Who Should Attend

• Researchers, graduate students and advanced undergraduates in mathematics, computer science and engineering.
• The only prerequisites are linear algebra and an interest in computation.

Course Fee

The participation fees for taking the course is as follows:
• Participants from abroad : US $150 (additional charges may apply)
• Industry/ Research Organizations: INR 10,000 + 18% GST
• Academic Institutions: INR 5,000 + 18% GST
• Students: INR 1,000 + 18% GST
• The above fee includes all instructional materials, computer use for tutorials and assignments, laboratory equipment usage charges, 24 hr. free internet facility.
• The participants will be provided with accommodation on a payment basis.

Mode of Instruction

Offline (IISc Campus, Bengaluru)

Last date to apply : 25-July-2025

We now explain these problems in detail. The first problem is about graphical designs in an unweighted graph which are discrete analogs of spherical designs and Platonic bodies. It is a subset of vertices with weights so that the global average of a sufficiently smooth function on the graph agrees with the weighted average of the function on this subset. Via the powerful theory of Gale duality, designs can be organized on the faces of special polytopes. The second problem is about the geometry of graph sparsifiers (weighted subgraphs). Sparsifiers that share the initial part of the spectral information of the graph retain important global properties of the graph. The set of all such sparsifiers will be shown to be a special convex body called a spectrahedron. A combinatorial graph G= (V, E) can be thought of as a weighted graph G= (V, E, w) with the edge weights 1 for all edges. The algebraic connectivity of G is defined as the smallest non-trivial eigenvalue of the Laplacian matrix. A natural question is whether one can increase connectivity by changing the edge weights. This would, for instance, lead to faster mixing Markov chains on G. Analogously, one can look for weights that minimize the largest eigenvalue. An orthogonal take is to ask which combinatorial graphs (with edge weights 1) are already optimal for the problems of maximizing and minimizing the above eigenvalues. Such conformally rigid graphs include edge-transitive graphs, distance-regular graphs and some (but not all) Cayley graphs. Conformal rigidity can be checked via semidefinite programming (whose feasible regions are spectrahedra!), and pose several interesting open problems.

Objectives

• The key objective of this course is to expose students to several different parts of mathematics that interact with problems on graphs, namely combinatorics, convex geometry and convex optimization.
• A second goal will be to connect to applications.
• Last goal will be hands-on computation using software such as Mathematica, allowing students to experiment and discover new results.