Workshop on Contemporary Topics in Analysis, Algebra and Geometry (August 15-19, 2017)

SMS Weekly Talk
Thursday, May 25 at 2:30pm in the SMS Seminar Room
Speaker: Haniya Azam (LUMS, Lahore)
Title: What is Schubert calculus?
AbstractNearly 150 years ago Schubert published his book on ‘Calculus of Enumerative Geometry’. As the name suggests, his work was about finding number of points, lines, planes etc., and was the first systematic study of what is now called Intersection Theory. Highly intuitive to begin with, this work became rigorous later on by the introduction of Schubert Varieties.One beautiful aspect of this subject is that geometric information about these varieties lies in the combinatorics of partitions.

In this talk we will introduce Schubert varieties inside a Grassmanian (more generally, a Flag) manifold. This will be followed by Schubert Calculus- an application of intersection theory to problems in enumerative geometry. Lastly, we will be demonstrating what generates the cohomology of this manifold. Hopefully, we will have enough time to give concrete examples.

PhD Thesis Defense
Friday, May 26 at 11:00 am in the SMS Seminar Room
Candidate: Maria Naseem (AS-SMS)
Title: Total and Entire Labeling of Graphs

Week: May 15 — May 19, 2017

SMS Weekly Talk
Thursday, May 18 at 2:30pm in the SMS Seminar Room
Speaker: Muhammad Imran Qureshi (LUMS, Lahore)
Title: Algebraic Varieties via unprojections: Tom & Jerry
Abstract: Many interesting classes of algebraic varieties can be studied in terms of their corresponding graded rings. In many interesting cases the corresponding ring is a Gorenstein ring. One can construct algebraic varieties in codimension less than and equal to 3 by using the structure theorems of the corresponding Gorenstein rings. In codimension greater than 3, one method of constructing algebraic varieties is the technique of unprojection. The aim of this talk is to explain the idea of unprojection by explicit examples. In particular, I will focus on two matrix formats, Tom and Jerry which gives the construction of algebraic varieties in codimension 4 via unprojection of 3 dimensional algebraic varieties.

PhD Thesis Defense
Friday, May 19 at 11:00 am in the SMS Seminar Room
Candidate: Shehraz Akhtar (AS-SMS)
Title: Natural Convection Flows of Viscous Fluids and Different Motions of Couple Stress Fluids

Week: May 08 — May 12, 2017

SMS Weekly Talk
Tuesday, May 09 at 2:30pm in the SMS Seminar Room
Speaker: Shamsa Ishaq (AS-SMS)
Title: Thermodynamics formalism in Ergodic theory
Abstract: The Introduction of Thermodynamics formalism within mathematical field is due to Yakov Sinai, David Ruelle (1935- ) and Rufus Bowen (1947-1978) in early ’70. The aim of this talk is to provide a brief introduction of elements of Thermodynamics formalism in Ergodic theory. Throughout the talk our dynamical system will remain Symbolic dynamical systems. The main idea is to introduced the concept of pressure function, equilibrium state, potential and phase transition etc in the language of invariant probability measure.

SMS Weekly Talk
Thursday, May 11 at 2:30pm in the SMS Seminar Room
Speaker: Sohail Zafar (UMT, Lahore)
Title: Binomial edge ideals of k-deficient graphs
Abstract: In this talk, we will study binomial edge ideals, some of its combinatorial and algebraic properties.The class of k-deficient graphs came into existence, when the sequentially Cohen-Macaulay property of binomial edge ideals was discussed. If we remove k edges (with one fixed vertex) from the complete graph, such a connected graph is called k-deficient graph.
We will study primary decomposition, krull dimension, depth, Hilbert series and Castelnuovo-Mumford regularity of binomial edge ideal of k-deficient graphs.
There is not so much work done so far in the direction of the minimal free resolution of binomial edge ideals. The ideal consists of binomials so the problem to find minimal free resolution looks hard. The minimal free resolution of the binomial edge ideal of simplest classes like complete graph, line graph, complete bipartite graph and cycle graph is known. We will discuss brand new idea of computing the minimal free resolution of k-deficient graphs by using the concept of initial ideals (by converting binomial edge ideal to monomial edge ideal).

