Minerva Foundation Lectures

This page is an archive. See the current page for the Minerva Lectures here.

These special lecture series in probability and mathematical finance are made possible by the generous support of the Minerva Research Foundation. Time and location vary.

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Spring Semester 2016

March 21 — March 23

• Scott Sheffield (Massachusetts Institute of Technology)
  Universal Randomness in 2D

  I will introduce several universal and canonical random objects that are (at least in some sense) two dimensional or planar, along with discrete analogs of these objects. In particular, I will introduce probability measures on the space of paths, the space of trees, the space of surfaces, and the space of growth processes. I will argue that these are in some sense the most natural and symmetric probability measures on the corresponding spaces.
  I will then describe several surprising relationships between these canonical objects. Many of these ideas have been historically motivated by physics —- especially string theory, conformal field theory, and statistical mechanics. On the math side, we will also draw inspiration from combinatorics, complex dynamics, stochastic processes, functional analysis, and representation theory.
  

• Monday, March 21, 5:30–7pm, Math 507

• Tuesday, March 22, 2:40–3:55pm, Math 417

• Wednesday, March 23, 4:30—6pm, Math 520

Fall Semester 2015

September 28 — October 2

• Frank den Hollander (Leiden University)
  Metastability for interacting particle systems

  Metastability is the phenomenon where a system, under the influence of a stochastic dynamics, moves between different subregions of its state space on different time scales. Metastability is encountered in a wide variety of stochastic systems, in physics, chemistry, biology and economics. The challenge is to devise realistic models and to explain the experimentally observed universality in the behaviour of metastable systems, both qualitatively and quantitatively.
  
  In statistical physics, metastability is the dynamical manifestation of a first-order phase transition. An example is condensation. When water vapour is cooled down slightly below 100 degrees Celsius, it persists for a very long time in a metastable vapour state before transiting to a stable liquid state under the influence of random fluctuations. The crossover occurs after the system manages to create a critical droplet of liquid inside the vapour, which once present grows and invades the system. While in the metastable vapour state, the system makes many unsuccessful attempts to form a critical droplet.
  
  Lectures 1+2 provide an overview of the theory of metastability for interacting particle systems, with excursions into the past, the present and the future.
  
  Lectures 3+4+5 focus on metastability for lattice systems at low temperature, outlining the general theory and discussing two specific applications.
  
  Literature: Anton Bovier and Frank den Hollander, Metastability - A Potential-Theoretic Approach, Grundlehren der mathematischen Wissenschaften, Springer, 2015,xxi + 569 pp.
  
  (pdf-file available on request)
  
  
  • Monday, September 28, 4-5:30pm, 507 Math
    
    Introduction: history, questions, approaches
    
    
    
    
  • Tuesday, September 29, 6-7:30pm, 507 Math
    
    Key results and key challenges
    
    Slides from Lectures 1 and 2
    
    
  • Wednesday, September 30, 4:30-6pm, *520* Math
    
    Metastability at low temperature: general theory.
    
    
    
    
  • Thursday, October 1, 5:30-7pm, 507 Math
    
    Glauber dynamics: Ising spins with random spin-flips.
    
    
    
    
  • Friday, October 2, 11:30am-1pm, *520* Math
    
    Kawasaki dynamics: lattice gas with random hopping.
    
    
    
    

October 13—22

• Constantinos Kardaras (LSE)
  

  • Title: Monotone convex analysis on $L^0_+$.
    
  
  
  • Abstract:
    
    Of importance in the field of Mathematical Finance and Economics is the space $L^0$, consisting of all equivalence classes of measurable functions built over a sigma-finite measure space equipped with the topology of convergence in measure. The absence of local convexity of the $L^0$-topology renders the use of traditional functional-analytic methods---especially those relying on the Hahn-Banach theorem and its offspring---impossible. In this series of talks, we shall develop a theory for monotone (nondecreasing) convex and concave functionals on $L^0_+$, the nonnegative orthant of $L^0_+$. Classical results in convex analysis, such as Fenchel-Moreau duality, have appropriate equivalents through a natural, but non-classical, ``duality'' pairing of $L^0_+$ with itself. Standard topics such as arbitrage theory, utility maximization and existence of equilibrium will be presented as applications of the abstract theory. If time permits, certain deeper results on the structure of $L^0_+$ will also be discussed.
  
