# Departmental Colloquium 2020-2021

## Logistics:

###
- Typically Thursdays on Zoom, for an hour.
- Most will be at 4:00pm or 9:00am (some flexibility is necessary to allow for speaker timezones/schedules).
- Colloquia will be preceded by a virtual tea using gather.town.
- Access information for Zoom and the tea will be distributed weekly through the department listserv with the colloquium announcement.

## Spring 2021

**April 22 (Thursday), 4:00pm**

**Speaker: ** Greg Forest, The University of North Carolina at Chapel Hill

**Title:**
*Modeling insights into SARS-CoV-2 respiratory tract infections*

**Abstract: **
*Insights into the mechanisms and dynamics of human respiratory tract (HRT) infections from the SARS-CoV-2 virus can inform public awareness as well as guide medical prevention and treatment for COVID-19 disease. Yet, the complex physiology of the human lung and the inability to sample diverse regions of the HRT pose fundamental roadblocks, both to discern among potential mechanisms for infection and disease and to monitor progression of infection. My group has explored lung biology and disease for over 2 decades in an effort called the UNC Virtual Lung Project, spanning many disciplines. We further explored how viruses “traffic” in mucosal barriers coating human organs, including the upper and lower respiratory tract, for the last decade, focusing on natural and synthetic antibody protection.
Then along came the novel coronavirus SARS-CoV-2, for which we have no immune protection, requiring a step back to a pre-immunity scenario. We developed a computational model that incorporates: detailed physiology of the HRT, and best current knowledge about the mobility of SARS-CoV-2 virions in airway surface liquids (ASL) as well as epithelial cell infectability and replication of infectious virions throughout the HRT. The model simulates outcomes from any dynamic deposition profile of SARS-CoV-2 throughout the HRT, and tracks the propagation of infectious virions in the ASL and infected epithelial cells. We focus this lecture on two clinical observations, their respective likelihoods, and open questions raised: an upper respiratory tract infection following inhaled exposure to SARS-CoV-2; and, progression to alveolar pneumonia. Our baseline modeling platform is poised to superimpose interventions, from adaptive immune responses to any form of medical or drug treatment, at any point from pre-exposure to disease progression, with several new collaborations to do so. The results presented highlight the urgency to understand the underlying physical and physiological conditions that facilitate transmission, including self-transmission, which we absolutely do not yet understand. *

**April 8 (Thursday), 4:00pm**

**Speaker: ** Aaron Pollack, University of California San Diego

**Title: **
*Modular forms on exceptional groups*

**Abstract: **
*Classical holomorphic modular forms are intensely studied objects. They are generalized to automorphic forms, which are associated to reductive groups G. When G is an exceptional group, such as E_8, one can ask if there are special automorphic forms on G, which can in some ways take the place of the classical holomorphic modular forms. Gan, Gross, Savin, and Wallach singled out a class of such automorphic forms, called the quaternionic modular forms. I will explain what these objects are, and what is known about them.*

__ April 6 (Tuesday), 4:00pm__ - AWM Speaker Series

**Speaker:**Ryan Hynd, University of Pennsylvania

**Title:**

*A conjecture of Meissner*

**Abstract:**

*A curve of constant width has the property that any two parallel supporting lines are the same distance apart in all directions. A fundamental problem involving these curves is to find one which encloses the smallest amount of area for a given width. This problem was resolved long ago and has a few relatively simple solutions. The volume minimizing shapes in three-dimensions have yet to be determined. We’ll discuss the two shapes which are conjectured to be volume minimizing and an approach to this outstanding problem.*

**April 1 (Thursday), 4:00pm**

**Speaker: ** Tatiana Toro, University of Washington

**Title:**
*How do Partial Differential Equations detect Geometry in Euclidean space?*

**Abstract: **
*In this talk we will present an area of analysis that is concerned with the relationship between differential operators, the properties of their solutions, and the geometry of the domain on which they are considered. The goal is to highlight how analytic properties of solutions to PDEs determine the geometry of the domain where they are considered. The tools used in this area come from analysis of partial differential equations, harmonic analysis and geometric measure theory.
*

