Seminars

This will be a hybrid meeting. Use the following link to join virtually. https://zoom.us/j/99670507398 Abstract: In vertex deletion problems, we have an input graph, and we aim to find a set of k vertices such that the rest of the graph belongs to some graph class. These problems are well-studied with examples including VERTEX COVER, FEEDBACK VERTEX SET, and ODD CYCLE TRANSVERSAL. In this thesis, we investigate vertex deletion problems in the following directions. In the first direction, we initiate the study of a natural variation of the problem of deletion to scattered graph classes. Here, we want to delete at most k vertices so that each connected component belongs to at least one of a fixed number of graph classes in the resulting graph. We give FPT algorithms for deletion to scattered graph classes when deletion to each class is FPT, and the property that a graph belongs to each class is CMSO expressible. We then give a faster FPT algorithm for the case when each class has a finite forbidden set. Later, we do a deep dive on pairs of specific graph classes to which we would like the connected components of the resulting graph to belong and design simpler and more efficient FPT algorithms. In the second direction, we look at several optimization problems where the parameter is the deletion distance to a graph class. Generally, such problems are studied assuming we are given a deletion set of size k as input. However, there are cases where the running time for finding the modulator is significant enough to pay attention to. We focus on one such parameter: deletion distance to chordal graphs. We develop an algorithmic framework for problems such as VERTEX COVER, FEEDBACK VERTEX SET and ODD CYCLE TRANSVERSAL that either identifies that there is no chordal deletion set of size k or gives a 2^{O(k)}n^{O(1)} algorithm to solve them. We also give FPT algorithms for other problems like LIST COLORING and variants of DOMINATING SET when parameterized by deletion distance to specific graph classes. In the third direction, we investigate vertex deletion problems where the deletion set is also required to satisfy additional constraints. Specifically, we study the parameterized complexity of SET-COVER where there is an underlying graph on the sets, and we require that the solution is also an independent set. We also study vertex deletion problems where the deletion set is required to form a fair set, where we don't want too many neighbours of any vertex in the set.

Ultracold atoms placed in optical lattices are well known emulators of  bosonic and spin Hamiltonians. Thus they provide us with clean experimental platforms for studying both ground states and quantum dynamics of several paradigmatic  models of strongly correlated bosons or spins. In this talk, I shall provide a pedagogical introduction to these systems focussing on quantum phase transitions observed experimentally  using several such platforms. These  include ultracold bosons placed in a tilted optical  lattice and arrays and ladders Rydberg atoms where such transitions are both  theoretically predicted and experimentally observed.   

Min-max theory for the area functional was developed by Almgren and Pitts to construct closed minimal hypersurfaces in arbitrary closed Riemannian manifolds. There is an alternate PDE based approach to the construction of minimal hypersurfaces. This approach is based on the study of the limiting behaviour of solutions to the Allen-Cahn equation. In my talk, I will briefly describe the Almgren-Pitts min-max theory and the Allen-Cahn min-max theory and discuss the question to what extent these two theories agree.

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The success of the standard model of particle physics is based on the quantum theory of fields where the basic assumptions require us to deal with concepts of infinity. While the theory of renormalization helps us understand why these infinities are in fact harmless, they are still difficult to handle computationally, especially within the strongly interacting sector of the standard model that describes nuclear physics. Recent effort to overcome these computational bottlenecks using quantum computers is motivating us to think of new ways to build our universe with qubits. I will discuss the basic ideas behind this recent effort and show some recent results that suggest that asymptotic freedom, which is considered a holy grail of fundamental quantum field theories, may also emerge with qubits.

Cosmic inflation is an important paradigm of the early Universe which is so far developed in two equivalent ways, either by geometrical modification of Einstein's general relativity (GR) or by introducing new forms of matter beyond the standard model of particle physics. Starobinsky's R+R^2 inflation based on a geometric modification of GR is one of the most observationally favorable models of cosmic inflation based on a geometric modification of GR. In this talk, I will discuss in detail the fundamental motivations for Starobinsky inflation and present how certain logical steps in the view of its UV completion lead to the emergence of a gravity theory that is non-local in nature. Then I will establish how one can perform studies of the early Universe in the context of non-local gravity and what are the observational consequences in the scope of future CMB and gravitational waves. I will discuss in detail how non-local R^2-like inflation can be observationally distinguishable from the local effective field theories of inflation. Finally, I will comment on the prospects of non-local gravity as a promising candidate for quantum gravity.

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TBA