  
  • Dates and Times:
    
    Tuesday, October 13th, 4:10—5:25pm, 507 Math
    
    Friday, October 16th, 10:30-11:45am, 507 Math
    
    Tuesday, October 20th, 4:10—5:25pm, 507 Math
    
    Thursday, October 22nd, 4:10—5:25pm, 903 SSW
    
    

November 2 — November 11

• Timo Seppalainen (University of Wisconson—Madison)
  

  • Title: KPZ fluctuation exponents for a class of zero-range processes.
    
  
  
  • Abstract:
    
    The one dimensional zero-range process (ZRP) whose particle flux is a strictly convex or concave function of particle density is believed to be a member of the Kardar-Parisi-Zhang (KPZ) universality class. These lectures develop a proof that a certain class of these processes possess fluctuation exponents that agree with KPZ universality. In particular, we show that an observer who moves at the characteristic speed observes fluctuations of order t^{1/3} in the net particle current. The class of ZRPs we discuss is defined by exponential decay of the slope of the jump rate function.
    
    The proof relies on a microscopic version of the assumption of strictly concave flux. The derivative of the flux is the characteristic speed of the system at a given density. This is the speed at which a small discrepancy propagates under the scalar conservation law that governs the evolution of the macroscopic particle density. Thus for a concave flux, the speed of a discrepancy decreases with density. The microscopic counterpart of a discrepancy is a second class particle. The microscopic property that makes the proof work is that we can couple together two processes so that a second class particle in the higher density system is kept behind a second class particle in the lower density system, and so that the motions of these second class particles relative to the discrepancy particles between the two processes can also be suitably controlled. The assumption on the jump rate function enables us to perform this construction. This is joint work with Marton Balazs and Julia Komjathy.
  
  
  • Dates and Times:
    
    Monday, November 2nd, 4-5:30pm, 507 Math
    
    Sign up for dinner after the seminar
    
    Wednesday, November 4th, 5:30-7pm, 507 Math
    
    Monday, November 9th, 4-5:30pm, 507 Math
    
    Wednesday, November 11th, 5:30-7pm, 507 Math
    
    

Spring Semester 2015

• April 20--24
  

• Fabio Toninelli (Lyon)
  
  Poster
  
  

  • Monday April 20 and Wednesday April 22, 4:10-6pm, 417 Math
    
    Height fluctuations in interacting dimers
    
    Abstract:
    
    Perfect matchings of Z^2 (also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance (see e.g. Kenyon, Okounkov, Sheffield '06). As soon as dimers mutually interact, the model is not solvable any more. However, tools from "constructive field theory" allow to prove that, as long as the interaction is small, the height field still behaves like a gaussian log-correlated field. In these 4 hours, I will try to explain the main ideas and some mathematical tools of the proof (Grassmann representation for the partition function, plus some ideas of "constructive renormalization group"). Work in collaboration with A. Giuliani and V. Mastropietro.
    
    
    
  • Friday April 24, 12noon-1pm, 520 Math
    
    A class of (2+1)-dimensional growth process with explicit stationary measure
    
    We introduce a class of (2 + 1)-dimensional random growth processes, that can be seen as asymmetric random dynamics of discrete interfaces. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. “Asymmetric” means that the interface has an average non-zero drift. When the asymmetry parameter p − 1/2 equals zero, the infinite-volume Gibbs measures pi_\rho (with given slope \rho) are stationary and reversible. When p\neq 1/2, \pi_\rho is not reversible any more but, remarkably, it is still stationary. In such stationary states, one finds that the height function at a given point x grows linearly with time t with a non-zero speed, := <(hx(t) − hx(0))> = v t and that the typical fluctuations of Q_x(t) are smaller than any power of t.
    
