Seminar

 

Speaker: Prof. Eknath Ghate, School of Mathematics, 
TIFR Mumbai.

Title:Generalized
Schwarz triples, Hypergeometric functions and Monodromy.

Time: 3 PM 29th April 2016
Venue: Committee room MZ 168
Abstract: We give a classification of all generalized Schwarz tuples,
using elementary number theory and group theory. This recovers Schwarz's
classical 1873 results showing there are 15 families of 3-tuples (exactly
the triplets for which the associated Gauss Hypergeometric function is
algebraic). We deduce some results on the finiteness of certain
monodromy representations. This is joint work with T. N. Venkataramana.

 

 

Speaker: Prof. Jon Aaronson,
Dept. Of Mathematics,  Tel-Aviv University, ISRAEL.
Time: 3PM – 4:30PM on 28th April 2016 
Title:Rational ergodicity of "discrepancy" skew products and
the asymptotics of affine random walks.

Abstract: It is not hard to show that if
(X(n): n=1,2,..) is given by a RAT (=random affine transformation) of
the form X(n+1)=a(n+1)X(n)+b(n+1) where {(a(n),b(n)):n=1,2,...} are iid rvs
taking values in {-1,+1} x R with E(b)=0 & E(b^2) finite, then
X(n) satisfies both central and local limit theorems. We'll establish a 
``weak, rough, local limit theorem" for certain non-stationary, 
multidimensional  versions of this and use this to show rational ergodicity 
of "discrepancy" skew products T:X=[0,1) x Z--> X by 
T(x,z)=(x+A mod 1, z+D(x)) where A is irrational and the 
"discrepancy function" D=1 on [0,1/2) and D=-1 on [1/2,1).

For each A there is a RAT sequence depending on the "minus-sign" continued 
fraction expansion of A. For badly approximated A, this RAT sequence satisfies
the ``weak, rough, local limit theorem", whence bounded rational ergodicity of 
T_A.

 

Speaker: K. Sandeep.

Title : Moser-Trudinger and Adams Inequalities.

Venue : Committe room.

Time: 4PM, April 18, 2016.

Abstract : Embeddings of Sobolev spaces play an important role in the

analysis of partial differential equations. We will discuss some of these sharp embeddings known as Moser-Trudinger and Adams Inequalities and
present some of the recent results obtained.


Speaker: Prof. Carlos Escudero, Departamento de Matem ́aticas 
Universidad Aut ́onoma de Madrid, email: carlos.escudero@uam.es 
Time: 4PM on 6th April 2016

Title: On a “rather critical“boundary value problem”

 Abstract: This talk is devoted to the analysis of a higher order 
semilinear elliptic problem. In particular, we consider a polyharmonic 
operator acting on the solution equal to the k-Hessian of the solution 
plus some datum. We pose the problem in the whole of the N-dimensional 
space and look for solutions with certain decay at infinity. We will first 
state our general existence result and then move to the possibly most 
interesting case: for a critical value of N the problem becomes especially 
hard as usual tools in nonlinear analysis seem to fail in the search for an 
existence proof. In order to build such a proof we develop an abstract tool 
consisting in a topological fixed point theorem. Existence of at least one 
solution to our differential problem follows from the combination of this 
theorem and a refinement of the classical critical Sobolev embedding. 
Regularity issues will also be discussed and, if time allows, we will extend 
our arguments to the nonlocal setting

Speaker:Dr Anish Ghosh, School of Mathematics, 
TIFR Mumbai

Title: Diophantine approximation andhomogeneous 
 dynamics 

Time:12 PM (NOON) 26thFeb 2016 

Venue:Committee room MZ 168.

Abstract:I will discuss the interplay between number theory 
and theergodictheory of group actions on homogeneous spaces 
of algebraic groups.




Speaker: Prof. Jacques Giacomoni, LMAP (UMR CNRS 5142), 
Universite de Pau, Pau, France

Title: Uniqueness results by p(x)-extension of Diaz-Saa Inequality.

Time: 12 PM (Noon), 25th Feb 2016.

Venue: Committee room MZ 168.

Abstract: I will present a new extension of the well-known 
inequality by Dıaz and Saa which, in our case, involves an 
anisotropic operator, such as the p(x)- Laplacian, 
 ∆p(x)u ≡ div(|∇u|^{p(x)−2}∇u). 
Our present extension of this inequality enables us to establish 
some new results on the uniqueness of solutions and comparison 
principles for some anisotropic quasilinear elliptic and parabolic 
equations.

