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Mathematics Colloquium

The colloquium runs Thursday afternoons at 4 PM in Phillips Hall 332 with a tea preceding at 3:30 PM in Phillips Hall 330. The talks are aimed at general audiences with different backgrounds.
In the year 2021-2022, most talks will be held on zoom with a virtual tea time preceding at 3:45pm.

Colloquium committee: Jiuzu Hong, David Rose, Andrey Smirnov
Zoom meeting ID: 985 5728 2488, Password: The first six digits of pi

2022 spring

Date Speaker Affiliation Mode Title
April 14 Andrey Negut MIT In-person Quantum algebras, shuffle algebras and Hall algebras
April 7 Tamas Hausel IST Austria Zoom Ubiquity of systems of homogenous polynomial equations with a unique solution
March 24 Ivan Loseu Yale In-person Unipotent representations and quantization
Feb 24 Ken Ono UVA Zoom New results in arithmetic statistics
Feb 10 Lisa Piccirillo MIT Zoom Knot concordance and 4-manifolds

Andrei Negut
Abstract: The Hall algebra of coherent sheaves on a genus g curve over F_q is an important object in geometric representation theory: when g=0 it gives rise to the positive half of U_q(Lsl_2), while its g=1 case (known as the elliptic Hall algebra) has recently found numerous applications, ranging from the study of categorical knot invariants to the study of derived categories of Hilbert schemes of surfaces. In the present talk, I will review a program that realizes the genus g Hall algebra as a quantum loop algebra, using shuffle algebra techniques. Based on joint work with Francesco Sala, Olivier Schiffmann and Alexander Tsymbaliuk.

Tamas Hausel
Abstract: Following Macaulay we will analyse systems of equations as in the title leading to marvelous properties of its multiplicity algebra. Examples include isolated surface singularities, equivariant cohomology and fixed point sets of group actions as well as the Hitchin integrable system on very stable upward flows.

Ivan Loseu
Abstract: A fundamental question in the representation theory of semisimple Lie groups is to classify their irreducible unitary representations. A guiding principle here is the
Orbit method, first discovered by Kirillov in the 60’s for nilpotent Lie groups. It states that the irreducible unitary representations should be related to coadjoint orbits, i.e.,
the orbits of the Lie group action in the dual of its Lie algebra. Passing from orbits to representations could be thought of as a quantization problem and it is known that in this setting this is very difficult. For semisimple Lie groups it makes sense to speak about nilpotent orbits, and one could try to study representations that should correspond to these orbits via the yet undefined Orbit method. These representations are called unipotent: they are expected to be nicer than general ones, while one hopes to reduce the study of general representations to that of unipotent ones. I will concentrate on the case of complex Lie groups. I will explain how recent advances in the study of deformation quantizations of singular symplectic varieties allow to define unipotent representations and obtain some results about them. The talk is based on the joint work with Lucas Mason-Brown and Dmytro Matvieievskyi.

Ken Ono
Abstract: Studying the statistical behavior of number theoretic quantities is presently in vogue. This lecture will begin with a new look at classical results in number theory from the perspective of arithmetic statistics, which then naturally lead to point counts for elliptic curves and K3 surfaces over finite fields. This lecture will use the celebrated Sato-Tate Conjecture (now theorem thanks to Richard Taylor and his collaborators) as motivation for refinements in several directions. One of the results will feature the exotic Batman distribution.

Lisa Piccirillo
Abstract: There is a rich interplay between the fields of knot theory and 3- and 4-manifold topology. In this talk, I will describe a weak notion of equivalence for knots called concordance, and highlight some historical and recent connections between knot concordance and the study of 4-manifolds, with a particular emphasis on applications of knot concordance to the construction and detection of small 4-manifolds which admit multiple smooth structures.

