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Aareyan Manzoor

Metrics on Polish groups

We investigate various compatible metrics that a Polish group admits. By Birkhoff-Kakutani, every Polish group admits a compatible left-invariant metric, and by being a Polish space, there is also a compatible metric which is complete. We discuss the relationship between these metrics, and introduce the class of CLI Polish groups, those Polish groups which admit a single compatible metric which is both left invariant and complete. We will mostly follow Section 2.2 of Gao's IDST book.

MC5403

William Gollinger

The Adams Spectral Sequence

After having a week off, in this talk we will review concepts in homotopy theory. This will include: fibrations and cofibrations; CW complexes; Eilenberg-MacLane spaces; homotopy fibers and related constructions; the Postnikov tower of a space.

MC5417

Gauree Wathodkar (University of Mississippi)

Partition regularity in commutative rings.

Let A ∈ Mm×n(Z) be a matrix with integer coefficients. The system of equations A⃗x = ⃗0 is said to be partition regular over Z if for every finite partition Z \ {0} = ∪ri =1Ci, there exists a solution ⃗x ∈ Zn, all of whose components belonging to the same Ci. For example, the equation x + y − z = 0 is partition regular. In 1933 Rado characterized completely all partition regular matrices. He also conjectured that for any partition Z \ {0} = ∪ri =1Ci, there exists a partition class Ci that contains solutions to all partition regular systems. This conjecture was settled in 1975 by Deuber. We study the analogue of Rado’s conjecture in commutative rings, and prove that the same conclusion holds true in any integral domain.

MC5403

Cynthia Dai

Global Heights

In this talk, we will wrap up local heights with respect to a presentation of a line bundle, then define global heights. If time permits, we will also talk about Weil heights.

MC 5417

Filip Milidrag

The relation between the Wythoff construction and abstract polytopes

In this talk we will use the Wythoff construction of a geometric polytope to describe its face lattice and then use this to make the connection between geometric polytopes and the notion of an abstract polytope. We will then go on to speak a bit about abstract polytopes and some related definitions.

MC 5417

Utkarsh Bajaj

An introduction to the Mckay correspondence

The McKay correspondence is a bijection between the finite subgroups of SL(2,C) and the Dynkin diagrams of the type  A_r, D_r, E_6, E_7, E_8. One bijection takes a subgroup G, constructs the orbit space C^2/G, resolves the singularities by inserting Riemann spheres multiple times, sees how the spheres intersect, and then constructs a graph to represent this information. Another bijection constructs irreducible representations of G. We will see how these bijections are related.

MC 5417

Aiden Suter

Mathematical aspects of Higgs and Coulomb branches

3d mirror symmetry is a duality between topological twists of 3 dimensional quantum field theories (QFTs) with N=4 supersymmetry. One of the most salient features of this duality is the symplectic duality between the branches of the moduli space of vacua of the full physical theory know as the “Higgs” and “Coulomb” branches. These branches are singular hyperkahler varieties that are interchanged under the duality. In this talk, I will primarily discuss two results regarding these varieties:

The first utilises a construction due to Costello and Gaiotto allowing one to associate a vertex operator algebra (VOA) to certain boundary conditions of these twisted QFTs and it has been conjectured that the associated variety of this VOA is the Higgs branch of the theory. In this talk I will outline a proof of this conjecture in the case of U(1) gauge theory acting on n>3 hypermultiplets, building on prior work of Beam and Ferrari who conjectured that the boundary VOA is the simple quotient of the psl(n|n) affine VOA.

In the second part of this talk I will outline a construction of a tilting generator for the derived category of sheaves on T*Gr(2,4). This space is the Coulomb branch for a certain quiver gauge theory and the construction is a realisation of a result due to Webster who proved the existence of such a tilting generator. In the case of quiver gauge theories such as this, the Coulomb branch algebra can be described in terms of a cyldrinical KLRW algebra, a type of diagrammatic algebra. Using these diagrammatic methods we explicitly describe the tilting generator and find that it differs to those previously known in the literature.

Join online at: https://pitp.zoom.us/j/97622507197

Ty Ghaswala

More on automatic continuity

I will talk about some specific Polish groups arising naturally (for some definition of natural) from low-dimensional topology and dynamical systems. The talk will then continue, all on its own, to a discussion about automatic continuity.

MC 5403

Michael Xu

Applications of large sieve in variance analysis

Due to its analytic nature, large sieve is considered one of the popular sieves in many number theory estimates, free from many combinatorial constraints. Thanks to the same reason, large sieve is also considered one of the difficult sieves as it lacks combinatorial nature and a proper visualization of sieving.

In this talk, I will attempt to showcase both characteristics of this sieve with applications, demonstrating its capability, in process of proving results about distribution of primes in arithmetic progression, namely Barban-Davenport-Halberstam bound.

MC 5403

Jason Fang

Large sieve inequality and classical large sieve

As its name suggests, the large sieve inequality has many applications in sieve theory, but it is not easy to construct the idea about the techniques involved. In this talk I will prove the large sieve inequality and classical large sieve result using tricks such as Parseval identity and Cauchy-Schwarz Inequality.

MC 5403