 

                  Speaker: J.-P. Raymond, Université Paul Sabatier Toulouse III & CNRS, Institut de Mathématiques de Toulouse 31062 Toulouse Cedex - France raymond@math.univ-toulouse.fr

Title: Local existence of strong solutions of fluid-structure models

Time: 3PM – 4PM on 22nd and 27th January 2016.

Venue: Committee or Seminar room (MZ168/ MZ163).

Abstract: We shall study a system coupling the incompressible Navier-Stokes equations with the Lam é system of linear elasticity. We prove the existence of strong solutions, local in time, when the fluid structure interface is flat. This is a joint work with M. Vanninathan. Next we shall see how we can adapt this analysis to the so-called immersed boundary method. This last part is a joint work in progress with Sylvain Ervedoza and Marc Savel.

 

          Speaker: Prof. M. Vanninathan, TIFR, Center for  Applicable Mathematics, Bangalore.

Title: Homogenization and Calculus of Variations

Time: 12 PM on 22nd and 27th January 2016.

Venue: Committee or Seminar room, (MZ 168 / MZ 163).

Abstract: Optimal Shape Problems form a sub­class of problems of Calculus of Variations. Typically,they do not admit classical solutions. Hence relaxation of these problems is called for. Homogenization convergence offers a method to do it. Recently, another class of problems called "Optimal Oscillation­ Dissipation Problems" have been introduced. A new notion of convergence is required for the relaxation of this new problem. In my discussion, I plan to present main issues and ideas involved in the resolution of above problems.

                 Speaker: Biswa Nath Datta, IEEE Fellow, Distinguished Research Professor, Northern Illinois University, USA. Email: dattab@math.niu.edu

Title: COMPUTATIONAL AND OPTIMIZATION METHODS FOR QUADRATIC INVERSE EIGENVALUE PROBLEMS ARISING IN MECHANICAL VIBRATION AND STRUCTURAL DYNAMICS

Time: 11AM – 12PM on 15th January 2016.

Venue: Committee room MZ 168.

Abstract: The Quadratic Eigenvalue Problem is to find eigenvalues and eigenvectors a quadratic matrix pencil of the form P (λ) = M λ 2 + Cλ + K , where the matrices M, C, andK are square matrices. Unfortunately, The problem has not been widely studied because of the intrinsic difficulties with solving the problem in a numerically effective way. Indeed, the state-of-the-art computational techniques are capable of computing only a few extremal eigenvalues and eigenvectors, especially if the matrices are large and sparse, which is often the case in practical applications. The inverse quadratic eigenvalue problem, on the other hand, refers to constructing the matrices M, C, and K, given the complete or partial spectrum and the associated eigenvectors. The inverse quadratic eigenvalue problem is equally important and arises in a wide variety of engineering applications, including mechanical vibrations, aerospace engineering, design of space structures, structural dynamics, etc. Of special practical importance is to construct the coefficient matrices from the knowledge of only partial spectrum and the associated eigenvectors. The greatest computational challenge is to solve the partial quadratic inverse eigenvalue problem using the small number of eigenvalues and eigenvectors which are all that are computable using the state-of-the-art techniques. Furthermore, computational techniques must be able to take advantage of the exploitable physical properties, such as the symmetry, positive definiteness, sparsity, etc., which are computational assets for solution of large and sparse problems.

                    Speaker: Prof. Paola Tardelli, Department of Industrial and
Information Engineering and Economics, University of L'Aquila, Italy.
Title: Filtering for a Probabilistic Prediction of Credit
Ratings.
Time: 3PM – 4PM on 12 th January 2016.
Venue: Committee room MZ 168.
Abstract: To analyze the credit quality of firms, this note proposes a dynamical model. The population of firms is divided into a finite number of classes depending on their credit status. The cardinality of the population can increase during the time, since new firms can enter in the financial market. Due to changes in credit quality and to the defaults, each firm can move from a class to another, or can go to the class of the defaulted firms. Different rating agencies are considered, each of them defines its own partition of the population. Aim of this note is to find the probabilistic prediction of the actual partition of the population, and of the conditional distribution of the distance to defaults. In a partial observing setting, under some markovianity assumptions, this topic is discussed using stochastic filtering techniques.

