## Yang Li : On the Donaldson-Scaduto conjecture

- Geometry and Topology ( 688 Views )Motivated by G2-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends in X \times R^3, where X is an A2-type ALE hyperkähler manifold. We prove this conjecture by solving a real Monge-Ampère equation with singular right hand side. The method produces many other asymptotically cylindrical U(1)-invariant special Lagrangians in X \times R^2, where X arises from the Gibbons-Hawking construction. This is joint work in progress with Saman Habibi Esfahani.

## Zane Li : Interpreting a classical argument for Vinogradovs Mean Value Theorem into decoupling language

- Applied Math and Analysis ( 136 Views )There are two proofs of Vinogradov's Mean Value Theorem (VMVT), the harmonic analysis decoupling proof by Bourgain, Demeter, and Guth from 2015 and the number theoretic efficient congruencing proof by Wooley from 2017. While there has been some work illustrating the relation between these two methods, VMVT has been around since 1935. It is then natural to ask: What does previous partial progress on VMVT look like in harmonic analysis language? How similar or different does it look from current decoupling proofs? We talk about a classical argument due to Karatsuba that shows VMVT "asymptotically" and interpret this in decoupling language. This is joint work with Brian Cook, Kevin Hughes, Olivier Robert, Akshat Mudgal, and Po-Lam Yung.

## Chun-Hsien Hsu : Weyl algebras on certain singular affine varieties

- Number Theory ( 131 Views )The module theory of the Weyl algebra, known as the theory of $D$-modules, has profound applications in various fields. One of the most famous results is the Riemann-Hilbert correspondence, establishing equivalence between holonomic $D$-modules and perverse sheaves on smooth complex varieties. However, when dealing with singular varieties, such correspondence breaks down due to the non-simplicity of Weyl algebras on singular varieties. In our ongoing work, we introduce a new ring of differential operators on certain singular affine varieties, whose definition is analytically derived from harmonic analysis. It should contain the Weyl algebra as a proper subring and shares many properties with the Weyl algebra on smooth varieties. In the talk, after a brief review of the Weyl algebra, I will explain how the new ring of differential operators arises as a consequence of an explicit form of the Poisson summation conjecture and discuss its properties.

## Wenzhao Chen : Negative amphicheiral knots and the half-Alexander polynomial

- Geometry and Topology ( 101 Views )In this talk, we will study strongly negative amphicheiral knots - a class of knots with symmetry. These knots provide torsion elements in the knot concordance group, which are less understood than infinite-order elements. We will introduce the half-Alexander polynomial, an equivariant version of the Alexander polynomial for strongly negative amphicheiral knots, focusing on its applications to knot concordance. In particular, I will show how it facilitated the construction of the first examples of non-slice amphicheiral knots of determinant one. This talk is based on joint work with Keegan Boyle.

## Robert Bryant : A Weierstrass representation for affine Bonnet surfaces

- Geometry and Topology ( 99 Views )Ossian Bonnet (1819–1892) classified the surfaces in Euclidean 3-space that can be isometrically deformed without changing the mean curvature function H, showing that there are two types: the surfaces of constant mean curvature and a 4-dimensional ‘exceptional family’ (with variable mean curvature) that are now known as Bonnet surfaces. The corresponding problem in affine 3-space is much more difficult, and the full classification is still unknown. More than 10 years ago, I classified the affine surfaces that can isometrically deformed (with respect to the induced Blaschke metric) while preserving their affine mean curvature in a 3-dimensional family (the maximum dimension possible), showing that they depend on 2 functions of 1 variable in Cartan’s sense. When I gave a talk* in this seminar about these results on September 10, 2013, I only knew that these surfaces corresponded to pseudoholomorphic curves in a certain almost-complex surface. However, I have recently shown that the structure equations for these mysterious surfaces can be interpreted as describing holomorphic Legendrian curves in CP^3 subject to a natural positivity condition, and the integration corresponds to a flat sp(2,R) connection, i.e., they can be interpreted as a Lax pair, but of a very special kind, for which the integration can be effected explicitly. I’ll explain these results and use them to show how the classical problem of determining the affine surfaces with constant affine mean curvature and constant Gauss curvature of the Blaschke metric can be explicitly integrated, which, heretofore, was unknown. * https://www4.math.duke.edu/media/watch_video.php?v=6948e657e69cadbaa1a6915335e9ea87

## Ashvin Swaminathan : Geometry-of-numbers in the cusp, and class groups of orders in number fields

- Number Theory ( 96 Views )In this talk, we discuss the distributions of class groups of orders in number fields. We explain how studying such distributions is related to counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We introduce two new methods to solve this counting problem, and as an application, we demonstrate how to determine the average size of the 2-torsion in the class groups of cubic orders. Much of this work is joint with Arul Shankar, Artane Siad, and Ila Varma.