__ March 16 (Tuesday), 4:00pm__ - AWM-RTG Speaker Series

**Speaker:**Ila Varma, University of Toronto

**Title:**

*Number field asymptotics and Malle's Conjecture*

**Abstract:**

*Malle's conjecture can be thought of as a generalization of the inverse Galois problem, which asks for every finite group G, is there a number field K such that their Galois group over Q is isomorphic to G? Although open, this question is widely believed to be true, and Malle went further to predict the asymptotics of how many number fields there are with a given Galois group that only depended on the group structure of G and the degree of the number field. In this talk, we will discuss the history as well as recent results and techniques surrounding these conjectures.*

**March 11 (Thursday), 9:00am**

**Speaker: ** Gilad Lerman, University of Minnesota

**Title:**
*Group Synchronization via Cycle-Edge Message Passing*

**Abstract: **
*The problem of group synchronization asks to recover states of objects associated with group elements given possibly corrupted relative state measurements (or group ratios) between pairs of objects. This problem arises in important data-related tasks, such as structure from motion, simultaneous localization and mapping, Cryo-EM, community detection and sensor network localization. Two common groups in these problems are the rotation and symmetric groups. We propose a general framework for group synchronization with compact groups. The main part of the talk discusses a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios. Under our mathematical model of adversarial corruption, it can be used to infer the corrupted group ratios and thus to solve the synchronization problem. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish competitive theoretical results under a uniform corruption model. Finally, we discuss the MPLS (Message Passing Least Squares) or Minneapolis framework for solving real scenarios with high levels of corruption and noise and with nontrivial scenarios of corruption. We demonstrate state-of-the-art results for two different problems that occur in structure from motion and involve the rotation and permutation groups.
*

**March 4 (Thursday), 4:00pm**

**Speaker: ** Jinchao Xu, Pennsylvania State University

**Title:**
*Deep Learning, Finite Element and Multigrid*

**Abstract: **
*In this talk, I will first give an elementary introduction to models
and algorithms from two different fields: (1) machine learning,
including logistic regression and deep neural networks, and (2)
numerical PDEs, including finite element and multigrid methods. I
will then explore mathematical relationships between these different
models and algorithms and demonstrate how such relationships can be
used to understand, study and improve different aspects of deep neural
networks, finite element and multigrid methods. We will show that
ReLU-DNN corresponds exactly to the traditional piecewise linear
finite functions and [ReLU]^k-DNN leads to new finite element of
piecewise polynomials of degree k with remarkable
approximation properties. We will demonstrate how a new convolutional
neural network (CNN), known as MgNet, can be derived by making very
minor modifications of a classic geometric multigrid method for the
Poisson equation and then discuss the theoretical and practical
potentials of MgNet.*

**February 25 (Thursday), 9:00am **

**Speaker: ** Frank Calegari, University of Chicago

**Title: **
*Recent progress in the arithmetic of the Langlands Program*

**Abstract: **
*Choose a random polynomial with integer coefficients, say x^3 + 1 or x^5 + x + 1.
If you choose a prime number p, what is the probability that the values of these polynomials will be squares modulo p?
Half the time? More than half the time? Less? Some of these elementary questions ultimately lead to difficult problems in the Langlands program.
The goal of this colloquium is to show how recent advances in the Langlands program can provide new answers to these questions,
and how they are linked to a possible notion of a “Galois group” of systems of polynomial equations in higher dimensions.
*

**February 18 (Thursday), 4:00pm**

* In this special colloquium event, we will be rebroadcasting the MAA-SIAM-AMS Hrabowski-Gates-Tapia-McBay Lecture from the 2021 Joint Mathematics Meeting, available here.*

