Seminario de Matemáticas

 

El Seminario de Matemáticas está dirigido tanto a profesores e investigadores como a estudiantes. Se presentan temas de investigación y de divulgación de diversas áreas de las matemáticas. Los expositores suelen pertenecer de otras instituciones o ser profesores y/o investigadores del ITAM. Las pláticas están estructuradas a manera que alguien que no sea especialista del tema pueda entender las ideas principales de la exposición con una sesión de preguntas al final. Semanalmente se actualiza la información sobre los seminarios y también se pueden consultar los seminarios de semestres anteriores. Estos seminarios son coordinados por los profesores Pablo Roldán González y Daniel Vera Rea.

Seminarios del Presente Semestre

 

FechaHoraSalón PonenteTítulo
20 - Ene13:00 - 14:00

B-3

 

Antonio Martinez Sykora
University of Southampton
 

Metaheuristics for the Irregular Bin Packing Problem with free rotations

25 - Ene13:00 - 14:00

110

 

Julia Bennell
University of Southampton
 

Get packing! Key concepts and future directions in cutting and packing problems

3 - Feb13:00 - 14:00

B-3

 

Svetlin G. Georgiev
Sorbonne University, Paris 6, France y Sofia University, Sofia, Bulgaria
 

Iso-Differential Calculus

10 - Feb13:00 - 14:00

B-3

Daniele Avitabile
University of Nottingham

Numerical computation of coherent structures in stochastic neural networks: from interface dynamics to coarse-grained bifurcation analysis

17 - Feb13:00 - 14:00

B-3

Andreas Wachtel
ITAM

Finite Elements on anisotropic meshes

24 - Feb13:00 - 14:00

B-3

Víctor Breña Medina
Centro de Ciencias Matemáticas, UNAM Campus Morelia

Formación de parches bioquímicos en pelos radiculares; dinámica de estructuras localizadas.

10 - MAr13:00 - 14:00

B-3

Carlos García Azpeitia
Facultad de Ciencias, UNAM

 Estabilidad y bifurcación en la configuración elíptica de Maxwell

24 - Mar13:00 - 14:00

B-3

Rafael de la Llave
GeorgiaTech

Quasi-periodic solutions for state dependent delay equations

7 - Abr13:00 - 14:00

B-3

Luis Alberto Cisneros Ake
ESFM, IPN

Efecto anarmónico en el solitón de Davydov en medios continuo y discreto.

Historial de seminarios pasados [+]

Resúmenes de los Seminarios del Semestre:

Quasi-periodic solutions for state dependent delay equations

Viernes 24 de Marzo, 13:00 - 14:00, Salón B-3

Rafael de la Llave
GeorgiaTech

Resumen: We consider delay differential equations in which the delay depends on  the state of the system. These equations appear naturally in  electrodynamics for particles interacting with retarded potentials (the delay is proportional to the distance) as well as in several biological models.  We note that for these equations the phase space is in?finite dimensional and not easy to describe.  Questions such  as existence, uniqueness, dependence on parameters are still puzzling. We develop a theory of quasi-periodic solutions that bypasses the questions of existence for general initial data.  We develop a functional equation for the quasi-periodic equations and study them by functional analysis methods.  The main results are stated in an a-posteriori format that states that given approximate solutions that satisfy some explicit non-degeneracy solutions, there are true solutions nearby.  This can be used to justify some numerical solutions that have been produced.  We show that in a one-parameter family, there are smooth solutions.  Furthermore, we can find a large
measure set where the quasi-periodic solution are analytic.  We conjecture that this regularity is optimal.  We also develop a theory of stable/unstable manifolds.  This is joint work with Xiaolong  He.

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Estabilidad y bifurcación en la configuración elíptica de Maxwell

Viernes 10 de Marzo, 13:00 - 14:00, Salón B-3

Dr. Carlos García Azpeitia
Facultad de Ciencias, UNAM.

Resumen: La configuración de Maxwell consiste de n cuerpos con masas iguales (anillo de Saturno) girando alrededor de una masa central (Saturno). También se puede considerar la misma configuración de Maxwell, pero cuando los n cuerpos siguen órbitas elípticas en lugar de órbitas circulares. Usando una simetría del tipo Hip-Hop, probamos existencia de lenguas de Arnold en el conjunto de parámetros para la masa central y la excentricidad de las órbitas elípticas. Dentro de las lenguas la configuración de Maxwell es inestable. Además, la configuración elíptica tiene una bifurcación de soluciones periódicas cada vez que la excentricidad cruza una de las lenguas. Este es un trabajo en progreso, en colaboración con Abimael Bengochea y Renato Calleja.