Speaker: Prof Pramod Kanwar, Ohio University,
Title: On some recent developments on clean elements in certain ring extensions.
Time: 11:30AM – 12:30PM.
Venue: Committee room MZ 168.
 

Abstract: An element a of a ring R is called clean if a = e + u for some 
idempotent e and some unit u in R. A ring is called clean if each of its 
elements is clean. Among other things, we discuss clean elements in certain 
group algebras and also in certain polynomial rings. It is shown that if the 
group algebra of a finitely generated solvable is clean then G is a finite group. 
We also obtain the set of clean elements in a polynomial ring and give conditions 
under which clean elements in a polynomial ring form a subring. It is shown that
for a ring R, the set Cl(R[x]) of clean elements of R[x] forms a subring of
R[x] if and only if Cl(R) is a subring of R and Cl(R[x]) = Cl(R) + N(R)[x]
(where N(R) is the upper nil radical) and that a positive solution to the
Köthe’s problem is equivalent to for any clean ring R, the set Cl(R[x]) of
clean elements of R[x] forms a subring of R[x] if and only if R/N(R) is a
reduced ring.
(This is a joint work with Andre Leroy and Jerzy Matczuk.)

Speaker: Dr. Prem Prakash Pandey.

Title: Square free values of polynomials

 Date and Time: 4PM – 5PM on 17th Nov and 11AM-12 PM 18th Nov.

 Venue: Classroom MZ-158.

Abstract: It is conjectured that all the separable polynomials with integers coefficients, under some local conditions, take infinitely many (in fact positive density) square free values on integer arguments. But not a single polynomial of degree greater than $3$ is proven to exhibit this property. In this article we will study some properties of roots of unity, which separate a primitive $m^{th}$ root of unity with $1$, and use them to generate polynomials of degree $m-1$ which take square free values at infinitely many arguments, in fact on positive density of arguments. A very widely talked polynomial, in this direction, is $X^4+2$. In this article we prove that $X^4+2$ takes infinitely many square free values. The ideas of this article give some new openings, in particular, towards the conjecture.


Speaker: Prof. K. R. Parthasarathy, ISI Delhi.

Title: What is quantum computation?

Date & Time: 3PM – 4PM on 2nd and 9th November 2015.

Location: MZ 168, Committee Room, Mathematics Department..

Abstract:

This subject may be of interest to students of mathematics, computer science and electrical engineering. In these two lectures we shall cover the following elementary ideas leading to the notion of a quantum computer and circuits for algorithms on a quantum computer.
1. Logic gates in a classical computer and the universality theorem.
2. n-qubit Hilbert space, states, unitary transformations as gates, measurements, dirac notation.
3. Elementary gates: 1-qubit, Pauli, CNOT,Toffoli, Swap gates, universality theorem.
4. Circuits for algorithms, two examples: addition of n-bit numbers, Fourier transform in the group {0,1,2,…, n-1} with addition modulo n.


Speaker: Dr. Shanta Laishram, ISI Delhi.

Title: On some Diophantine equations involving terms of binary recurrence sequences.

Date & Time: 22nd September 2015, 3:00 PM.

Location: MZ 163, Department of Mathematics.

Abstract:

Fibonacci and other binary sequences come up in number theory and many other areas of mathematics. They have interesting properties and have been well studied. Most of the properties and problems in integers have an analogue in binary recurrence sequences. In this talk, I will discuss about some Diophantine equations involving terms of binary recurrence sequences. I will also report on some recent results on a joint work with P. Das. The talk will be accessible to general mathematical audience.


 

Speaker: Prof. Krishna B. Athreya, Iowa State University, USA.

Title: Introduction to Markov Chain Monte Carlo methods.

Date & Time: 17th July 2015, 11:00 AM.

Location: MZ 168, Committee Room, Mathematics Department.


Speaker: Prof. Gerald Beer, Emeritus Professor at California State University, Los Angeles.

Talk 1: On locally Lipschitz functions.

Date and Time: 21st April 2015, 11:00 AM to 12 noon.

Talk 2: Uniform continuity of the product of real functions.

Date and Time: 23rd April 2015, 11:00 AM to 12 noon.

Location: MZ 168, Committee Room, Mathematics Department.

