari asociat evoluției în timp a unei observabile în Mecanica Cuantică

   - semigrupul de tranziție asociat unui proces Markov în analiza stocastică.

Ca un pas intermediar, în primul seminar vom discuta și despre aplicarea metodei dilatărilor (P.Halmos) pentru a pune în corespondență dinamica unui lanț Markov și dinamica unui sistem cuantic reprezentat pe o sferă complexă. Aspectele acestei conexiuni nu au doar o valoare pur-teoretică, ele pot fi aplicate în modelarea de qubits - pe sfera Bloch, respectiv în calculul cuantic sau în procedee precum tomografia cuantică. 

 În plus pe această cale este devoalată o legătură subtilă între grafuri (asociate dinamicii unui lanț Markov) și spațiile proiective complexe, respectiv sferele cuantice (asociate dinamicii cuantice unitare).

• 1/09/2023 (Ionel Popescu, FMI and IMAR) Ergodic Properties of Markov Processes III

This is intended as a working seminar to understand some of the powerful results related to convergence of Markov processes from some of the papers of Hairer:  http://www.hairer.org/notes/Markov.pdf and https://hairer.org/notes/Convergence.pdf

• 12/19/2022 (Alexandra Andriciuc, IMAR) Ergodic Properties of Markov Processes II

This is intended as a working seminar to understand some of the powerful results related to convergence of Markov processes from some of the papers of Hairer:  http://www.hairer.org/notes/Markov.pdf and https://hairer.org/notes/Convergence.pdf

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• 12/12/2022 (Alexandra Andriciuc, IMAR) Ergodic Properties of Markov Processes I

This is intended as a working seminar to understand some of the powerful results related to convergence of Markov processes from some of the papers of Hairer:  http://www.hairer.org/notes/Markov.pdf and https://hairer.org/notes/Convergence.pdf

• 11/07/2022 (Jiaming Chen, ETH Zurich) Randomized algorithm and its convergence to Schramm-Loewner evolution

How do we simulate a singular stochastic diffusion with high efficiency? In the context of Schramm- Loewner evolution, we study the approximation of its traces via the Ninomiya-Victoir Splitting Scheme. We prove a strong convergence in probability with respect to the sup-norm to the distance between the SLE trace and the output of the Ninomiya-Victoir Splitting Scheme when applied in the context of the Loewner differential equation.

• 06/06/2022 (Ionel Popescu, FMI and IMAR) The fundamental theorem of Machine learning III

We will rediscuss the fundamental Theorem of machine learning and introduce the VC dimension with some examples.  

• 30/05/2022 (Ionel Popescu, FMI and IMAR) The fundamental theorem of Machine learning II

After introducing the machine learning problem and starting the proof of it, we will see how the Rademacher complexity and the VC dimension appear naturally.  

• 23/05/2022 (Ionel Popescu, FMI and IMAR) The fundamental theorem of Machine learning

I will discuss some aspects of the fundamanetal theorem of machine learning.  There is a natural framework in which one can define the machine learning problem.  One of the key ingredients in there is the appearance of VC dimension, which seems very misterious.  The interesting fact is that this VC dimension appears naturally in the process of the proof of the main theorem.  

• 16/05/2022 (Mihai Alin Neascu) Masuri de risc

Mihai ne va povesti ce sunt masurile de risc in sectorul financiar si cum sunt modelate probabilist.  De asemenea ne va povesti cum sunt introduse, ce anume exemple avem si cum arata aceste masuri de risc in general pe spatii finite.  Aceasta este parte din lucrarea de licenta pe care o va prezenta in vara.  

• 09/05/2022 (Diana Conache, TU Munchen) O mica introducere la lanturi de ordin infinit

In acest seminar vom vorbi despre lanturi de ordin infinit (cunoscute in literatura si sub numele de lanturi cu conexiuni complete sau masuri-g). In particualr, vom discuta despre un rezultat de existenta si cateva criterii de unicitate.

• 02/05/2022 (Ionel Popescu)  O mica introducere la lanturile Markov si mixing IV

Dupa ce am introdus un pic problema de lanturi Markov, vom incerca intelegerea teoremi Perron Frobenius si formularea precisa a lantului Markov de card shuffling.  