Week: May 02 — May 05, 2017

SMS Weekly Talk
Tuesday, May 02 at 2:30pm in the SMS Seminar Room
Speaker: Samina Mazhar (Queen Mary College, Lahore)
Title: Fuzzy Lattice and Statistical Metric
Abstract: In this talk, I will discuss some concepts from lattice theory aiming to introduce [0,1] as a complete bounded lattice possessing an involution but unfortunately failing to follow some desirable laws. Then, I will relate the above stated lattice as range of the mapping known as “Fuzzy Set”. Afterwards, I will introduce some mappings from Statistical Metric Spaces to enrich our theory on fuzzy sets. Lastly, I will attempt to show some marvelous applications of this area.

SMS Weekly Talk
Thursday, May 04 at 2:30pm in the SMS Seminar Room
Speaker: Waqas Ali Azhar (AS-SMS)
Title: Introduction to One Dimensional Dynamics
Abstract: The goal of this talk is to introduce many of the basic techniques from the theory of dynamical systems in a setting as simple as possible. For this reason, all the dynamics will be considered one dimensional i.e. on the real line or unit circle in the plane. We introduce topics such as hyperbolicity, topological conjugacy and structural stability. The main idea is to understand in the light of examples what it means for a dynamical system to be chaotic.

Week: April 24 — April 28, 2017

PhD Thesis Defense
Tuesday, April 25 at 2:30pm in the SMS Seminar Room
PhD Thesis Defense: Imran Abbas Baloch (AS-SMS)
Title of the thesis: Harmonically (s,m)-Convex Functions and Related Inequalities

Wednesday, April 26 at 3:00 pm in the SMS Seminar Room
Speaker: Mumtaz Hussain (La Trobe University, Australia)
Title: Diophantine approximation on manifolds: theory and applications
Abstract: The well-known theorems of Khintchine and Jarnik in Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalizations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this talk, I will explain my recent Jarnik type results for a parabola in homogeneous settings. This result is the first of its kind.

 Week: April 17 — April 22, 2017 Advanced Topics in Theoretical Physics Friday, April 21 at 3:00 pm in the SMS Seminar Room Speaker: Amer Iqbal (AS-SMS & LUMS) Title: Topological Strings-I (Spinors in various Dimensions) Abstract:  Spanning over a few lectures, Amer Iqbal will introduce the world of topological strings which are some extremely useful and fascinating objects in a theoretical physicist’s inventory. In our quest to understand topological strings, their properties and applications we will begin with a review of spinors in various dimensions.We will discuss the spin groups, their relation with orthogonal groups and irreducible spin representations. We will pay special attention to spinors in two dimensions.
 Week: April 10 — April 14, 2017 SMS Weekly Talk Tuesday, April 11 at 2:30pm in the SMS Seminar Room Speaker:  Ambreen Ahmed (AS-SMS) Title: Riemann Surfaces and its Triangulation Abstract:  A Riemann surface is a one dimensional complex manifold. It arises naturally in the study of complex functions. The study of maps from Riemann surfaces to other spaces is a rich and very active field. In this talk we will discuss examples of Riemann surfaces and holomorphic maps on them. We will also discuss the triangulation of Riemann surfaces, Hurwitz formula, the construction of Riemann surface associated with $\sqrt{z}$ and the one with genus one. SMS Weekly Talk Thursday, April 13 at 2:30pm in the SMS Seminar Room Speaker: Shamsa Kanwal (AS-SMS) Title: On Standard Basis of an Ideal Abstract: A standard basis of an ideal is nothing but a special set of generators, which allows the computation of many invariants of the ideal just from its leading monomials. In this talk I will give a brief introduction about the importance and inception of the concept of standard basis in commutative algebra. I will discuss different algorithms to compute standard basis.
 Week: March 27 — March 31, 2017 Advanced Topics in Theoretical Physics Tuesday, March 28 at 5:30 pm in the SMS Seminar Room Speaker: Pervez Hoodbhoy (FCCU & QAU) Title: Instantons and Quantum Tunneling Abstract:  This lecture will be on a fairly simple but surprisingly rich topic – tunneling in a double well potential using the method of pseudoparticles or instantons. Although instantons are crucial in quantum field theory, even in quantum mechanics they are an important means of systematically improving non-perturbative calculations. It turns out that there are tunneling phenomena in condensed matter physics that people seem not to have paid attention to but with possibly important consequences. Students will benefit from this lecture only if they read about Feynman’s path integral method before coming to the lecture.
 SMS Weekly Talk Thursday, March 30 at 2:30 pm in the SMS Seminar Room Speaker: Asma Khalid (AS-SMS) Title: Completeness and Morphisms of Algebras Abstract: A completion is any of several related functors on rings and modules that result in complete topological rings and modules. It is among the most basic tools in analysing commutative rings. The completion of a ring is useful in commutative algebra because it sometimes allows us to bring methods of analysis to bear on problems in algebra. One of the main aim of this talk is to explain smooth morphisms of rings which are one of the main ingredients in Desingularization. Moreover, we will also discuss number of morphism and regular morphisms of Noetherian rings and some of their interesting properties.
 Week: March 20 — March 24, 2017 SMS Weekly Talk Thursday, March 21 at 2:30 pm in the SMS Seminar Room Speaker: Numan Amin (AS-SMS) Title: Introduction to Tropical Geometry Abstract: Tropical geometry is a fascinating area of mathematics that is relatively new with connection to various other fields including physics and geometry. It has been studied quite vigorously in the last few years. I will discuss the basics of tropical geometry by explaining the arithmetic in tropical. I will define tropical polynomial, its hypersurface and explain the connection between the newton polygon and the hypersurface of tropical polynomial. We will also see the relation between the tropical variety with variety in classical geometry.
 Week: March 13 — March 17, 2017 SMS Weekly Talk Tuesday, March 14 at 2:30 pm in the SMS Seminar Room Speaker: Azeem Khadam (AS-SMS) Title: Grothendieck’s Local Cohomology — II Abstract: I will continue my discussion of local cohomology which was introduced by Grothendieck in the early sixties. Local cohomology modules can be used to measure the depth of a module on an ideal, and as a way to test the Cohen-Macaulay properties. Moreover, the cohomology of coherent sheaves on projective varieties can be recovered from graded components of local cohomology modules, providing useful insights into theorems about projective varieties that were originally proved in other ways. SMS Weekly Talk Thursday, March 16 at 2:30 pm in the SMS Seminar Room Speaker: Shamsa Kanwal (AS-SMS) Title: Introduction to Border Bases Abstract: Gröbner bases has become an essential tool in algebra in the last few decades. Theory of Border bases is an extension of the theory of Gröbner bases in the case of zero-dimensional ideals. I will give a brief introduction to Border bases, its importance and some of its applications. Moreover, I will discuss the algorithm for the calculation of Border bases using some examples.
Week: March 06 — March 10, 2017