**Speaker: (rebroadcast) ** Erica Graham, Bryn Mawr University

**Title:**
*Anti-racism in mathematics: Who, what, when, where, why, and how? *

**Abstract: **
*The Black Lives Matter movement, and many like it, has garnered widespread support for dismantling the racist structures woven into the fabric of our society at large. The academic discipline of mathematics--alongside many institutions of higher education--has also reached a point of reckoning in its history of institutionalizing racism. We must acknowledge the necessity, not choice, of persistent and active anti-racist work in realizing transformative, long-lasting change. In this talk, I will discuss the ‘Five Ws and How’ for anti-racism as my vision for the mathematical community.*

**February 2 (Tuesday), 9:00am**

**Speaker: ** Tony Yue Yu, Université Paris-Sud

**Title:**
*Non-archimedean mirror symmetry and its applications*

**Abstract: **
*Mirror symmetry is one of the most mysterious dualities in mathematics. My research explores a new approach to mirror symmetry, via non-archimedean geometry. I will give an overview of the recent progress in this direction. I will talk about non-archimedean enumerative geometry, the Frobenius structure conjecture, and derived non-archimedean geometry. I will also describe an application towards cluster algebras in representation theory, and another application towards the moduli spaces of Calabi-Yau pairs and their compactifications.*

**January 28 (Thursday), 4:00pm**

**Speaker: ** Junliang Shen, Massachussetts Institute of Technology

**Title:**
* The P=W conjecture and hyper-Kähler geometry*

**Abstract: **
*Topology of Hitchin's integrable systems and character varieties play important roles in many branches of mathematics. In 2010, de Cataldo, Hausel, and Migliorini discovered a surprising phenomenon which relates these two very different geometric objects in an unexpected way. More precisely, they predict that the topology of Hitchin systems is tightly connected to Hodge theory of character varieties, which is now called the "P=W" conjecture. In this talk, we will discuss recent progress of this conjecture. In particular, we focus on general interactions between topology of Lagrangian fibrations and Hodge theory in hyper-Kähler geometries. This hyper-Kähler viewpoint sheds new light on both the P=W conjecture for Hitchin systems and the Lagrangian base conjecture for compact hyper-Kähler manifolds.*

**January 26 (Tuesday), 4:00pm**

**Speaker: ** Harold Blum, Stony Brook University

**Title:**
*Moduli spaces of Fano varieties and K-stability*

**Abstract: **
*Algebraic geometry is focused on the study of algebraic varieties, which are shapes cut out by polynomial equations. A fundamental problem in the field is to construct and study spaces that parametrize algebraic varieties of a given type, referred to as moduli spaces. The quintessential example of such a parameter space is the moduli space of curves, which is a space whose points are in bijection with isomorphism classes of Riemann surfaces. After surveying these ideas, I will discuss the problem of constructing moduli spaces parametrizing Fano varieties, which are a class of complex manifolds that form one of the three main building blocks of varieties in algebraic geometry. A new approach to this problem relies on the notion of K-stability, which is an algebraic definition introduced by differential geometers to characterize when a Fano variety admits a certain canonical metric.*

**January 21 (Thursday), 4:00pm**

**Speaker: ** Man-Wai Cheung, Harvard University

**Title:**
*Counting tropical curves by quiver representation*

**Abstract: **
*Mikhalkin established the correspondence between holomorphic curves and tropical curves on toric surfaces. Tropical curve (disk) counts are then seen as the algebro-geometric analogue holomorphic disk countings in mirror symmetry. The invariants are heavily used in the mirror construction. On the other hand, spaces of quiver representations have been extensively studied in the representation theory world. Together with Travis Mandel, we have linked up these two appearingly independent areas together and developed an expression of tropical disks countings in terms of quiver representations.*

**January 19 (Tuesday), 4:00pm**

**Speaker: ** Kristin DeVleming, University of California San Diego

**Title:**
*Comparing compactifications of moduli spaces*

**Abstract: **
*A foundational goal in algebraic geometry is the classification of all varieties, which are locally vanishing loci of polynomials. To make this problem tractable, one fixes discrete invariants like dimension or genus of the variety, and then studies continuous families of varieties with those invariants. The parameters for those continuous families are called moduli, and the associated parameter space is called a moduli space. Classically, there are many techniques to construct moduli spaces and compactifications of moduli spaces, from geometric invariant theory to the minimal model program, and recently there has been great progress constructing them using K-stability. I will motivate each of these techniques and ultimately focus on comparing the different results. There will be many pictures and examples! *