## Yao Xiao : Equivariant Lagrangian Floer theory on compact toric manifolds

- Geometry and Topology ( 93 Views )We define an equivariant Lagrangian Floer theory on compact symplectic toric manifolds for the subtorus actions. We prove that the set of Lagrangian torus fibers (with weak bounding cochain data) with non-vanishing equivariant Lagrangian Floer cohomology forms a rigid analytic space. We can apply tropical geometry to locate such Lagrangian torus fibers in the moment map. We show that these Lagrangian submanifolds are nondisplaceable by equivariant Hamiltonian diffeomorphisms.

## Mariana Olvera-Cravioto : Opinion dynamics on complex networks: From mean-field limits to sparse approximations

- Probability ( 90 Views )In a world of polarized opinions on many cultural issues, we propose a model for the evolution of opinions on a large complex network. Our model is akin to the popular Friedkin-Johnsen model, with the added complexity of vertex-dependent media signals and confirmation bias, both of which help explain some of the most important factors leading to polarization. The analysis of the model is done on a directed random graph, capable of replicating highly inhomogeneous real-world networks with various degrees of assortativity and community structure. Our main results give the stationary distribution of opinions on the network, including explicitly computable formulas for the conditional means and variances for the various communities. Our results span the entire range of inhomogeneous random graphs, from the sparse regime, where the expected degrees are bounded, all the way to the dense regime, where a graph having n vertices has order n^2 edges.

## Kai Xu : pi_2-systolic inequalities for 3-manifolds with positive scalar curvature

- Geometry and Topology ( 88 Views )We discuss the following recent result of the speaker. Suppose a closed 3-manifold M has scalar curvature at least 1, and has nontrivial second homotopy group, and is not covered by the cylinder (S^2)*R. Then the pi_2-systole of M (i.e. the minimal area in the second homotopy group) is bounded by a constant that is approximately 5.44pi. If we include quotients of cylinder into consideration, then the best upper bound is weakened to 8_pi. This shows a topological gap in the pi_2-systolic inequality. We will discuss the ideas behind this theorem, as well as the proof using Huisken and Ilmanen’s weak inverse mean curvature flow.

## Chun-Hung Liu : Assouad-Nagata dimension of minor-closed metrics

- Mathematical Biology ( 81 Views )Assouad-Nagata dimension addresses both large-scale and small-scale behaviors of metric spaces and is a refinement of Gromov’s asymptotic dimension. A metric space is a minor-closed metric if it is defined by the distance function on the vertices of an edge-weighted graph that satisfies a fixed graph property preserved under vertex-deletion, edge-deletion, and edge-contraction. In this talk, we determine the Assouad-Nagata dimension of every minor-closed metric. It is a common generalization of known results about the asymptotic dimension of H-minor free unweighted graphs, about the Assouad-Nagata dimension of complete Riemannian surfaces with finite Euler genus, and about their corollaries on weak diameter coloring of minor-closed families of graphs and asymptotic dimension of minor-excluded groups.

## Zack Bezemek : Large Deviations and Importance Sampling for Weakly Interacting Diffusions

- Probability ( 81 Views )We consider an ensemble of N interacting particles modeled by a system of N stochastic differential equations (SDEs). The coefficients of the SDEs are taken to be such that as N approaches infinity, the system undergoes Kac’s propagation of chaos, and is well-approximated by the solution to a McKean-Vlasov Equation. Rare but possible deviations of the behavior of the particles from this limit may reflect a catastrophe, and computing the probability of such rare events is of high interest in many applications. In this talk, we design an importance sampling scheme which allows us to numerically compute statistics related to these rare events with high accuracy and efficiency for any N. Standard Monte Carlo methods behave exponentially poorly as N increases for such problems. Our scheme is based on subsolutions of a Hamilton-Jacobi-Bellman (HJB) Equation on Wasserstein Space which arises in the theory of mean-field control. This HJB Equation is seen to be connected to the large deviations rate function for the empirical measure on the ensemble of particles. We identify conditions under which our scheme is provably asymptotically optimal in N in the sense of log-efficiency. We also provide evidence, both analytical and numerical, that with sufficient regularity of the solution to the HJB Equation, our scheme can have vanishingly small relative error as N increases.