• 18/04/2022 (Ionel Popescu)  O mica introducere la lanturile Markov si mixing III

Dupa ce am introdus un pic problema de lanturi Markov, vom inainta in intelegerea teoremei 3.33 de aici:  http://www.hairer.org/notes/Markov.pdf.  Vom incerca intelegerea teoremi Perron Frobenius si formularea precisa a lantului Markov de card shuffling.  

• 11/04/2022 (Ionel Popescu)  O mica introducere la lanturile Markov si mixing II

Dupa ce am introdus un pic problema de lanturi Markov, vom inainta in intelegerea teoremei 3.33 de aici:  http://www.hairer.org/notes/Markov.pdf.

• 4/04/2022 (Ionel Popescu)  O mica introducere la lanturile Markov I

Voi incerca sa prezint ce este un lant Markov si cum se analizeaza el din punct de vedere analitic si probabilist.  Scopul mai larg ar fi sa incercam sa intelegem teorema 3.33 de aici:  http://www.hairer.org/notes/Markov.pdf.

• 28/03/2022 (Oana Lang, Imperial College London) Solutii maximale pentru ecuatii cu derivate partiale stocastice II

In acest seminar vom vorbi despre ce este o "solutie maximala" pentru un sistem specific de ecuatii cu derivate partiale stocastice si vom arata ca timpii maximali care corespund la doua norme Sobolev diferite sunt P - aproape sigur egali.  

• 21/03/2022 (Oana Lang, Imperial College London) Solutii maximale pentru ecuatii cu derivate partiale stocastice I

In acest seminar vom vorbi despre ce este o "solutie maximala" pentru un sistem specific de ecuatii cu derivate partiale stocastice si vom arata ca timpii maximali care corespund la doua norme Sobolev diferite sunt P - aproape sigur egali.  

• 15/03/2022 (Ionel Popescu)  Cand o functie este polinom si la ce ne foloseste?  

 Stim ca daca o functie este polinom pe un interval, atunci derivata repetata se anuleaza de la un rang incolo.   Intrebarea inversa, daca avem o functie $f$ pe un interval (deschis) I al axei reale are proprietatea ca pentru orice punct $x\in I$ exista un numar natural $n=n_x$ pentru care $f^{(n)}(x)=0$, rezulta ca $f$ este polinom?  Raspunsul la aceasta intrebarea are niste consecinte foarte interesante pentru retelele neuronale! 

• 21/02/2022, 28/02/2022 si 7/03/2022 (Ionel Popescu)  Metoda probabilista 

Metoda probabilista este o metoda propusa de Paul Erdos care este foarte folosita mai ales in combinatorica.  Voi introduce metoda, si voi discuta cazul grafurilor Erdos Reyni G(n,p).  Exista o legatura foarte profunda cu procesele de branching ce apar natural in aceasta zona si care descriu ce se intampla cu componentele conexe mari in aceasta grafuri aleatoare cand n tinde la infinit si p=c/n.  Avem o tranzitie foarte interesanta pentru cazul c<1, c=1, c>1.  Referinta de baza este cartea lui Noga Alon si Joel Spencer, The Probabilistic Method.

• 14/02/2022 (Iulian Cimpean)  O abordare probabilista a modelului SIR

Modelul SIR este un model de ecuatii diferentiale pentru propagarea unei epidemii.  Aici vom discuta varianta probabilista de interactiune la nivel de individ.  Modelul SIR va fi vizibil ca o varianta in medie a acestor interactii individuale.  

• 25/01/2022  si 01/02/2022 si 08/02/2022 (Ionel Popescu)  ICA, independent component analysis

Voi prezenta o conjectura si o abordare partiala pentru aceasta problema.  ICA este acronimul pentru independent component analysis, care isi are originea intr-o metoda de a recupera vocile individuale din inregistrari ale uni numar de vorbitori. Articolul de baza este aici:  https://arxiv.org/abs/1901.08334

• 18/01/2022 (Ionel Popescu) O abordare pentru analiza evolutiei Covid in Romania

Voi povesti o abordare a analizei evolutiei Covid in Romania pe care Marian Petrica si cu mine am pus-o la punct recent.  Modelul de baza este SIRD, o varianta a SIR care tine cont de morti ca fiind o categorie separata.  Articolul de baza este: https://arxiv.org/abs/2203.00407 

• 03/1/2022 (Alexandra Andriciuc) Swarm particle Optimization 

Aceasta metoda de optimizare este bazata pe idea ca mai multe particule testeaza individual valoarea unei functii insa la fiecare iteratie are loc o evaluare bazata pe performanta individuala si pe cea globala ceea ce duce la o optimizare destul de rapida.  Din pacate partea matematica ce justifica aceasta eficienta nu este lamurita.  

