IMPAN Colloquium
Wednesdays at 14:15 in Room 321, before the lecture there are cookies and tea in room 409 at 13:45
Organizers:
Marcin Lara, Grigor Sargsyan, Mateusz Wasilewski, Aneta Wróblewska-Kamińska
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UPCOMING MEETINGS
UPCOMING MEETINGS
JUNE 11
Mateusz Kwaśnicki (Wrocław University of Science and Technology)
Caffarelli–Silvestre extension technique: analytic and probabilistic perspectives
Consider a path of the reflected Brownian motion in the half-plane {y ≥ 0}, and erase its part contained in the interior {y > 0}. What is left is, in an appropriate sense, the path of a jump-type stochastic process on the line {y = 0} — the boundary trace of the reflected Brownian motion. It is a classical result that this process is in fact the 1-stable Lévy process, also known as the Cauchy process.
The PDE interpretation of the above fact is the following. Consider a bounded harmonic function u in the half-plane {y > 0}, with sufficiently smooth boundary values f. Let g denote the normal derivative of u at the boundary. The mapping f ↦ g is known as the Dirichlet-to-Neumann operator, and it is again well known that this operator coincides with the square root of the 1-D Laplace operator −Δ. Thus, the Dirichlet-to-Neumann operator coincides with the generator of the boundary trace process.
Molchanov and Ostrovskii proved that isotropic stable Lévy processes are boundary traces of appropriate diffusions in half-spaces. Caffarelli and Silvestre gave a PDE counterpart of this result: the fractional Laplace operator is the Dirichlet-to-Neumann operator for an appropriate second-order elliptic equation in the half-space. Again, the Dirichlet-to-Neumann operator turns out to be the generator of the boundary trace process.
During my talk I will discuss boundary trace processes and Dirichlet-to-Neumann operators in a more general context. My main goal will be to explain the connections between probabilistic and analytical results. Along the way, I will present an answer to the question posed by Caffarelli and Silvestre in their seminal paper about the class of operators that arise as Dirichlet-to-Neumann maps in a half-space, obtained jointly with Jacek Mucha, as well as more recent results about processes and operators in a half-plane.
PAST MEETINGS (2024/2025)
OCTOBER 2
Janusz Grabowski (IMPAN Warszawa)
Multiplication by reals on manifolds
The playground for differential geometry is formed by manifolds. They are geometric structures that locally are like Euclidean spaces, e.g., spheres, tori, or projective spaces. What is crucial is that we can develop a differential calculus on manifolds like on Rn.
Particularly important, also in applications, are linear structures generalizing vector spaces and called vector bundles. They are locally products U x Rn, where the base U is an open subset of a Euclidean space and the fibers {p} x Rn carry vector space structures. The multiplication by reals is defined globally on every vector bundle, while the vector addition is defined on each fiber separately.
The main message is that one can forget about the addition, since every vector bundle is completely determined by its multiplication by reals. This approach, originated by Mikołaj Rotkiewicz and me, simplifies radically big parts of differential geometry. In particular, the concept of the compatibility of a vector bundle structure with other geometric structures on the manifold is then very easy, as we can use only the multiplication by reals and forget the other parts of the vector bundle structure. For instance, two vector bundle structures are compatible if the corresponding multiplications by reals commute.
We can define also abstract multiplications by reals on manifolds. Unexpectedly, such structures are very rigid and have been fully described locally by our tandem with Rotkiewicz. Canonical examples, that are not vector bundles, are higher tangent bundles, i.e. bundles of jets of curves on manifolds.
OCTOBER 9
Artem Dudko (IMPAN Warszawa)
On spectrum of self-similar groups and graphs
Studying spectra of self-similar groups and related questions led to many important discoveries. Self-similar groups contain examples of groups solving various important open questions, including the Grigorchuk group (the first example of finitely generated groups of intermediate growth), the Basilica group (first example of amenable but not subexponentially amenable groups), and the Lamplighter group (a counterexample to the strong Atiyah conjecture). In this talk I will give an introduction to self-similar groups, discuss methods of studying their spectral properties and spectral properties of associated graphs, and present some recent results on this topic (joint with Rostislav Grigorchuk).