## Farid Hosseinijafari : On the Special Values of Certain L-functions: G_2 over a Totally Imaginary Field

- Number Theory ( 72 Views )In this talk, I will present an overview of the framework originally proposed by Harder and further developed in collaboration with Raghuram to address rationality problems for special values of certain automorphic L-functions. I will then proceed to state my main results on the rationality of the special values of Langlands-Shahidi L-functions appearing in the constant term of the Eisenstein series associated with the exceptional group of type G_2 over a totally imaginary number field. This study marks the first instance where rank-one Eisenstein cohomology is employed to investigate the arithmetic of automorphic L-functions in the presence of multiple L-functions.

## Erik Bates : The Busemann process of (1+1)-dimensional directed polymers

- Probability ( 66 Views )Directed polymers are a statistical mechanics model for random growth. Their partition functions are solutions to a discrete stochastic heat equation. This talk will discuss the logarithmic derivatives of the partition functions, which are solutions to a discrete stochastic Burgers equation. Of interest is the success or failure of the “one force-one solution principle” for this equation. I will reframe this question in the language of polymers, and share some surprising results that follow. Based on joint work with Louis Fan and Timo Seppäläinen.

## Lenny Ng : New algebraic invariants of Legendrian links

- Geometry and Topology ( 62 Views )For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.

## Dean Bottino : Evaluating Strategies for Overcoming Rituximab (R) Resistance Using a Quantitative Systems Pharmacology (QSP) model of Antibody-Dependent Cell-mediated Cytotoxicity & Phagocytosis (ADCC & ADCP): An Academic/Industrial Collaboration

- Mathematical Biology ( 57 Views )Despite the impressive performance of rituximab (R) containing regimens like R-CHOP in CD20+ Non-Hodgkin’s Lymphoma (NHL), 30-60% of R-naïve NHL patients are estimated to be resistant, and approximately 60% of those patients will not respond to subsequent single agent R treatment. Given that antibody dependent cell mediated cytotoxicity (ADCC) and phagocytosis (ADCP) are thought to be the major mechanisms of action of Rituximab, increasing the activation levels of natural killer (NK) and macrophage (MP) cells may be one strategy for overcoming R resistance.

During (and after) the Fields Institute Industrial Problem Solving Workshop in August 2019, academic participants and industry mentors developed and calibrated to literature data a quantitative systems pharmacology (QSP) model of ADCC/ADCP to interrogate which mechanisms of R resistance could be overcome by increased NK or MP activation, and how much effector cell activation would be required to overcome a given degree and mechanism of R resistance.

This work was motivated by a real-world pharmaceutical drug development question, and the academic-industry interactions during and after the workshop resulted in sharknado plots as well as a published QSP model (presented at American Association of Cancer Research Annual Meeting, 2021) that was able to address some of the key questions around overcoming R resistance. The published model was then incorporated into an in-house QSP model supporting the development of a Takeda investigational drug which is being developed to restore R sensitivity in an R-resistant patient population.

## Leo Darrigade : Modelling G protein-coupled receptors (GPCRs) compartmentalized signaling

- Mathematical Biology ( 49 Views )G protein-coupled receptors (GPCRs) are membrane receptors that play a pivotal role in the regulation of reproduction and behavior in humans. Upon binding to specific ligands, they trigger a local cAMP production. Activated receptor are then internalized to different endosomal compartments where they can continue signaling before being recycled or destroyed. Recent studies showed that the different pools of cAMP have different effect on the cell.