SMS Weekly Talk
Thursday, March 07 at 2:30 pm in the SMS Seminar Room
Speaker: Seyedeh Zahra Rezaei Lalami (University of Leicester)
Title: Actuarial Science: A Success Story

Abstract: This talk aims at discussing the actuarial science as an emerging academic discipline. Actuarial science couples mathematics with other fields of science to consider real world scenarios. Part of the talk will be aimed at advertising the university of Leicester actual sciences program. I will discuss actuarial qualifications a student enrolled in the program can get. I will also discuss its wider scope in industry since university of Leicester has many top employers on its panel for student placement.

SMS Weekly Talk
Thursday, March 07 at 2:30 pm in the SMS Seminar Room
Speaker:  Itrat Abbas (AS-SMS)
Title: Introduction to Fractional Order Viscoelasticity
Abstract:  The fractional calculus and fractional differential equations are used quiet often to describe many physical phenomena. I will explain, using some examples, the use of fractional calculus to model some materials which lie between Hook’s solid and Newtonian fluids.

Friday, March 10 at 3pm in the SMS Seminar Room
Speaker: Babar Qureshi (LUMS)
Title: Lorentz Invariance in Quantum Field Theory-II
Abstract: In the second lecture on the topic, I will discuss Dirac, Weyl and Majorana Lagrangians, coupling of electromagnetic field to matter, equations of motion,
discrete symmetries, Noether’s theorem and conserved currents associated with various symmetries of these Lagrangians.

Week: February 27 — March 03, 2017

SMS Weekly Talk
Tuesday, February 28 at 2:30 pm in the SMS Seminar Room
Title: Grothendieck’s Local Cohomology
Abstract: Local cohomology was introduced by Grothendieck in the early sixties, in part to answer a conjecture of Pierre Samuel about when certain types of commutative rings are unique factorization domains. Local cohomology modules can be used to measure the depth of a module on an ideal, and as a way to test the Cohen-Macaulay properties. Moreover, the cohomology of coherent sheaves on projective varieties can be recovered from graded components of local cohomology modules, providing useful insights into theorems about projective varieties that were originally proved in other ways.

SMS Weekly Talk
Thursday, March 02 at 2:30 pm in the SMS Seminar Room
Speaker: Laila Naqvi
Title: Mesh-free Methods in Numerical Analysis