OCTOBER 16
Martin Bridson (University of Oxford & Clay Mathematics Institute)
Chasing finite shadows of infinite groups through geometry
There are many situations in geometry and group theory where it is natural, convenient or necessary to explore infinite groups via their actions on finite objects – i.e. via the finite quotients of the group. But how much understanding can one really gain about an infinite group by examining its finite images? Which properties of the group can one recognise, and when does the set of finite images determine the group completely? How hard is it to decide what the finite images of an infinite group are?
In this talk, I will sketch some of the rich history of these problems and present results that illustrate how the subject has been transformed in recent years by input from low-dimensional topology and the study of non-positively curved spaces.
OCTOBER 23
Masha Vlasenko (Kyiv School of Economics & IMPAN Warszawa)
Varieties in the mirror and arithmetic
In physics, string theory considers elementary particles as one-dimensional objects called strings. In this theory spacetime has more than four dimensions, and the extra dimensions are described by complex manifolds of a special type, the so-called Calabi-Yau manifolds. Mirror symmetry is a phenomenon when two Calabi-Yau manifolds look very different geometrically, but are nevertheless equivalent when used as extra dimensions to describe interactions of particles. I plan to tell a story from the beginnings of mirror symmetry, when Candelas, de la Ossa, Green and Parkes discovered that difficult problems in enumerative geometry could be solved quickly ''in the mirror'' using differential equations. Physicists also noticed that the so-called instanton numbers observed on the other side of the mirror turned out to be integers. Together with Frits Beukers we were able to verify this conjecture in several key examples of mirror symmetry.
OCTOBER 30
Barbara and Jaroslav Zemanek Prize Award Ceremony
14:15–15:15 Introductory lecture
Stuart White (University of Oxford)
Introduction to the classification of simple nuclear C*-algebras
In this talk I'll give an introduction to simple nuclear C*-algebras and their classification. This will be illustrated by examples coming from group actions. My aim will be to describe at a high level the paradigm shift from using internal structure to classify, to obtaining internal structure from classification that Chris Schafhauser's work made possible. No prior knowledge of operator algebras will be assumed.
15:45–16:45 Lecture of the laureate
Christopher Schafhauser (University of Nebraska–Lincoln)
Lifting problems in C*-algebras and applications
A classical problem of Halmos asks which essentially normal operators (those commuting with their adjoint modulo a compact operator) on Hilbert space are compact perturbations of normal operators (those commuting with their adjoint). A complete solution was obtained by Brown, Douglas, and Fillmore in the early 70s, and their solution led to the introduction of algebraic topological methods in operator algebras. In particular, for a compact metric space X, they considered all embeddings of C(X) into the quotient B(H)/K(H) of bounded operators on a Hilbert space modulo the compact operators and showed that homotopy classes of such embeddings form an abelian group K_1(X), which is the degree one term for a generalized homology theory dual to topological K-theory. Building on this, Kasparov developed a much more general extension theory, studying lifting problems along more general quotient maps up to a stabilized notion of homotopy. I will discuss some recent progress in `non-stable’ extension theory with applications to embedding problems and classification problems for simple nuclear C*-algebras.
NOVEMBER 6
Maciej Borodzik (IMPAN Warszawa)
Slice knots
Understanding knot concordance is one of the most difficult and most intricate problem in low-dimensional topology. It has connections with deformation of singularities, geometry of plane curves on the one side, and the smooth Poincare conjecture in dimension four, on the other. In this talk I will present classical and modern tools used for studying knot concordance.
