Title: structure of finite generated Lambda module and its adjoint

Date: 10/01/2021

Time: 3:00 PM - 4:00 PM

Place: Online (virtual meeting)

Contact: Chuangtian Guan ()

I will introduce the $\Lambda$-module and talk about the structure and its adjoint in a classical language. If possible, I will talk about the generalized version of it and the reason that why we study it.

The Reshetikhin-Turaev invariants of a knot are topological invariants built through the representation theory of certain Hopf algebras, such as quantum groups. In the early 2000s, Turaev introduced a G-graded version of this construction that produces invariants of knots equipped with representations of their fundamental group into the group G.
This talk will be about a special case of the G-graded construction. We will show that a graded Drinfeld double construction leads to Reshetikhin-Turaev invariants of knots which are “twisted” via the usual Fox calculus. This construction applies to a wide class of Hopf algebras, and in the case of an exterior algebra, it specializes to the twisted Reidemeister torsion of the complement of a knot.

The study of permutations is both ancient and modern. They can be viewed as the integers $1,2,\ldots,n$ in some order or as $n\times n$ permutation matrices. They can be regarded as data which is to be sorted. The explicit definition of the determinant uses permutations. An inversion of a permutation occurs when a larger integer precedes a smaller integer. Inversions can be used to define two partial orders on permutations, one weaker than the other. Partial orders have a unique minimal completion to a lattice, the Dedekind-MacNeille completion. Generalizations of permutation matrices determine related matrix classes, for instance, alternating sign matrices (ASMs) which arose independently in the mathematics and physics literature. Permutations may contain certain patterns, e.g. three integers in increasing order; avoiding such patterns determines certain permutation classes. Similar restrictions can be placed more generally on $(0,1)$-matrices. The convex hull of $n\times n$ permutation matrices is the polytope of $n\times n$ doubly stochastic matrices. In a similar way we get ASM polytopes. We shall explore these and other ideas and their connections.

The Siegel modular variety $\mathcal{A}_2(3)$, which parametrizes abelian surfaces with full level $3$ structure, was recently shown to be rational over $\mathbf{Q}$ by Bruin and Nasserden. What can we say about its twist $\mathcal{A}_2(\rho)$ that parametrizes abelian surfaces $A$ whose $3$-torsion representation is isomorphic to a given representation $\rho$? While it is not rational in general, it is always unirational over $\mathbf{Q}$ showing that $\rho$ arises as the $3$-torsion representation of infinitely many abelian surfaces. We will discuss how we can obtain an explicit description of the universal object over such a unirational cover of $\mathcal{A}_2(\rho)$ using invariant theoretic ideas, thus parametrizing families of abelian surfaces with fixed $3$-torsion representation. Similar ideas work in a few other cases, showing in particular that whenever $(g,p) = (1,2)$, $(1,3)$, $(1,5)$, $(2,2)$, $(2,3)$ and $(3,2)$, the necessary condition of cyclotomic similitude is also sufficient for a mod $p$ Galois representation to arise from the $p$-torsion of a $g$-dimensional abelian variety.

Matrix Concentration inequalities such as Matrix Bernstein inequality have played an important role in many areas of pure and applied mathematics. These inequalities are intimately related to the celebrated noncommutative Khintchine inequality of Lust-Piquard and Pisier. In the middle of the 2010's, Tropp improved the dimensional dependence of this inequality in certain settings by leveraging cancellations due to non-commutativity of the underlying random matrices, giving rise to the question of whether such dependency could be removed.
In this talk we leverage ideas from Free Probability to fully remove the dimensional dependence in a range of instances, yielding optimal bounds in many settings of interest. As a byproduct we develop matrix concentration inequalities that capture non-commutativity (or, to be more precise, ``freeness''), improving over Matrix Bernstein in a range of instances. No background knowledge of Free Probability will be assumed in the talk.
Joint work with March Boedihardjo and Ramon van Handel, more information at arXiv:2108.06312 [math.PR].

Grid diagrams are a type of knot diagram which allow us to tackle problems in knot theory combinatorially. After introducing them we will explore some examples of their use in traditional knot invariants and hint at their place in understanding manifolds.