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Formación de parches bioquímicos en pelos radiculares; dinámica de estructuras localizadas.

Viernes 24 de Febrero, 13:00 - 14:00, Salón B3

Víctor Breña Medina
Centro de Ciencias Matemáticas, UNAM Campus Morelia

Resumen: Una amplia diversidad de fenómenos biológicos son susceptibles de ser estudiados desde el punto de las matemáticas. Desde esta manera, la biomatemática y la biología matemática, transdisciplina la primera e interdisciplina la segunda, son áreas del conocimiento que han cobrado particular interés en la comunidad científica. En particular, esto se debe a que la biología ofrece de manera natural un amplio espectro de problemas abiertos a las matemáticas; a su vez, las teorías y métodos matemáticos proporcionan una capacidad rigurosa de análisis de hipótesis fuera del alcance experimental.

En esta charla se expondrán los ingredientes clave del análisis numérico y analítico de las propiedades dinámicas de un modelo de iniciación en una célula localizada en la epidermis de la raíz de la planta Arabidopsis thaliana. Este modelo consiste en un sistema no homogéneo de reacción-difusión. Se mostrarán algunos resultados obtenidos por medio del análisis de bifurcación numérico en una y dos dimensiones espaciales. De igual manera, se presentará un bosquejo del análisis realizado a través de la teoría de interacciones semi-fuertes y de problemas de valores propios no locales y no lineales. Los resultados obtenidos arrojan luz en el papel que juegan algunas características físicas en interacciones que se consideran de origen puramente genético.

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Numerical computation of coherent structures in stochastic neural networks: from interface dynamics to coarse-grained bifurcation analysis

Viernes 10 de Febrero, 13:00 - 14:00, Salón B3

Daniele Avitabile
University of Nottingham

Resumen: I will discuss the formation of coherent structures in spatially-extended neural networks posed on a ring or a torus. Phenomenological models of neural networks have been intensively studied in the past and are known to support a variety of coherent structures observed experimentally (localised bumps of activity, travelling fronts, travelling bumps, lurching waves, rotating waves). These models are typically written as integro-differential equations, where the integral term is a convolution between a synaptic kernel, specifying the neural anatomical connectivity, and a sigmoidal firing rate. Successful strategies for the analysis of these systems includespecial choices of the synaptic kernels (leading to equivalent PDE formulations) and interface methods. The latter are obtained via approximating the firing rates with Heaviside distributions. In this case, the evolution of the system is described entirely by the loci of points where the neural activity attains the firing rate threshold value.

In this talk, I will initially consider a 1D model with heterogeneous synaptic kernel and Heaviside firing rate and show that interface methods allow for the explicit construction of a bifurcation equation for localised steady states, so that analytical expressions for a classical "snakes and ladders” bifurcation scenario can be derived.

I will then consider a 2D model which does not admit an equivalent PDE formulation.  In this (and other neural field models featuring a convolution structure) it is advantageous to combine FFT and Newton-Krylov solvers to perform numerical bifurcation analysis directly on the integral model. In particular, I will show that steep sigmoidal firing rates give rise to well-conditioned linear systems, thereby eliminating the need of a preconditioner in the Krylov step, even in large-dimensional discretisations of the network.

Finally, I will discuss how the insight gained in the Heaviside limit may be used to perform coarse-grained bifurcation analysis on neural networks, even in cases where the network does not evolve according to an integro-differential equation. As an illustrative example, a heterogeneous neural network in the form of a discrete Markov chain with discrete ternary state space, posed on a lattice, will be considered. The model supports coarse bumps, multi-bumps and travelling waves, but the derivation of a coarse evolution equation is nontrivial. Hence, the emerging states are followed in parameter space, using numerical coarse bifurcation analysis. At the core of the coarse bifurcation analysis algorithm is a lifting step which provides an ensemble of
microscopic realisations compatible with a given macroscopic observable of the system. I will show that, by choosing the interface as the coarse variable, it is possible to construct efficient lifting steps for this system. Numerical results indicate that refractoriness in this network is responsible for the transition from stationary states to travelling states.