    For the specific case p = 1 and in the case of the hexagonal lattice, the dynamics coincides with the “anisotropic KPZ growth model” studied by A. Borodin and P. L. Ferrari. For a suitably chosen initial condition (that is very different from the stationary state), they were able to determine the hydrodynamic limit and the interface fluctuations, exploiting the fact that some space-time correlations can be computed exactly, and predicted stationarity of Gibbs measures.
    

• April 8--10
  

• Wendelin Werner (ETH Zurich)
  
  Poster
  
  

  • Wednesday April 8, 4:10-6pm, 417 Math, and
    
    Thursday April 9, 5:40-7pm, 520 Math
    
    Critical phenomena within (random) planar carpets
    
    Abstract:
    
    This series of two lectures will be a survey of some recent developments concerning some natural two-dimensional fractals (that bear some similarities with the Sierpinski carpets) called Conformal Loop Ensembles and more specifically how to understand and describe a process that can be interpreted as critical phenomena within these carpets. This exhibits a notion of conformal invariance within such domains, even if their fractal dimension is strictly smaller than two. Part of these lectures will be based on joint work with Jason Miller and Scott Sheffield.
    
    
    
    
  • Friday April 10, 11am-12noon, 520 Math
    
    A simple graph-valued Markov processes and renormalization
    
    Abstract:
    
    We describe a simple graph-valued Markov process and will explain how it can provide an elementary approach to some renormalization group ideas and questions. In particular, a (small) perturbation of the conjectured continuous scaling limits of critical models appear as stationary distributions for these Markov chains. We also point out how to relate this approach to some of the known results concerning scaling limits for some two dimensional models (the relation to uniform spanning tree is joint work with Stéphane Benoist and Laure Dumaz).
    

• March 9--11
  

• Philippe Di Francesco (University of Illinois at Urbana-Champaign)
  
  

• Title
  
  Integrable Combinatorics
  
  
  
• Abstract
  
  Integrability occurs in physical problems with sufficiently many symmetries, and allows for exact, often elegant solutions with deep geometric and algebraic meaning. Such problems often boil down to that of enumerating weighted configurations of particular systems, which can be rephrased in purely combinatorial or probabilistic terms.
   
  In these lectures, we review manifestations of integrability in various enumeration problems such as Lorentzian Triangulations, Planar Maps, Alternating Sign Matrices, Domino Tilings, Plane Partitions, Current Algebra Tensor Product Multiplicities, and their interplay with the underlying structure of Cluster Algebra and its quantum deformation.
  
  
  
• Time and location:
  
  Monday March 9, 5:30-7:00pm, 507 Math [w/ the Informal Math-Physics seminar]
  
  Tuesday March 10, 5:30-7:00pm, 622 Math
  
  Wednesday March 11, 4:10-6pm, 417 Math (tbc)
  
  
  
• Slides: Lecture 1; Lecture 3.
  

Spring Semester 2014

• April 21--25
  

• Alexei Borodin (MIT)
  
  Poster
  

• Mon Apr 21 and Thu Apr 24, 5:30-7pm, Math 507
  
  Macdonald processes
  
  Abstract:
  
  Our goal is to explain how certain basic representation theoretic ideas and constructions encapsulated in the form of Macdonald processes lead to nontrivial asymptotic results in various `integrable' probabilistic problems. Examples include dimer models, general beta random matrix ensembles, and various members of  the (2+1)d anisotropic KPZ and (1+1)d KPZ universality classes, such as growing stepped surfaces, q-TASEP, q-PushASEP, and directed polymers in random media.
  
  

• Fri Apr 25, 11am-12noon, School of Social Work 903
  
  Gaussian Free Field in beta ensembles and random surfaces
  
  Abstract:
  
  The goal of the talk is to argue that the two-dimensional Gaussian Free Field is a universal and unifying object for global fluctuations of spectra of random matrices and random surfaces. This viewpoint leads to natural Gaussian processes on larger spaces which, despite their explicit covariance structure, so far lack conceptual understanding.
  