2021 fall

Date Speaker Affiliation Mode Title
Nov 18 Dima Arinkin Wisconsin Zoom Moduli spaces and their compactifications
Nov 11 Greg Forest UNC In-person Modeling insights into SARS-CoV-2 respiratory tract infections
Nov 4 David Nadler UC Berkley In-person Skeleta of Weinstein manifolds
Oct 28 Lilian Pierce Duke Zoom Counting problems: open questions in number theory
Spe 9 Chenyang Xu Princeton Zoom Canonical metrics, stability and moduli space

Dima Arinkin
Abstract: Very broadly speaking, geometry is the study of spaces. Here `space’ is a placeholder: different flavors of geometry work with spaces such as differentiable manifolds (differential geometry), topological spaces (topology), varieties (algebraic geometry, my favorite), and so on. But what makes a space an interesting object of study? One class of `interesting’ spaces is the so-called moduli spaces (the word `moduli’ goes back to Riemann and means `parameters’). Moduli spaces parametrize objects of some type: say, moduli space of triangles parametrizes triangles, moduli space of differential equations parametrizes differential equations, and so on. Thus, geometry of moduli spaces encodes interesting features of the totality of objects that they parametrize.

Unfortunately, many naturally arising moduli spaces are not compact, which makes them harder to work with. Sometimes, it is possible to compactify a moduli space by enlargening the corresponding class of objects. In my talk, I will present some examples of compactification of moduli spaces, with emphasis on examples that are relevant to the geometric Langlands program. Time permitting, I will also mention a more categorical approach to compactification, which generalizes the procedure to important `non-commutative’ spaces.

Greg Forest
Abstract: I and many collaborators, postdocs, and students from many disciplines have explored lung mechanics and disease pathology for over 2 decades in a pan-university effort called the UNC Virtual Lung Project. In the last decade with the Sam Lai lab we have explored how viruses “traffic” in mucosal barriers, including the human respiratory tract (RT). These efforts have focused on understanding the primary respiratory defense — mucociliary clearance – and secondary defense from antibodies. Then along came the novel coronavirus SARS-CoV-2, requiring a major pivot to a pre-immunity study of the human RT. Over the past 20 months we developed a computational model that incorporates detailed anatomy and physiology of the RT and best, evolving, knowledge about SARS-CoV-2. We model virus mobility in airway surface liquids (ASL), infectability of epithelial cells and their replication rates and duration of infectious virions. We then simulate outcomes from inhaled SARS-CoV-2 depositions anywhere in the RT, likelihoods of clearance versus infection, and propagation of the viral load and infection. Results shed mechanistic insights into clinical observations prior to immune protection from SARS-CoV-2. Next we give some insights / predictions about the  variant and antibody protection.

This work is a collaboration w/ Ric Boucher, Director of the Marsico Lung Institute (MLI), Ronit Freeman of APS, Sam Lai of the School of Pharmacy, and Ray Pickles of the MLI. The mathematical modeling results are with Alex Chen, Cal State Dominguez Hills, Tim Wessler, UNC and U. Michigan postdoc, and Kate Daftari and Jason Pearson, UNC PhD students.

David Nadler
Abstract: I’ll survey some history and motivation for the study of Weinstein manifolds and their skeleta. Then I’ll discuss recent and ongoing work with Dani Alvarez-Gavela and Yasha Eliashberg devoted to understanding polarized Weinstein manifolds in terms of their skeleta.

Lilian Pierce
Abstract: Many questions in number theory can be phrased as counting problems. How many primes are there? How many elliptic curves are there? How many integral solutions to this system of equations are there? How many number fields are there? Sometimes the answer is “infinitely many,” and then we want to understand the order of growth for the “family” of objects we are counting. In other settings, we might want to show the answer really is “very few indeed.” In this talk, we will explore how both types of counting problem can be related to each other. In particular, we will trace one “simple” problem from the 1600’s as it has evolved into a deeply connected web of mysterious “counting problems” that may take decades (centuries?) to solve. This talk intends to be radically accessible to students and researchers in all areas of math.

Chenyang Xu
Abstract: In the last half century, the interplay between canonical metrics and stability for various algebraic objects has been a central topic in geometry. Many new theories are developed to understand each side, as well as their relation. In my talk, I will survey the recent progress on one example of this kind: the complete solution of the Yau-Tian-Donaldson Conjecture for varieties with a positive Chern class. While this is a differential geometry problem, its solution largely depends on the development of a new algebraic theory. We will focus on this algebraic theory, and its application on a new moduli construction. Video Passcode: WFi*a7MH