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Efecto anarmónico en el solitón de Davydov en medios continuo y discreto.

7 de Abril 13:00 - 14:00, Salón B3

Luis Alberto Cisneros Ake
ESFM, IPN

Resumen: Consideramos el modelo de Davydov para la transferencia de energía en un medio cristalino con interacciones no lineales del tipo
cúbico. En el límite continuo encontramos ecuaciones efectivas descritas por el acoplamiento entre las ecuaciones NLS y KdV. En dicho sistema acoplado encontramos una variedad de soluciones que van desde el solitón de Davydov de primera y segunda especies hasta los solitones obscuros (aquí llamados de primera y segunda especie, también). Finalmente, hacemos una búsqueda por medios variacionales y numéricos de éste tipo de soluciones en el medio discreto originalmente planteado.

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Iso-Differential Calculus

Viernes 3 de Febrero, 13:00 - 14:00, Salón B-3

Svetlin G. Georgiev
Sorbonne University, Paris 6, France y Sofia University, Sofia, Bulgaria

Resumen: In this lecture are introduced iso-numbers, iso-functions and isoderivatives. They are made comparisons with the real numbers, realvalued functions and their derivatives. They are given some applications of iso-functions.


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Finite Elements on anisotropic meshes

Viernes 17 de Febrero, 13:00 - 14:00, Salón B-3

Andreas Wachtel
ITAM

Resumen: Finite Element Methods (FEM) are used to approximate solutions of (partial) dierential equations. The field of applications is wide, for example, membrane equations, plate equations, fluid  flow problems, etc. However, a good quality of the approximation is in general only guaranteed for isotropic meshes. This is the most common limitation of such methods. We start this talk gently with a denition of such meshes and methods. Then, we dive into what happens when this condition is violated.

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Metaheuristics for the Irregular Bin Packing Problem with free rotations

Viernes 20 de Enero, 13:00 - 14:00, Salón Por Confirmar

Antonio Martinez Sykora
University of Southampton

Resumen:  We present a number of variants of a constructive algorithm able to solve a wide variety of variants of the Two-Dimensional Irregular Bin Packing Problem (2DIBPP). The aim of the 2DIBPP is to pack a set of irregular pieces,which may have concavities, into stock sheets (bins) with fixed dimensions in such a way that the utilization is maximized. This problem is inspired by a real application from a ceramic company in Spain. In addition, this problem arises in other industries such as the garment industry or ship building. The constructive procedure presented in this paper allows both free orientation for the pieces, as in the case of the ceramic industry, or a finite set of orientations as in the case of the garment industry. We explicitly model the assignment of pieces to bins and compare with the more common strategy of packing bins sequentially. There are very few papers in the literature that address the bin packing problem with irregular pieces and to our knowledge this is the first to additionally consider free rotation of pieces with bin packing. We propose several Integer Programming models to determine the association between pieces and bins and then we use a Mixed Integer Programming model for placing the pieces into the bins. The computational results show that the algorithm obtains high quality results in sets of instances with different properties. We have used both industry data and the available data in the literature of 2D irregular strip packing and bin packing problems.

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Get packing! Key concepts and future directions in cutting and packing problems

Miércoles 25 de Enero, 13:00 - 14:00, Salón por Confirmar

Julia Bennell
University of Southampton

Resumen: A huge number of products we use, wear and consume begin as raw material that requires cutting as part of the production process. Clothes, furniture, tools, pipes, shoes and windows are just a few examples. Moreover, the transportation of products efficiently, and safely, require an intelligent methodology for packing and loading taking into account many complex constraints. Cutting and packing problems cover a wide range of applications and with these comes many diverse and interesting challenges. Researchers have been tackling these problems since the 1960s and there is a strong legacy of methodological and application focused contributions. Despite this, there still remains many interesting open problems as well as rich opportunities for working at the interface of cutting and packing with other domains such as transportation and production planning. In this talk I will give a flavour of the diverse scope of cutting and packing problems and discuss some of the emerging application areas. The talk will review some of the key concepts and methodologies used for cutting and packing and highlight some of the current challenges. Moreover, I hope, the talk will enthuse researchers to engage with cutting and packing problems and its research community.

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Lista de seminarios de semestres pasados

Las siguientes ligas le llevarán a los resúmenes de los seminarios que se llevaron a cabo en semestres anteriores: