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27 seminars found


, Tuesday

Geometria em Lisboa


, Université de Neuchâtel.

Abstract

Certain simple symplectic manifolds (symplectic vector space, Milnor fibres of certain complex surface singularities,...) contain sets of symplectically distinct Lagrangian tori which have the following remarkable property: they remain symplectically distinct under embeddings into any reasonable (i.e. geometrically bounded) symplectic manifold. This leads to a vast extension of the class of spaces in which the existence of exotic tori is known, especially in dimensions six and above. In this talk we mainly focus on recent joint work with Johannes Hauber and Joel Schmitz which treats the more intricate case of dimension four.


, Wednesday

Probability and Stochastic Analysis


Hindy Drillick, Columbia University.

Abstract

In this talk, we will consider two models for diffusing particles in time-dependent random environments: the discrete random walk in random environment (RWRE) and a continuum scaling limit of the RWRE called sticky Brownian motion. We will present some recent results on the weak convergence of both models to the KPZ equation in the moderate deviation regime. We will also discuss an application to the fluctuations of the maximal particle in these models. This is joint work with Sayan Das and Shalin Parekh.


, Tuesday

String Theory

Unusual schedule


Watse Sybesma, University of Iceland & Isaac Newton Institute for Mathematical Sciences.

Abstract

Starting from the Polyakov action we consider two distinct Carroll limits in target space, keeping the string worldsheet relativistic. The resulting magnetic and chiral Carroll string models exhibit different symmetries and dynamics. Both models have an infinite dimensional symmetry algebra with Carroll symmetry included in a finite dimensional subalgebra. For the magnetic model, this is the so-called string Carroll algebra. The chiral model realises an extended version of the string Carroll algebra. The magnetic model does not have any transverse string excitations. The chiral model is less restrictive and includes arbitrary left-moving modes that carry transverse momentum but do not contribute to the energy in target space.


, Wednesday

Topological Quantum Field Theory


, University of Leeds.

Abstract

The coloured Jones and Alexander polynomials are quantum invariants that come from representation theory. There are important open problems in quantum topology regarding their geometric information. Our goal is to describe these invariants from a topological viewpoint, as intersections between submanifolds in configuration spaces. We show that the Nth coloured Jones and Alexander polynomials of a knot can be read off from Lagrangian intersections in a fixed configuration space. At the asymptotic level, we geometrically construct a universal ADO invariant for links as a limit of invariants given by intersections in configuration spaces. The parallel question of providing an invariant unifying the coloured Jones invariants is the subject of the universal Habiro invariant for knots. The universal ADO invariant that we construct recovers all of the coloured Alexander invariants (in particular, the Alexander polynomial in the first term).


, Thursday

Applied Mathematics and Numerical Analysis

Room P3.10, Mathematics Building, Instituto Superior TécnicoInstituto Superior Técnico


Alberto Girelli, Department of Mathematics and Physics Università Cattolica del Sacro Cuore Brescia, Italy.

Abstract

Lymph nodes (LNs) are organs scattered throughout the lymphatic system which play a vital role in our immune response by breaking down bacteria, viruses, and waste; the interstitial fluid, called lymph once inside the lymphatic system, is of fundamental importance in this process as it transports these substances inside the lymph node. The main mechanical features of the lymph node include the presence of a porous bulk region (lymphoid compartment, LC), surrounded by a thin layer (subcapsular sinus, SCS) where the fluid can flow freely.

These nodes are vital for filtering and processing lymph, which contains immune cells, antigens, and other molecules. Understanding the fluid dynamics within lymph nodes is essential for elucidating immune mechanisms and developing therapies for lymphatic disorders. Despite its significance, few models in the literature attempt to describe lymph behavior from a mechanical perspective.

In this talk, we will introduce a mathematical model, derived using the asymptotic homogenization technique, to describe fluid flow within a lymph node, considering its multiscale nature. We will discuss how this model can elucidate flow patterns, pressure distribution, and shear stress within the node.

, Thursday

Colloquium of Logic


, Departement of Mathematics and LASIGE, Ciências ULisboa.

Abstract

I will give a board-and-chalk-and-informal-talk presentation about the difficulty of proving lower-bounds in computational complexity.

The P vs NP problem is one of the most famous unsolved problems in mathematics. One may phrase the P vs NP question in various equivalent ways. One way, which is not completely equivalent, but almost, is the following ("P/poly vs NP problem"). Does there exist a small Boolean circuit which solves the CLIQUE problem? I.e., does there exist a $\operatorname{poly}(N)$-size Boolean circuit which, when given as input the $N\times N$ adjacency matrix of an undirected graph, decides whether the graph has a clique of size $N/2$?

Complexity theorists, me included, believe that the answer is no. We believe that there exists a super-polynomial "lower-bound" on the complexity of CLIQUE. Many people have tried proving such a lower-bound, and so far all have failed. But why? Why is the problem so difficult?