  
  Also note the propaedeutic talk:
  

• Mon Apr 14, 5:30-7pm, Math 507 - Informal Mathematical Physics seminar
  
  Macdonald polynomials for everyone
  

• February 24 -- February 28
  

• Martin Hairer (Warwick)

• Title: Regularity Structures
  

• Abstract:
  
  When considering the large-scale behaviour of physical models arising
  naturally in statistical mechanics, one is often lead to stochastic partial
  differential equations that seem to be nonsensical: they are typically
  nonlinear with nonlinearities involving products of distributions. Examples
  of such models are the KPZ equation, the dynamical $\Phi^4_3$ model,
  the continuous parabolic Anderson model, etc.
  
  The theory of regularity structures provides a unified way of interpreting
  and analysing these equations in a robust way. It also provides a clean
  separation of the problem into an algebraic, an analytic, and a probabilistic
  component. I will give an introduction to the main concepts and results
  of the theory, as well as a number of applications.
  

• Time & location
  
  Mon, Tue, Wed, Thu: Math 507, 5:30-7pm.
  Fri: 11am-1pm, School of Social Work 903
  

Fall Semester 2013

• October 21-November 1, 2013

• Nizar Touzi (CMAP Polytechnique)

• Title: Viscosity solutions of path-PDEs and applications

• Abstract:

  The notion of classical solutions of path-dependent PDEs was introduced by Bruno Dupire in the context of the pricing and hedging of path-dependent derivatives. This can be seen as a classical version of the martingale representation in a Brownian filtration. Based on this, we introduce a notion of viscosity solu- tions for path-dependent parabolic nonlinear PDEs. We provide the corresponding well-posedness and stability results. Similar to the standard corresponding PDE literature, uniqueness is implied by a com- parison result. We also adapt this notion in the context of path-dependent elliptic PDEs. Examples in- clude backward SDEs, non-Markov stochastic control and stochastic differential games, and Monte Carlo methods for nonlinear path-PDEs.

Spring Semester 2012

• (Academic semester)
  

• Grigori Olshanski (Institute for Information Transmission Problems of the Russian Academy of Sciences)
  
  Title: Determinantal Processes and Related Topics

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Fall Semester 2011

• September 7--15, 2011
  
  

  • Denis Talay (INRIA, Sophia-Antipolis)
    Model Risk: Modeling, Analysis, Control and Numerics
    
    Poster
    
    All lectures are 90 minutes.
    Lecture I: 9 am September 7 in Room 903 SSW
    Lecture II: 9 am September 8 in Room 507 Math
    Lecture III: 9 am September 12 in Room 903 SSW
    Lecture IV: 9 am September 14 in Room 903 SSW
    Lecture V: 9 am September 15 in Room 507 Math
    
    Abstract for lectures:
    The objective of these lessons is to show that model risk analysis, particularly financial model risk analysis, opens new interesting stochastic analysis problems, to present recent mathematical and numerical techniques to tackle them, and to analyze mathematically some robust strategies which, issued from the technical analysis, do not rely on a specific mathematical model. We will also present a selection of challeng- ing open questions.
    
    Various theories will be used, such as statistics of random processes, stochastic control, Malliavin calculus, backward stochastic differential equations, viscosity solutions of nonlinear Partial Differential equations. However the course will be self-contained and, whenever possible, the proofs will be fully detailed.
    
    

  
  

  September 22--October 27, 2011
  
  

  • Vladas Sidoravicius (IMPA)
    Phase transitions in stochastic processes with long-range interactions: a multiscale analysis approach
    
    Poster
    
    Thursdays 9:10-10:40am in Mathematics Hall 507.
    
    Abstract: Stochastic processes with long-range interactions play an important role in many areas of science, ranging from abstract questions of Probability Theory and Statistical Mechanics to applied problems in biology, genetics, chemistry and many others.
    