In the first part of the talk, I will present a piecewise deterministic Markov process (PDMP) of intracellular signaling. The stochastic part of the model accounts for formation, coagulation, fragmentation and recycling of intracellular vesicles which contain the receptor, whereas the deterministic part of the model represents evolution of chemical reactions due to signaling activity of the receptor. We are interested in the existence of and convergence to a stationary measure. I will present different cases for which we were able to obtain results in this direction.

In the second part of the talk, I will present the numerical workflow (SBML, PEtab and PyPESTO) we use to fit ODEs model of GPCR signaling to longitudinal measure of chemical concentrations (BRET data).

## Vakhtang Poutkaradze : Lie-Poisson Neural Networks (LPNets): Data-Based Computing of Hamiltonian Systems with Symmetries

- Applied Math and Analysis ( 47 Views )Physics-Informed Neural Networks (PINNs) have received much attention recently due to their potential for high-performance computations for complex physical systems, including data-based computing, systems with unknown parameters, and others. The idea of PINNs is to approximate the equations and boundary and initial conditions through a loss function for a neural network. PINNs combine the efficiency of data-based prediction with the accuracy and insights provided by the physical models. However, applications of these methods to predict the long-term evolution of systems with little friction, such as many systems encountered in space exploration, oceanography/climate, and many other fields, need extra care as the errors tend to accumulate, and the results may quickly become unreliable. We provide a solution to the problem of data-based computation of Hamiltonian systems utilizing symmetry methods. Many Hamiltonian systems with symmetry can be written as a Lie-Poisson system, where the underlying symmetry defines the Poisson bracket. For data-based computing of such systems, we design the Lie-Poisson neural networks (LPNets). We consider the Poisson bracket structure primary and require it to be satisfied exactly, whereas the Hamiltonian, only known from physics, can be satisfied approximately. By design, the method preserves all special integrals of the bracket (Casimirs) to machine precision. LPNets yield an efficient and promising computational method for many particular cases, such as rigid body or satellite motion (the case of SO(3) group), Kirchhoff's equations for an underwater vehicle (SE(3) group), and others. Joint work with Chris Eldred (Sandia National Lab), Francois Gay-Balmaz (CNRS and ENS, France), and Sophia Huraka (U Alberta). The work was partially supported by an NSERC Discovery grant.

## Duncan Dauvergne : Random planar geometry and the Kardar-Parisi-Zhang universality class

- Presentations ( 44 Views )Consider the lattice Z^2, and assign length 1 or 2 to every edge by flipping a series of independent fair coins. This gives a random weighted graph, and looking at distances in this graph gives a random planar metric. This model is expected to have a continuum scaling limit as we decrease the spacing between lattice points. Moreover, most natural models of random planar metrics and random interface growth (the so-called `KPZ universality class') are expected to converge to the same limiting geometry. The goal of this talk is to introduce this limit, known as the directed landscape, and describe at least one model where we can actually prove convergence.

## Cheng Chen : Progresses of the local Gan-Gross-Prasad conjecture

- Number Theory ( 44 Views )The classical branching rules describe the spectrum of an irreducible complex representation of a compact Lie group to its subgroup. The local Gan–Gross–Prasad conjecture generalizes the branching problem to classical groups over local fields of characteristic zero. After the pioneering work of Waldspurger, there has been significant progress on the conjecture using various approaches. In my talk, I will introduce a relatively uniform approach to prove the conjecture, including joint work with Z. Luo and joint work with R. Chen and J. Zou.

## Duncan Dauvergne : Geodesic networks in random geometry

- Presentations ( 40 Views )The directed landscape is a random directed metric on the plane that is the scaling limit for models in the KPZ universality class. In this metric, typical pairs of points are connected by a unique geodesic. However, certain exceptional pairs are connected by more exotic geodesic networks. The goal of this talk is to describe a full classification for these exceptional pairs. I will also discuss some connections with other models of random geometry.

## Chen Wan : A local twisted trace formula for some spherical varieties

- Number Theory ( 40 Views )In this talk, I will discuss the geometric expansion of a local twisted trace formula for some special varieties. This generalizes the local (twisted) trace formula for reductive groups proved by Arthur and Waldspurger. By applying the trace formula, we prove a multiplicity formula for these spherical varieties. And I will also discuss some applications to the multiplicity of the Galois model and the unitary Shalika model. This is a joint work with Raphael Beuzart-Plessis.