Abstracts


Speaker: Dr. Christopher Anand, McMaster University, Canada

Title: High Performance Medical Image Reconstruction. What could go wrong?

Date & Time: 1st April 2015, 3:00 PM.

Location: MZ 168, Committee Room, Mathematics Department.

Abstract


 

Speaker: Dr. Sanjiban Santra, Faculdade de Matematica, Universidade Federal do Para 66.075-110 – Belem – Para – Brazil.

Title: On some fourth order problems associated to Physics.

Date & Time: 31th March 2015, 4:00 PM.

Location: MZ 168, Committee Room, Mathematics Department.

Abstract:

We consider a fourth order traveling wave equation associated to the Suspension Bridge Problem (SBP). This equations are modeled by the traveling wave behavior on the Narrows Tacoma and the Golden Gate bridge. We prove existence of homoclinic solutions when the wave speed is small. We will also discuss the associated  fourth order Liouville theorem to the problem and possible link with the De Giorgi’s conjecture.  This is an attempt to prove the McKenna-Walter conjecture which is open for the last two decades.


Speaker: Dr. Gautam Borisagar, Zakir Husain Delhi College.

Title : Iwahori-Hekce model for supersingular representation for GL2(Qp).

Iwahori-Hekce model for supersingular representation for GL2(Qp)

- See more at: http://sms.niser.ac.in/news/seminar-27#sthash.evWcPUUe.dpu

Iwahori-Hekce model for supersingular representation for GL2(Qp)

- See more at: http://sms.niser.ac.in/news/seminar-27#sthash.evWcPUUe.dpuf.

Date & Time: 17th March 2015, 3:00 PM.

Location: MZ 168, Committee Room, Mathematics Department.

Abstract


Speaker: Prof. M. L. Chaudhry, Royal Military College of Canada, Ontario, Canada.

Title: Roots, Stochastic Processes and Computers.

Date & Time: 10th March 2015, 3:00 PM.

Location: MZ 168, Committee Room, Mathematics Department 

Abstract:

Roots of equations involving high degree polynomials and transcendental functions play an important role in many stochastic processes such as queueing, reliability and random walks. Several researchers have expressed concerns regarding the evaluation of such roots. Kendall (1964) states that interesting features of queueing theory are obscured by “the Laplacian Curtain”. Kleinrock (1975) states in his book “one of the most difficult parts of this method of spectrum factorization is to solve for the roots.” Neuts states in a paper by Shaler Stidham “In discussing matrix-analytic solutions, I had pointed out that when the Rouché roots coincide or are close together, the method of roots could be numerically inaccurate.” In a recent paper, while dealing with a complex batch service queueing system with batch size dependent service, Bar-Lev et al. state that, in practice, the general solution of their problem requires calculation of roots of a complicated equation which, in practice, can result in numerical inaccuracies when the decision variable assumes a large value. The purpose of this talk will be to trace the history of root finding and respond to above questions as well as show how easily and accurately the roots can be found using recent advances in computer science and software packages provided enough precision, which is readily available on such packages, is used. If time permits, we will discuss how roots are used in inverting probability generating functions and Laplace transforms.


Speaker: Dr. Manoj Verma, Post-doctoral fellow, IMSc, Chennai.

Title: On a form of degree $d$ in $2d+1$ variables.

Date & Time: 25th Feb 2015, 3:45 PM.

Location: MZ 168, Committee Room, Mathematics Department 

Abstract:

The square-root barrier implies that the circle method is unlikely to give an asymptotic formula for the number of representations of zero by a form of degree $d$ if the number of variables is less than $2d+1$. Such asymptotic formulas are known in case of some cubic forms in seven variables but no example of such an asymptotic formula for a form of degree $d$ in $2d+1$ variables with $d\geq 4 $ seems to exist in literature. For each $d\geq 4$, we derive such an asymptotic formula for a particular form of degree $d$ in $2d+1$ variables.


 

Speaker: Dr. Durga Prasad Challa, Postdoctoral researcher, INHA University, South Korea.

Title: Wave propagation by sound-soft small bodies.

Date & Time: 19th Feb 2015, 3:00 PM.