    During my lectures I will focus on three classical one-dimensional models: long-range percolation, Ising model and Markov chains with complete connections. Processes with long-range interaction even in one dimensional case present very rich and complex behavior, and undergo phase transitions. However due to infinite range of interactions the behavior of systems is very complex and traditional techniques fail to capture the picture. One of aims of this course is to introduce and explain some very powerful multi-scale analysis techniques (with their roots in Renormalization Group approach), which are remarkably efficient in the present situation.
    
    The course will follow nearly chronologically the development of a beautiful chapter of modern Probability (and Statistical Mechanics), which started with W. Doeblin, and I will explain fundamental works of Freeman Dyson, J. Frohlich and T. Spencer, C. Newman and L. Schulman, M. Bramson and S. Kalikow, whose deep ideas shaped this field.
    
    During the course I will also explain all necessary techniques and basic concepts from Percolation Theory and Random Spatial Processes. The course requires knowledge of basic Probability and Markov Chains.
    
    

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  Spring Semester 2011

  • February 1--8, 2011
    
    

    • Johannes Muhle-Karbe (ETH Zurich)
      Shadow Prices in Portfolio Optimization with Transaction Costs
      
      Poster
      
      Lecture I: 4--6 pm February 1 in Room 312 Math
      Lecture II: 4--6 pm February 3 in Room 312 Math
      Lecture III: 4--6 pm February 7 in Room 622 Math
      Lecture IV: 4--6 pm February 8 in Room 312 Math.
      
      Abstract for lectures:
      Talk I, Feb 1: Motivation, History, and Existence (based on http://arxiv.org/abs/0911.4801 and some new results, joint work with Jan Kallsen, Mark Owen, and Luciano Campi)
      Abstract: A ``shadow price'' is a process evolving within the bid-ask spread of a market with proportional transaction cots, such that the maximal expected utility in this frictionless market is the same as in the original market with transaction costs. In this talk, we introduce this concept and also outline its origins, which go back to Jouini & Kallal (J. Econom. Theory, 1995), Kusuoka (Annals Appl. Probab., 1995), and Cvitanic & Karatzas (Math. Finance, 1996). We also present an elementary existence proof for finite probability spaces. Moreover, we discuss work in progress on existence in more general setups.
      
      Talk II, Feb 3: The Growth-Optimal Portfolio under Transaction Costs (based on http://arxiv.org/abs/1005.5105, joint work with Stefan Gerhold and Walter Schachermayer)
      Abstract: In this talk, we discuss how to use the idea of ``shadow prices'' for computations. More specifically, we determine a shadow price whose growth-optimal portfolio coincides with the one for proportional transaction costs in the Black-Scholes model. This provides a new simple proof for the results of Taksar et al. (Math. Oper. Res., 1988). Moreover, it also leads to asymptotic expansions of the optimal policy and the maximal growth rate for small transaction costs.
      
      Talk III, Feb 7: Maximizing Log-Utility from Consumption under Transaction Costs (based on http://arxiv.org/abs/1010.4989 resp. http://arxiv.org/abs/1010.0627, joint work with Jan Kallsen resp. Stefan Gerhold and Walter Schachermayer)
      Abstract: We revisit the problem of maximizing expected logarithmic utility from consumption over an infinite horizon in the Black-Scholes model with proportional transaction costs, as studied in the seminal paper of Davis and Norman (Math. Oper. Res., 1990). As in Talk II, we tackle this problem by determining a shadow price. Moreover, for small transaction costs, we again determine power series of arbitrary order for the optimal policy and the value function. This extends work of Janecek and Shreve (Finance Stoch., 2004), who determined the first-order terms.
      