NOVEMBER 13
Iwona Chlebicka (MIMUW)
Fast p-Laplace evolution
Being a natural nonlinear counterpart of the heat equation, p-Laplace evolution equation is studied since 1960s. It's difficult to say whether the significant interest it enjoys is driven more by its numerous applications or by its intricate and profound intrinsic mathematical properties, which vary with the exponent p. For p=2 one deals with the linear equation, which is well-understood. In the interval 1<p<2, numerous thresholds emerge, governing the dynamics. My talk will focus on the relaxation towards self-similarity of nonnegative solutions and its rate across various subranges of small p. It is based on a joint project arXiv:2405.05405 with Matteo Bonforte (UAM, Madrid) and Nikita Simonov (Sorbonne University, Paris).
NOVEMBER 20
Joanna Kułaga-Przymus (Nicolaus Copernicus University)
Randomness of the Möbius function and dynamics
The Möbius function, central to analytic number theory, is conjectured to exhibit randomness similar to independent sequences. During my talk I will concentrate on the interplay between number theory and dynamics, presenting two main strategies towards proving Sarnak's conjecture on the absence of correlations between the Möbius function and deterministic dynamical systems. I will include results on Veech's conjecture, joint with A. Kanigowski, M. Lemańczyk and T. de la Rue and provide insights into the current obstacles in establishing Sarnak's conjecture.
NOVEMBER 27
Błażej Wróbel (IMPAN Wrocław/University of Wrocław)
Three discrete maximal functions in high dimensions - Gaussians, balls and spheres
High-dimensional phenomena constitute an important research theme that has been present in harmonic analysis for several decades. It was initiated by E. M. Stein in 1980s yet still a number of open problems remain unresolved. The main task is usually to obtain dimension-free estimates for norms of various operators as the dimension goes to infinity. The study of dimension-free estimates in the discrete setting was initiated several years ago by J. Bourgain, E. M. Stein, M. Mirek and myself. It is an example of discrete analogues in harmonic analysis - a vibrant area of modern harmonic analysis.
During my talk I will discuss discrete maximal functions, for Gaussians, balls, and spheres. The study of dimension-free estimates for these maximal functions requires bounding certain multiplier symbols, which are highly symmetric exponential sums. This requires interdisciplinary tools from analytic number theory (Warring's problem) and combinatorics. A preliminary step in the case of balls and spheres is a count of lattice points in these sets which has to be performed in a uniform (dimension-free) manner. I will report on recent progress with M. Mirek and T.Z. Szarek (spheres and Gaussians), and with J. Niksiński (balls and spheres).
DECEMBER 4
Feliks Przytycki (IMPAN Warszawa)
Geometric pressure and periodic orbits for iteration of quadratic polynomials in the complex plane
Hyperbolic Hausdorff dimension of the Julia set J(f) for a rational mapping f (ratio of two polynomials) of degree at least 2 on the Riemann sphere is defined as the supremum of the Hausdorff dimensions of its invariant hyperbolic subsets (usually it is just the Hausdorff dimension of the Julia set itself). It is the first zero t = t0 of the geometric pressure function P(f, −tlog|f′|). There are various equivalent definitions of this pressure, e.g. variational. I will sketch the proof that it can be expressed via periodic trajectories in the case of quadratic polynomials, partially answering an old problem. The method I use is to show that there are not many periodic trajectories going in bunches, using Milnor’s Orbit Portraits for external rays and their arguments.
A special case is the question of how many (at most) periodic trajectories of period n can be entirely in the disc B(x, rn) for rn small enough, for Cremer’s fixed point x (or periodic trajectory). Cremer’s means that the linear part of f at x is multiplication by exp(2πiα) with α real irrational and f is not linearizable at x (that happens if α is fast approximated by its continued fraction rational convergents pn/qn). I can give an answer for quadratic polynomials and rn ≤ exp(−δn): at most one periodic trajectory for any δ > 0 and all n large enough.