Classically, the notion of symmetry is described by a group. In recent decades, we have seen the emergence of quantum mathematical objects whose symmetries are best described by tensor categories. Fusion categories simultaneously generalize the notion of a finite group and its category of finite dimensional complex representations, and we think of these objects as encoding quantum symmetries. We will give a basic introduction to the theory of fusion categories and describe applications to some areas of mathematics and physics, namely operator algebras and theoretical condensed matter. [Zoom passcode distributed upon request.]

In this series of talks we show a necessary and sufficient condition for the vanishing of the Hochschild cohomology of a uniform Roe algebra. Specifically, the n-dimensional continuous Hochschild cohomology vanishes if and only if every norm continuous n-linear map from the uniform Roe algebra to itself is equivalent to a weakly continuous n-linear map.
Hochschild cohomology was introduced by Gerhard Hochschild in his 1945 paper “On the Cohomology Groups of an Associative Algebra”. The Hochschild cohomology of associative algebras has become a useful object of study in many fields of mathematics such as representation theory, mathematical physics, and noncommutative geometry, to name a few.
Last time we showed that all bounded derivations on uniform Roe algebras are inner. The first Hochschild cohomology measures how close derivations are to being inner. Hence, our result from last time can be restated as the first Hochschild cohomology of the uniform Roe algebra vanishing. It is then natural to ask if the higher dimensional cohomologies also vanish.
We will begin with the definition and several properties of multilinear maps which are essential to building the Hochschild complex. We then define the Hochschild complex and Hochschild cohomology as they apply to multilinear maps from a C*-algebra A to a Banach A-bimodule V. We then review many properties of these cohomologies. Lastly, we will show that if all n-linear maps have a weakly continuous representation in the Hochschild cohomology then the n’th dimensional Hochschild cohomology vanishes.

In a series of joint works with Tian Yang, we made a volume conjecture and an asymptotic expansion conjecture for the relative Reshetikhin-Turaev invariants for a closed oriented 3-manifold with a colored framed link inside it. We propose that their asymptotic behavior is related to the volume, the Chern-Simons invariant and the adjoint twisted Reidemeister torsion associated with the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring.
In this talk, I will first discuss how our volume conjecture can be understood as an interpolation between the Kashaev-Murakami-Murakami volume conjecture of the colored Jones polynomials and the Chen-Yang volume conjecture of the Reshetikhin-Turaev invariants. Then I will describe how the adjoint twisted Reidemeister torsion shows up in the asymptotic expansion of the invariants. Especially, we find new explicit formulas for the adjoint twisted Reidemeister torsion for the fundamental shadow link complements and for the 3-manifold obtained by doing hyperbolic Dehn-filling on those link complements. Those formulas cover a very large class of hyperbolic 3-manifold and appear naturally in the asymptotic expansion of quantum invariants. Finally, I will summarize the recent progress of the asymptotic expansion conjecture of the fundamental shadow link pairs.

Abstract: The hook length formula for the number of Standard Young Tableaux of a partition lambda is one of the few miraculous product formulas we see in combinatorics. While no such formula for the number of skew SYTs exists in general, the recent Naruse Hook Length Formula (NHLF) brings us close. I will explain this formula and generalizations, give some proofs and bijections, and discuss new extensions of NHLF for increasing tableaux originating in the study of Grothendieck polynomials.
Based on a series of papers with A. Morales and I. Pak

Legendrian links play a central role in low dimensional contact topology. A rigid theory uses invariants constructed via algebraic tools to distinguish Legendrian links. The most influential and powerful invariant is the Chekanov-Eliashberg differential graded algebra (Chekanov, Inventiones, 2002), which set apart the first non-classical Legendrian pair and stimulated many subsequent developments. The functor of points for the dga is a moduli space which acquires rich algebraic structures and can distinguish exact Lagrangian fillings. Such fillings are difficult to construct and to study, whereas the only known classification is the unique filling for Legendrian unknot (Eliashberg-Polterovich, Annals, 1996). For a long time, a folklore belief is that exact Lagrangian fillings are scarce and a Legendrian link can only have finitely many, based on the observation from limited examples.
In this talk, I will report a joint work with Roger Casals, where we applied the techniques from contact topology, microlocal sheaf theory and cluster algebras, and successfully found the first examples of Legendrian links with infinitely many Lagrangian fillings, reversing the general belief.