In the late 1980s, Alexander Razborov proved that there exist no $\operatorname{poly}(N)$-size "monotone" circuits for solving CLIQUE. Namely, if we forbid the Boolean circuit from doing negations, so they can only do ORs and ANDs, then polysize circuits cannot solve the CLIQUE problem. He (and many others) then tried to prove the same result for ordinary circuits (with negation gates). And he failed (and they all failed, too). But along the way he (and many others) proved many different lower-bounds. Lower-bounds for simpler kinds of circuits (e.g. constant-depth), lower-bounds for communication protocols (a different but related computational model), and lower-bounds for other models. Razborov proved these lower-bounds, and he also thought long and hard about why lower-bounds against Boolean circuits were so difficult to prove.

In 1994, Razborov and Stephen Rudich presented their paper, "natural proofs", which had a very reasonable explanation for why circuit lower-bounds were difficult to prove. They showed, remarkably, that every single lower-bound proof that was known at the time had a certain "logical structure" (or could be made to have such a structure by small changes to the argument). This logical structure made the proof very simple and natural, and they called proofs with such a structure "natural proofs". Then they showed that super-polynomial lower-bounds on CLIQUE cannot be shown using natural proofs, unless certain cryptographic primitives, such as factoring, are unsecure. This is a kind of informal independence result. (Based on the natural proofs result, Razborov also later proved formal independence results, showing that P vs NP is independent of certain weak systems of arithmetic, but I do now know the details of those.)

In one fell swoop, Razborov and Rudich ruled out every single lower-bound technique known at the time, saying: these techniques are not enough to solve the P vs NP problem (unless cryptography is insecure). To a very large extent this barrier still applies today, as almost all the lower-bound proofs that we know are natural proofs, i.e., they have the very same logical structure as the proofs known since the 1980s.

In this seminar, I will explain what is a "natural proof", and why it is reasonable to expect that no natural proof can solve the P vs NP problem. Only a few words will be said about some of my research and how it connects to this topic.


, Friday

Topological Quantum Field Theory

Unusual schedule
Room P3.10, Mathematics Building, Instituto Superior TécnicoInstituto Superior Técnico


, University of Cologne.

Abstract

Starting with a counting problem for elements of the symmetric group, we introduce the so-called shifted symmetric functions. These functions, which also occur naturally in enumerative geometry, have the remarkable property that the corresponding generating series are quasimodular forms. We discuss another family of functions on partitions with the same property. In particular, using certain Hamiltonian operators associated to cohomological field theories, we explain how this seemingly different family of functions turns out to be closely related to the shifted symmetric functions.


, Wednesday

Probability and Statistics


Taban Baghfalaki, Bordeaux University, Bordeaux, France.

Abstract

Dynamic event prediction, using joint modeling of survival time and longitudinal variables, is extremely useful in personalized medicine. However, estimating joint models that include multiple longitudinal markers remains a computational challenge due to the large number of random eff ects and parameters that need to be estimated. We propose a model-averaging strategy to combine predictions from several joint models for the event, including models with only one longitudinal marker or pairwise longitudinal markers. The prediction is computed as the weighted mean of the predictions from the one-marker or two-marker models, with the time-dependent weights estimated by minimizing the time-dependent Brier score. This method enables us to combine a large number of predictions issued from joint models to achieve a reliable and accurate individual prediction. The advantages and limitations of the proposed methods are highlighted by comparing them with the predictions from well-specifi ed and misspecifi ed all-marker joint models, as well as one-marker and two-marker joint models, using the available PBC2 dataset. The method is used to predict the risk of death in patients with primary biliary cirrhosis. The method is also used to analyze a French cohort study called the 3C data. In our study, seventeen longitudinal markers are being considered to predict the risk of death.


, Thursday

Probability in Mathematical Physics

Room P3.10, Mathematics Building, Instituto Superior TécnicoInstituto Superior Técnico


, Université Côte d'Azur.

Abstract

Ginzburg-Landau (GL) dynamics are popular interacting particle systems. Macroscopic fluctuations theory (MFT) is now considered as the cornerstone of non-equilibrium statistical mechanics for diffusive systems. In this talk I will consider GL dynamics with long range interactions so that the system is superdiffusive and hydrodynamic limits are given by (non-linear) fractional diffusion equations. I will discuss issues concerning the establishment of a MFT for these GL dynamics. Joint work with R. Chetrite, P. Gonçalves and M.Jara.


, Tuesday

Probability in Mathematical Physics

Room P3.10, Mathematics Building, Instituto Superior TécnicoInstituto Superior Técnico


Persi Diaconis, Stanford University.

Abstract

Imagine three gamblers with respectively $A$, $B$, $C$ at the start. Each time, a pair of gamblers are chosen (uniformly at random) and a fair coin is flipped. Of course, eventually, one of the gamblers is eliminated and the game continues with the remaining two until one winds up with all $A+B+C$. In poker tournaments (really) it is of interest to know the chances of the six possible elimination orders (e.g. $3,1,2$ means gambler $3$ is eliminated first, then gambler 1, leaving 2 with all the cash). In particular, how do these depend on $A,B,C$? For small $A,B,C$, exact computation is possible, but for $A,B,C$ of practical interest, asymptotics are needed. The math involved is surprising; Whitney and John domains, Carlesson estimates. To test your intuition, recall that if there are two gamblers with $1$ and $N-1$ the chance that the first winds up with all $N$ is $1/N$. With three gamblers with $1,1$ and $N-2$ the chance that the third is eliminated first is $\frac{\operatorname{Const}}{N^3}$. We don't know the answer for four gamblers. This is a report of joint work with Stew Ethier, Kelsey Huston-Edwards and Laurent Saloff-Coste.