Location: MZ 168, Committee Room, Mathematics Department 

Abstract: 

In this talk, we discuss the time harmonic wave propagation by the collection of small bodies embedded in a homogeneous acoustic medium in R^3. We model the small scatterers as impenetrable obstacles of arbitrary shapes. We assume no periodicity nor randomness in distributing the small scatterers. This collection of small scatterers is characterized by their number ‘M’, maximum among their diameters ‘a’ and the minimum distance between them ‘d’. The goal is to derive the asymptotic expansion of the scattered fields by taking into account all the 3 parameters ‘M; a and d’. The leading term appearing in this approximation is nothing but the exact formula derived, using Foldy-Lax method, for the far-field associated to the scattering by the point-like scatterers located on the locations of small scatterers where the scattering coefficients are replaced by the capacitances of the small scatterers.  Finally, as an application to our result we we will discuss the corresponding inverse problem of locating the scatterers, recovering their capacitances and then to estimate their sizes. We will also discuss, in the case that the number of obstacles is very large, the problem of reconstructing the coefficients which models the equivalent effective medium.


Speaker: Prof. Krishna Athreya, Iowa State University, USA.

Title: Introduction to Probability Theory.

Date & Time: 29th January 2015, 3:00 PM.

Location: MZ 168, Committee Room, Mathematics Department.

Abstract:

Probability theory provides the mathematical basis for the study of random phenomena. First one identifies the set of all possible outcomes,called the sample space, followed by a collection of subsets of the sample space called the events collection and finally a function mapping the event collection into the interval [0,1]. This triplet consisting of the sample space, event collection and the probability function is an example of a measure space. We illustrate this with examples when the sample space is finite, countable, the real line, the Euclidean space, the sequence space of real numbers, the function space of continuous functions and many more. We also introduce the concepts of random variables, random vectors, random functions and random fields. We also describe the basic law of large numbers and the central limit theorem.

Ref. Probability Theory  TRIM series Vol 41.


Speaker:  Professor Roberta Musina, Department of Mathematics, Università degli Studi di Udine, Italy

First talk:  20th January at 11:30 AM

Title of talk: A survey lecture on the classical obstacle problem

Second talk:  23rd January at 11:30 AM

Title of talk: The Dirichlet and Navier Fractional Laplacians

Location for both talks: MZ 168, Committee Room, Mathematics Department.


Speaker: Dr. Bhargab Chattopadhyay, Univ of Texas at Dallas, USA.

Title : Fixed-Width Confidence Interval for Gini Index.

Date & Time: 9th January 2015, 3:00 PM.

Location: MZ 168, Committee Room, Mathematics Department.

Abstract


Speaker: Prof. Bong Dae Choi, Sungkyunkwan University, Seoul, Korea.

Title : Performance Analysis of MAC Protocol of EDCA on Common Channel and Reservation On Service Channels for IEEE 802.11p/1609.4 WAVE.

 Date & Time: 24th December 2014, 11:00 AM.

 Location: MZ 168, Committee Room, Mathematics Department.

Abstract


Dr. Debajyoti Nandi, Rutgers University, USA

Title : Combinatorial identities arising from representation theory of affine Lie algebras using vertex-operator-theoretic techniques.

Date & Time: 25th November 2014, 3:00 PM

Location: MZ 168, Committee Room, Mathematics Department.

Abstract:

Lepowsky-Wilson’s remarkable vertex-operator-theoretic proof of the classical Rogers-Ramanujan identities initiated a fruitful area of “algebraic combinatorics” relating partition identities to the representation theory of vertex algebras. In this talk I will give a brief historic overview of this area and a few examples of such partition identities, including my recent discovery of a new set of (conjectured) partition identities arising from the standard level 4 representations of the affine Lie algebra $A_2^{(2)}$. These new partition identities have exciting new features that were not seen in any of the previous examples of this type. My result follows from a construction of a spanning set using certain “vertex operators” acting on a highest weight vector. I will also talk about how “experimental mathematics” can be used to gain insight into such problems.


Dr. Dhanya Rajendran, Post-Doctoral fellow, IISc, Bangalore.

Title : A three solution theorem for singular nonlinear elliptic boundary value problems.

Date & Time: 18th November 2014, 12:00 Noon.

Location: MZ 168, Committee Room, Department of Mathematics.