      Talk IV, Feb 8: Long-Run Optimal Portfolios under Transaction Costs (work in progress, joint with Stefan Gerhold, Paolo Guasoni, and Walter Schachermayer) Abstract: The computations in Talks II and III crucially exploited that the investor's preferences are modeled by a logarithmic utility function. In this talk, we describe how to determine shadow prices also for power utility functions. More specifically, we focus on the long-run optimal portfolio in the Black-Scholes model with proportional transaction costs and provide a rigorous proof for the results of Dumas and Luciano (J. Finance, 1991). Moreover, we again explain how to obtain full asymptotic expansions for the optimal policy and the optimal growth rate.
      
      

    
    

    March 29--April 1, 2011
    
    

    • Darrell Duffie (Stanford University)
      Dark Markets
      
      Poster
      
      Tuesday, March 29
      10:30am - 12:00pm, SSW 1025
      
      Wednesday, March 30
      10:30am - 12:00pm, SSW 1025
      
      Friday, April 1
      12:00pm - 1:00pm, SSW 903
      
      Abstract for lectures:
      
      The financial crisis of 2007-2009 brought significant concerns and regulatory action regarding the role of over-the-counter markets, particularly from the viewpoint of financial instability. Over-the- counter markets for derivatives, collateralized debt obligations, and repurchase agreements played particularly important roles in the cri- sis and in subsequent legislation in U.S. and Europe. This legisla- tion has also focused on increasing competition and transparency. The modeling of OTC markets, however, is still relatively undeveloped in comparison to the available research on central market mechanisms.
      
      Rather than trading through a centralized mechanism such as an auction, specialist, or limit-order book, over-the-counter mar- ket participants negotiate terms privately with other market partici- pants, often pair-wise. Over-the-counter investors may be largely un- aware of prices that are currently available elsewhere in the market, or of recent transactions prices. In this sense, OTC markets are relatively opaque; investors are somewhat in the dark about the most attractive available terms and about who might offer them. These lectures addresses how prices, asset allocations, and information transmission in OTC markets are influenced by this form of opaque- ness. The objective is to provide a brief introduction to OTC mar- kets, including some of the key conceptual issues and modeling tech- niques, and to provide a foundation for reading more advanced re- search in this topic area. The lectures assume a graduate-level background in probability theory.
      
      

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  Fall Semester 2010

  • December 3--20, 2010
    
    

    • Boris L. Rozovsky (Brown University)
      Generalized Malliavin Calculus and Stochastic PDEs
      
      Abstract for lectures: [pdf]
      
      
      • Friday Dec 3, 12--1 pm in Math 520: Probability Seminar.
      • Friday, Dec 3, 2.30--3.20 pm in Stat 1025: Lecture I. ``Stochastic Quantization and Navier-Stokes Equation.''
      • Friday, Dec 10, 10--11:30 am in Math 622: Lecture II. ``Introduction to Malliavin calculus''.
      • Friday, Dec 17, 2.30--4.00 pm in Stat 1025: Lecture III. ``Generalized Malliavin calculus''.
      • Monday, Dec 20, 10--11.30 am in Stat 1025: Lecture IV. ``Bi-linear stochastic PDEs driven by stationary noise''.

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    SPRING Semester 2010

  • April 12--22, 2010
    
    

    • Ofer Zeitouni (UMN & Weizmann Institute)
      Random Walks in Random Environments

           
  

  • Mar 25--29, 2010
    
    

    • Geoffrey Grimmett (Cambridge)
      Lattice models in probability
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  • Mar 22--25, 2010
    
    

    • Walter Schachermeyer (Vienna)
      The asymptotic theory of transaction costs
      

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  • FALL Semester 2009

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                    September 9--30, 2009

• Jean Bertoin (Universite Paris VI)
  Exchangeable Coalescents

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  •                                       Spring Semester 2009

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    • February-April, 2009
      

    •                 Hans Follmer (Berlin)
                      Convex Risk measures and their dynamics

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  •  January 20--27, 2009
    

  •                 Sara Biagini (Pisa)
                    Topics in Portfolio Optimization

  
  

• • FALL Semester 2008

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                    October--November, 2008

• Shige Peng (Sandong University)
  Coherent and convex risk measures and nonlinear expectations

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