, Tuesday

String Theory

Unusual schedule


Joris Raeymaekers, Czech Academy of Sciences.

Abstract

Superconformal ‘type B’ quantum mechanical sigma models arise in a variety of interesting contexts, such as the description of D-brane bound states in an $AdS_2$ decoupling limit. Focusing on $N= 2B$ models, we study superconformal indices which count short multiplets and provide an alternative to the standard Witten index, as the latter suffers from infrared issues. We show that the basic index receives contributions from lowest Landau level states in an effective magnetic field and that, due to the noncompactness of the target space, it is typically divergent. Fortunately, the models of interest possess an additional target space isometry which allows for the definition of a well-behaved refined index. We compute this index using localization of the functional integral and find that the result agrees with a naive application of the Atiyah-Bott fixed point formula outside of it’s starting assumptions. In the simplest examples, this formula can also be directly verified by explicitly computing the short multiplet spectrum.


, Wednesday

Lisbon WADE — Webinar in Analysis and Differential Equations

Room P3.10, Mathematics Building, Instituto Superior TécnicoInstituto Superior Técnico


, Rutgers University.

Abstract

This will be a survey talk about recent progress on norm and pointwise convergence problems for classical and multiple ergodic averages along polynomial orbits. A celebrated theorem of Szemeredi asserts that every subset of integers with nonvanishing upper Banach density contains arbitrarily long arithmetic progressions. We will discuss the significance of using ergodic theory and Fourier analysis in solving this problem. We will also explain how this problem led to the conjecture of Furstenberg-Bergelson-Leibman, which is a major open problem in pointwise ergodic theory. Relations with number theory and additive combinatorics will be also discussed.



, Wednesday

Lisbon WADE — Webinar in Analysis and Differential Equations

Room 6.2.33, Faculty of Sciences of the Universidade de Lisboa


Guy Bouchitté, Université de Toulon.

Abstract

The classical Kantorovich-Rubinstein duality theorem established a significant connection between Monge optimal transport and the maximization of a linear form on 1-Lipschitz functions. This result has been widely used in various research areas, particularly to demonstrate a bridge between Monge transport theory and some class of optimal design problems in mechanics.

The aim of this talk is to present a similar theory when the linear form is maximized over all real $C^{1,1}$ functions with a Hessian matrix spectral norm not exceeding one. It turns out that this new maximization problem can be viewed as the dual of a specific optimal transport problem. The task is to find a minimal three-point plan with given first two marginals, where the third is assigned to be larger than both in the sense of convex order. The existence of optimal plans allows to express solutions of the underlying Beckman problem as a combination of rank-one tensor measures supported by a graph. In the context of two-dimensional mechanics, this graph encodes the optimal location of a grillage to support a given bending load.

, Wednesday

Probability and Statistics

Room P3.10, Mathematics Building, Instituto Superior TécnicoInstituto Superior Técnico


Diogo Pereira, CEMAT, Instituto Superior Técnico.

Abstract

The maximum likelihood problem for Hidden Markov Models is usually numerically solved by the Baum-Welch algorithm, which uses the Expectation-Maximization algorithm to find the estimates of the parameters. This algorithm has a recursion depth equal to the data sample size and cannot be computed in parallel, which limits the use of modern GPUs to speed up computation time. A new algorithm is proposed that provides the same estimates as the Baum-Welch algorithm, requiring about the same number of iterations, but is designed in such a way that it can be parallelized. As a consequence, it leads to a significant reduction in the computation time. We illustrate this by means of numerical examples, where we consider simulated data as well as real datasets.


, Thursday

Colloquium of Logic


, Department of Computer Science and Engineering and INESC, Técnico ULisboa.

Abstract

In the last two decades, Craig interpolation has emerged as an essential tool in formal verification, where first-order theories are used to express constraints on the system, such as on the datatypes manipulated by programs. Interpolants for such theories are largely exploited as an efficient method to approximate the reachable states of the system and for invariant synthesis. In this talk, we report recent results on a stronger form of interpolation, called uniform interpolation, and its connection with the notion of model completion from model-theoretic algebra. We discuss how uniform interpolants can be used in the context of formal verification of infinite-state systems to develop effective techniques for computing the reachable states in an exact way. Finally, we present some results about the transfer of uniform interpolants to theory combinations. We argue that methods based on uniform interpolation are particularly effective and computationally efficient when applied to verification of the so-called data-aware processes: these are systems where the control flow of a process can interact with a data storage.






Instituto Superior Técnico
Av. Rovisco Pais, Lisboa, PT