DECEMBER 18
Jacek Jendrej (Institut de Mathématiques de Jussieu)
Dispersion, solitons, their stability and emergence
Dispersive partial differential equations are evolution equations (that is, involving the time variable) whose solutions preserve the energy, but can still decay in large time due to the fact that various frequencies propagate with distinct velocities. In some cases, there exist non-trivial special solutions called solitons, which do not change their shape as time passes. Usually they are not stable in the usual Lyapunov sense, and the study of their (appropriately defined) stability is a major challenge.
I will discuss these issues in the case of a nonlinear wave equation known as the φ4 model. I will introduce the solitons for this model and explain why the question of their stability is a difficult one. A special case was solved a few years ago by Kowalczyk, Martel and Muñoz. I will also mention related results on similar models.
JANUARY 8
Justyna Signerska-Rynkowska (Dioscuri Center in Topological Data Analysis, Gdansk University of Technology)
Low-dimensional dynamics enhancing modeling of neuronal activity
The talk will start with a brief introduction to the history of neuron modelling, including models of Lapicque, Hodgkin-Huxley and FitzHugh-Nagumo. These pioneering models were proposed decades ago and since then many other models arose, including hybrid systems combining continuous dynamics with discrete resets accounting for neuronal spiking.
Due to the variety of these models, it is important to indicate the most universal and reliable tools allowing understanding dynamical mechanisms shaping their properties, related to diversity of neuronal activity and biological relevance. We will see how the theory of low dimensional dynamics (e.g. rotation theory, S-unimodal maps, phase-space methods) can be effectively applied in the analysis of map-based and hybrid neuron models, commonly used in computational and theoretical neuroscience. I will also mention recent tools adapted from nonlinear time series analysis and computational topology facilitating, among others, classification of spike-trains and phase-portraits in neuron models.
JANUARY 15
Piotr Nowak (IMPAN Warsaw)
Rigid groups and spectral gaps
The goal of this talk is to discuss Kazhdan’s property (T), a classical, powerful rigidity property of groups, and a new approach to proving it via algebraic spectral gaps for the Laplacian in the group ring. These new methods grew out of a new description of property (T) in terms of noncommutative sums of square due to Ozawa, and in particular I will present how it was used to prove property (T) for Aut(F_n), the automorphism groups of free groups, for n at least 5. I will also discuss some applications and generalizations to higher cohomology and higher index theory.
JANUARY 22
Jacinta Torres (Jagiellonian University)
Algebraic combinatorics and geometry in representation theory
The first part of this talk will be a brief introduction to the representation theory of finite groups, Lie algebras and groups, and associative algebras, mostly over the field of complex numbers. In the second part of the talk I will explain how objects in representation theory can be modelled using combinatorics and geometry. Finally, I will highlight some of the landmark advances in the field and present some of my own contributions as well as various open problems.
JANUARY 29
Marcin Napiórkowski (University of Warsaw)
Bose-Einstein condensation: an ongoing mathematical challenge
Proving Bose-Einstein condensation in the thermodynamic limit remains a major open problem in mathematical physics. In my talk, I will explain the content of the conjecture and review recent progress in the study of bosonic many-body systems.
FEBRUARY 19
Jan Rozendaal (IMPAN Warszawa)
The local smoothing conjecture
The local smoothing conjecture for the Euclidean wave equation is one of the main open problems in harmonic analysis. It is concerned with determining, in a quantitative sense, the extent to which waves can collide and focus in regions of flat space. This conjecture is known to imply several other major and seemingly unrelated problems in harmonic and geometric analysis, all of which have been open for decades, and work on the conjecture has also led to advances in areas such as analytic number theory. In this talk I will discuss the local smoothing conjecture, versions of the conjecture on smooth manifolds, and some recent work on manifolds with rough metrics.
FEBRUARY 26
Karol Palka (IMPAN Warszawa)
Minimal models of singular algebraic surfaces
The Minimal Model Program gives a method to make algebraic varieties smaller by contracting or modifying their subvarieties. The outcomes of the program, the minimal varieties, are then of special importance, as all other varieties can be recovered from them. We will discuss the notion of minimality for singular (rational) algebraic surfaces. A description of possible minimal models, called del Pezzo surfaces of Picard rank 1, was not known until recently. We will explain some ideas of our classification program (a joint work with Tomasz Pełka).