• 12/13/2021(Catalin Sandu) MLP, mixing layer perceptron

MPL este o retea neuronala recent scoasa la iveala care are performante remarcabile comparat cu retelele mai consacrate.  Articolul de baza este aici:  https://arxiv.org/abs/2105.01601

• 23/11/2021 (Vlad Constatinescu) Structura retelelor neuronale si interpolarea

Pentru o retea neuronala, cu mai multi neuroni decat data disponibile pentru problemele de regresie regimul de parametrii pentru punctele de minim este o varietate de o anumita dimensiune.  Bazat pe articolulul:  https://arxiv.org/abs/2005.04210

• 16/11/2021 (Ionel Popescu) Elefant Random Walk with Delays 

In principiu este vorba de un random walk in care pasii inainte nu sunt independenti ci pot fi pasi pe care drumul le-a mai facut inainte.  Cum se modeleaza acest lucru si ce anume se intampla cand numarul de pasi este mare ramane de vazut.  Bazate pe articolul: https://arxiv.org/abs/1906.04930

• An interesting connection between the Gaussian moment conjecture (one of the many with this name) and the global invertibility of a polynomial map is presented here: THE GAUSSIAN MOMENTS CONJECTURE AND THE JACOBIAN CONJECTURE 

• An interesting concept, namely that of PL condition for a function leads to interesting convergence results of the gradient descent toward the optimal minima.  The paper which gives some results is Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Łojasiewicz Condition (click here for the pdf file) and in the arxiv form here 

• Thanks to Sorin Costiner who bought up this interesting topic which seems to be very hot now.  The topic is more related to combinatorics.  Here is a short introduction to it Elegant Six-Page Proof Reveals the Emergence of Random Structure and the real paper A Proof of the Kahn-Kalai Conjecture.  It would be very interesting to see what is going on here. 

• A probabilistic model for the Euler numbers appear here:  A probabilistic interpretation of Eulerian numbers This is an interesting way of analyzing asymptotic properties of numbers defined recursively for which an asymptotic analysis is hard to perform.

• A Markov model with labels and an interesting result without any data on the mixing time: Assessing significance in a Markov chain without mixing

• An intrguiging property of positive definite matrices and the application to numerical analysis:  A variant of Schur’s product theorem and its applications Interestingly this paper appeared recently in Advances in Mathematics, one respected journal.  

• This is an interesting approach on stochastic gradient descent arguments and its applications to machine learning.  In general gradient descent speed (at least for convex functions is not that great and speeding versions are very important):  A variational perspective on accelerated methods in optimization 

• This is a paper on the coin flipping and algebraic numer of computations for computations of polynomials: Coin Flipping Cannot Shorten Arithmetic Computations 

• An interesting refinment of Chebyshev inequality and its applications to Monte Carlo method:  Halving the Bounds for the Markov, Chebyshev, and Chernoff Inequalities Using Smoothing 

• When is a string of numbers random?  This question is fundamental in cryptography and statistics.  Here is an intresting article Higher-order dangers and precisely constructed taxa in models of randomness 

• What is the relationshio between the zeros of a random polynomial and its critical values?  One relatively accesible paper is here showing that in certain cases the critical values follows the same distribution of the zeros of a random polynomial: On the distribution of critical points of a polynomial.  There is more in this direction as for instance here:  CRITICAL POINTS OF RANDOM POLYNOMIALS WITH INDEPENDENT IDENTICALLY DISTRIBUTED ROOTS 

• Bernstein polynomials are related to probabilistic interpretation and has a serious extension here: Lower Estimates for Centered Bernstein-Type Operators