## Fall 2020

**December 3 (Thursday), 2:00pm-3:00pm**

**Speaker: ** Kirsten Wickelgren, Duke University

**Title:**
*An arithmetic count of rational plane curves*

**Abstract: **
*There is a unique line through 2 points in the plane, and a unique conic through 5. These counts generalize to a count of degree d rational curves in the plane passing through 3d-1 points. Surprisingly, the problem of determining these numbers turns out to be deep and connected to string theory, and it was not until the 1990's that Kontsevich determined them with a recursive formula. Such formulas are valid when you allow your curves to be defined with complex coefficients. For fields of characteristic not 2 or 3, we use A1-homotopy theory to show that by counting with bilinear forms, there is an invariant arithmetic count of rational plane curves. This is joint work with Jesse Kass, Marc Levine, and Jake Solomon.*

**November 19 (Thursday), 12:45pm-1:45pm**

*AWM-RTG Speaker Series*

**Speaker: ** Christine Berkesch, University of Minnesota

**Title:**
*Virtual resolutions for smooth toric varieties
*

**Abstract: **
*The minimal free resolution of a graded module encodes many geometric properties of the corresponding sheaf on projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety X, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In joint work with Daniel Erman and Gregory G. Smith, we propose considering instead virtual resolutions, which more closely reflect the geometry of sheaves on X. We will survey recent results which build this case. *

**November 5 (Thursday), 4:00pm-5:00pm**

*AWM Speaker Series*

**Speaker: ** Nilima Nigam, Simon Fraser University

**Title:**
*A modification of Schiffer's conjecture, and a proof via finite
elements*

**Abstract: **
*Approximations via conforming and non-conforming finite elements
can be used to construct validated and computable bounds on eigenvalues
for the Dirichlet Laplacian in certain domains. If these are to be used as
part of a proof, care must be taken to ensure each step of the computation
is validated and verifiable. In this talk we present a long-standing
conjecture in spectral geometry, and its resolution using validated finite
element computations. Schiffer's conjecture states that if a bounded
domain Ω in R^n has any nontrivial Neumann eigenfunction which is a
constant on the boundary, then Ω must be a ball. This conjecture is
open. A modification of Schiffer's conjecture is: for regular polygons of
at least 5 sides, we can demonstrate the existence of a Neumann
eigenfunction which does not change sign on the boundary. In this talk, we
provide a recent proof using finite element calculations for the regular
pentagon. The strategy involves iteratively bounding eigenvalues for a
sequence of polygonal subdomains of the triangle with mixed Dirichlet and
Neumann boundary conditions. We use a learning algorithm to find and
optimize this sequence of subdomains, and use non-conforming linear FEM to
compute validated lower bounds for the lowest eigenvalue in each of these
domains. The linear algebra is performed within interval arithmetic. This
talk is based on the following paper, which is a joint work with
Bartlomiej Siudeja and Ben Green at University of Oregon.*

**October 29 (Thursday), 9:00am-10:00am**

**Speaker: ** Shi Jin, Shanghai Jiao Tong University

**Title:**
*Random batch methods for classical and quantum N-body problems*

**Abstract: **
*We first develop random batch methods for classical interacting particle systems with large number of particles. These methods use small but random batches for particle interactions, thus the computational cost is reduced from O(N^2) per time step to O(N), for a system with N particles with binary interactions. For one of the methods, we give a particle number independent error estimate under some special interactions.
This method is also extended to molecular dynamics with Coulomb interactions, in the framework of Ewald summation, and to quantum Monte-Carlo methods for the N-body Schrodinger equation. In each case we will show its superior performance compared to the current state-of-the-arts methods for the corresponding problems, in the computational efficiency and parallelizability.
This talk is based on joint works with Lei Li, Jian-Guo Liu, Zhenli Xu, and Xiaotao Li.*