MARCH 5
Mateusz Wasilewski (IMPAN Warszawa)
What are noncommutative graphs?
Noncommutative graphs are objects appearing in quantum information while studying zero-error capacity. of channels I will explain what these objects are in mathematical terms and how they can be seen as a relaxation of graphs. Even though many notions readily generalise to this setting, many challenges remain, notably the ones pertaining to paths, which are very important for classical graphs. I will also try to indicate connections to neighbouring areas of mathematics, such as operator algebras, (quantum) groups or random matrices.
MARCH 12
Urszula Foryś (MIMUW)
On mathematical modelling of tumour growth and treatment
I will present two examples of mathematical models used to reflect clinical data and specific type of treatment. The first example is related to standard type of treatment for hormone sensitive prostate cancer patients which is androgen deprivation therapy. The other one describes non-standard treatment which is immunotherapy of glioblastoma (brain tumour).
MARCH 19
Piotr M. Hajac (IMPAN Warszawa)
Counting paths in directed graphs
Graph theory is considered one of the oldest and most accessible branches of combinatorics and has numerous natural connections to other areas of mathematics. In particular, directed graphs, or quivers, are fundamental tools in representation theory as well as in noncommutative geometry and topology. In this talk, I will consider the class of directed graphs with N ≥ 1 edges and without loops shorter than k. Using the concept of a labelled graph, I will show how to determine graphs from this class that maximize the number of all paths of length k. To end with, I shall pose a related open problem concerning the maximal dimension of the path algebra of an acyclic graph with N ≥ 1 edges, and compute an upper bound for this dimension. Based on joint work with Oskar Stachowiak.
MARCH 26
Marcin Sroka (Jagiellonian University)
Canonical metrics in complex geometry through Hessian equations
The endeavor to find a canonical metric on a given space, Riemannian manifold, is one of the cornerstones of differential geometry. Since the fundamental works of Yau and Aubin on Calabi's conjecture this grew up to an interesting and important piece of complex geometry. I will underline the state of art regarding the existence of Einstein, constant scalar curvature or extremal Kähler metrics on complex manifolds. This will include the Calabi-Yau theorem, Yau-Tian-Donaldson conjecture and Tian's properness conjecture. If time permits, we will dive into the more technical set up of singular metrics and/or spaces as well as more recent results on complete non-compact spaces. The main focus will be to underline how all these existence problems on metrics relate to existence, regularity and other properties of solutions to a certain class of global PDEs.
APRIL 2
Łukasz Stettner (IMPAN Warszawa)
Risk-Sensitive Control of Markov Processes: Motivations, Results, and Problems
We consider (for simplicity) controlled discrete-time Markov processes with a risk-sensitive functional. This functional is motivated by the goal of maximizing the average reward while minimizing the variance (with an appropriate weight), a measure of risk that has long been studied (e.g., H. Markowitz, 1954). Additionally, the functional is interpreted as a law-invariant, time-consistent dynamic measure of risk (see M. Kupper and W. Schachermayer, 2009). Another motivation arises from certainty-equivalent control problems, which are typically time-inconsistent (i.e., the Bellman optimality principle does not apply). Interestingly, for a wide class of utility functions, the optimal controls for certainty-equivalent problems correspond to those derived from exponential utility functions, which are directly linked to risk-sensitive control (see Ł.S., 2023).
To solve the average reward risk-sensitive control problem, we must address a corresponding Bellman equation. The existence of solutions to this equation can be established either through duality representations (see G. Di Masi and Ł.S., 1999) or by applying the celebrated Krein-Rutman theorem (see Ł.S., 2024). In the final part of the talk, computational aspects will be discussed. For finite state and control spaces, a solution to the so-called Blackwell problem will be presented (see M. Pitera, N. Bauerle, and Ł.S., 2024).