Abstract (PDF)


Dr. Lakshmi Sankar, University of West Bohemia, Czech Republic

Title : Semipositone Problems on Unbounded Domains

Date & Time: 17th November, 2014, 12:00 Noon

Location: MZ 168, Committee Room, Mathematics Department

Abstract (PDF)


Dr. Shrishendu Chaudhuri, TIFR, Centre for Applicable Mathematics, Bangalore

Title : Controllability of Linearized Compressible Navier Stokes equations.

Date & Time: 14th November 2014, 3:00 PM

Location: MZ 168, Committee Room, Mathematics Department.

Abstract:

Then we consider compressible Navier-Stokes equations in one dimension, linearized around a constant steady state $(Q_0,V_0)$, with $Q_0 > 0,V_0\geq 0$. It is a coupled system involving both transport and parabolic effects. We study the controllability of this linearized system in bounded interval $(0,L)$. We find that the properties of the two semigroups $(e^{tA})_{t\geq0} $ (the one when $V_0 = 0$ and the one when $V_0> 0$) and the spectrum of $A$ are completely different where $A$ is the corresponding linearized operator. We obtain several interesting positive and negative results for the null controllability and approximate controllability of the system using interior or boundary control in both the cases $V_0 =0$ and $V_0>0$.

Joint works with Mythily Ramaswamy, Jean-Pierre Raymond, Debanjana  Mitra and Michael Renardy.


Dr. Sutanu Roy, Visiting Scientist, ISI Kolkata

Title: A brief introduction to braided multiplicative unitaries

Date & Time: October 17, 2014, 11:00 AM.

Location: MZ 168 Committee Room, Mathematics Department.

Abstract
Multiplicative unitaries are one of the fundamental objects to study locally compact quantum groups. Roughly, it is a unitary operator that encodes all structures of a locally compact quantum group and its dual. In this talk we shall start from a multiplicative unitary of a locally compact group, and motivate the concept of more general objects namely braided multiplicative unitaries.

Dr. Krishan Gahalaut (Postdoctoral fellow), King Abdullah University of Science and Technology, Saudi Arabia

Title of the talk: Isogeometric Analysis: Condition Number Estimates and Multigrid Methods

Date & Time: October 13, 2014, 3:00 PM.

Location: MZ 168 Committee Room, Mathematics Department.

Abstract (PDF)


Professor S. S. Sritharan, Director, Center for DECISION, RISK, CONTROLS & SIGINT (DRCSI), Naval Postgraduate School,  Monterey, California, USA.

Title: The Three Astras of Tosio Kato: Harmonic Analysis Methods for Nonlinear PDEs of Fluid Dynamics & Physics

Date & Time: 11th Sept. 2014 at 3:00 PM

Location: Committee Room, Mathematics Department.

Abstract:

In this talk we will give an exposition of three powerful weapons utilized effectively by Tosio Kato in the solvability theory of important nonlinear partial differential equations such as the Navier-Stokes equations, Nonlinear Schrodinger equations, the Maxwell-Dirac equations of relativistic quantum theory, the Einstein field equations, etc. The three tools are linear and nonlinear accretive operators, commutator estimates and logarithmic Sobolev inequality. In this talk we will work out one or two specific nonlinear PDEs mentioned above to demonstrate how these tools come together.


Dr. Ashish K. Srivastava, Associate Professor, Saint Louis University, USA

Title: Modules invariant under automorphisms of their covers and envelopes.

Date & Time: 21st August  3:00 PM

Location: Committee Room (MZ 168)

Abstract:

In this talk we will discuss the theory of modules which are invariant under automorphisms of their covers and envelopes. When applied to specific cases like injective envelopes, pure-injective envelopes, cotorsion envelopes, projective covers, or flat covers, these results extend and provide a much more succinct and clear proofs for various results existing in the literature. We will discuss how additive unit structure of von Neumann regular rings plays a crucial role in this study.

Dr. Prosenjit Roy, TIFR-CAM, Bangalore

Title: Eigenvalue Problems with Dirichlet-Neumann Type  Boundary Conditions on Large Cylinders

Date & Time:     12th May 2014 at 4 PM

Abstract: We will consider eigenvalue problems with mixed (Dirichlet on some part and Neumann on the remaining part of the boundary) boundary type conditions. The domains (dimensions > 1) under consideration will be of cylindrical types, which will tend to become unbounded in one direction. We will study the asymptotic behavior of the eigenmodes of such problems. I will show how these problems are connected to “Problems of Dimension Reduction”.