APRIL 9
Jarosław Mederski (IMPAN Warszawa)
The nonlinear curl-curl problem
The nonlinear curl-curl problems have recently arisen in the search for exact propagation of electromagnetic waves in non-linear media modeled with classical Maxwell equations. For instance, the quintic effect leads to the critical partial differential equation with the strongly indefinite nature. Ground states solutions of the problem are related with the optimizers of a novel Sobolev-type inequality involving the curl operator. We present recent results concerning the existence of ground state and bound state solutions. Some symmetric properties of the problem and extensions to the p-curl-curl equation in the critical case with applications to zero modes of the Dirac equation will be also discussed.
APRIL 16
Marek Kuś (CFT PAN)
Symplectic Geometry of Quantum Entanglement
I will review a geometric approach to the classification and analysis of quantum correlations in composite systems. Since quantum information tasks are typically implemented by manipulating spin systems or, more generally, systems with a finite number of energy levels, classification problems are usually addressed within the framework of linear algebra. I propose to shift the focus towards a geometric description. By consistently treating quantum states as points in a projective space — rather than as vectors in a Hilbert space — we are able to apply powerful methods from differential, symplectic, and algebraic geometry to tackle the problem of classifying states according to the strength and nature of their correlations.
Such classifications can be interpreted as identifying states with equivalent correlation properties — that is, states which can serve the same information-processing purposes, or, from another perspective, states that can be transformed into one another via specific, experimentally accessible operations. This latter perspective addresses a fundamental question: what can be transformed into what using available resources? Crucially, this notion of mutual transformability can be naturally formulated in terms of the action of specific groups on the space of quantum states, providing the starting point for the methods I propose.
MAY 14
Lei Chen (Chinese Academy of Sciences), laureate of the Kamil Duszenko award for the year 2024
Actions of homeomorphism groups on manifolds
I will talk about my joint work with Kathryn Mann on actions of homeomorphism groups on manifolds. We prove an orbit classification problem which says that any action in some sense is similar to the diagonal action on products.
MAY 21
Otto Kong (National Central University, Taiwan)
Noncommutative Geometry from the Perspective of Quantum Physics
The phase space of quantum physics has been appreciated as a noncommutative symplectic geometry. Realizing Dirac's notion of q-numbers in a concrete manner, we have a picture of quantum mechanics as particle dynamics on a noncommutative geometric model of spacetime which we argue to be a candidate of the simplest q-number manifold that could serve as the starting point to construct a geometric language for the geometries. Quantum physics and quantum geometry of spacetime would then be simply the q-number version of the classical, real number, picture. Implications for a proper theory of quantum gravity will be discussed.
MAY 28
Jolanta Marzec-Ballesteros (Adam Mickiewicz University, Poznań)
On Fourier coefficients of modular forms
Modular forms have the remarkable property of appearing across diverse areas of mathematics. Often, their Fourier coefficients encode deep arithmetic information that sheds light on the original objects of study. In this talk, we begin by recalling a few examples of such situations involving modular forms of degree 1. We then turn to the case of higher degrees, focusing in particular on Siegel modular forms of degree 2. In this setting, understanding Fourier coefficients is considerably more challenging. We will explain why this is the case and highlight some of the known results.
JUNE 4
Wojciech Kucharz (Jagiellonian University)
Approximation of maps between real algebraic varieties
A nonsingular real algebraic variety Y (one can take Y to be an algebraic subset of Rn for some n) is said to have the approximation property if, for every nonsingular real algebraic variety X, the following holds: if f: X→Y is a C∞ map that is homotopic to a regular map, then f can be approximated in the C∞ topology by regular maps. I will briefly describe the history of the problem of characterizing the varieties Y with the approximation property, going back to the 1980s. Then, I will present a solution obtained in a recently published joint work with Juliusz Banecki.
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