Sketching and random projection methods are a powerful set of techniques to speed up computations in numerical linear algebra, statistics, machine learning, optimization and data science. In this talk, we will discuss some of our works on developing a "big data" asymptotic perspective on sketching in the fundamental problems of linear regression and principal component analysis. This can lead to remarkably clean and elegant mathematical results, which yield powerful insights into the performance of various sketching methods. To highlight one, orthogonal sketches such as the Subsampled Randomized Hadamard Transform are provably better than iid sketches such as Gaussian sketching. This is obtained by using deep recent tools from asymptotic random matrix theory and free probability, including asymptotically liberating sequences (Anderson & Farrell, 2014). This is based on joint works with Jonathan Lacotte, Sifan Liu, Mert Pilanci, David P. Woodruff, and Fan Yang.

I will give some sort of introduction to a cool quantum 3-manifold invariant called the Turaev-Viro invariant and connect it to hyperbolic geometry through the Volume Conjecture. Then we will talk about shadows of 4-manifolds and how they can be used to prove the asymptotic additivity (we made this term up) of the Turaev-Viro invariants for a certain family of link complements.

Where do eigenfunctions of the Laplacian concentrate as eigenvalues go to infinity? Do they equidistribute or do they concentrate in an uneven way? It turns out that the answer depends on the nature of the geodesic flow. I will discuss various results in the case when the flow is chaotic: the Quantum Ergodicity theorem of Shnirelman, Colin de Verdière, and Zelditch, the Quantum Unique Ergodicity conjecture of Rudnick–Sarnak, the progress on it by Lindenstrauss and Soundararajan, and the entropy bounds of Anantharaman–Nonnenmacher. I will conclude with a more recent lower bound on the mass of eigenfunctions obtained with Jin and Nonnenmacher. It relies on a new tool called "fractal uncertainty principle" developed in the works with Bourgain and Zahl.

Kinetic equations are mesoscale description of the
transport of particles such as neutrons, photons, electrons,
molecules as well as their interaction with a background
medium or among themselves, and they have wide applications in many areas of mathematical physics, such as nuclear engineering, fusion device, optical tomography, rarefied gas dynamics, semiconductor device design, traffic network, swarming, etc. Because the equations are posed in the phase space (physical space plus velocity space), any grid-based method will run into computational bottleneck in real applications that are 3D in physical space and 3D in velocity space.
This talk will survey three numerical solvers that we developed aiming at efficient computations of kinetic equations: the adaptive sparse grid discontinuous Galerkin method, the reduced basis method and the machine learning moment closure method. They aim at effective reduced order computations of such high dimensional equations. Benchmark numerical examples will be presented.
Finally, I will introduce WINASc: Women in Numerical Analysis and Scientific Computing, which is part of the AWM advance network.
Meeting Zoom password: awmams

This talk is motivated by surprising connections between two very different approaches to 3-dimensional topology, namely quantum topology and hyperbolic geometry. The Kashaev-Murakami-Murakami Volume Conjecture connects the growth of colored Jones polynomials of a knot to the hyperbolic volume of its complement. More precisely, for each integer n, one evaluates the n-th Jones polynomial of the knot at the n-root of unity exp(2 pi i/n). The Volume Conjecture predicts that this sequence grows exponentially as n tends to infinity, with exponential growth rate related to the hyperbolic volume of the knot complement.
I will discuss a closely related conjecture for diffeomorphisms of surfaces, based on the representation theory of the Kauffman bracket skein algebra of the surface, a quantum topology object closely related to the Jones polynomial of a knot. I will describe the mathematics underlying this conjecture, which involves a certain Frobenius principle in quantum algebra. I will also present experimental evidence for the conjecture, and describe partial results obtained in work in progress with Helen Wong and Tian Yang.

Many combinatorial objects with strikingly good enumerative formulae and dynamical behavior (such as cyclic sieving) have underlying algebraic meaning. We first review classical results on promotion of standard Young tableaux, rotation of matchings/webs, and related invariant polynomials and symmetric group actions. We then discuss recent joint work with Rebecca Patrias and Oliver Pechenik involving the more general setting of increasing tableaux and noncrossing partitions.

I will present recent joint work with N. Chen about dominant rational maps from products of curves to surfaces with p_g=q=0. The gonality of an algebraic curve C is the minimal degree of a non-constant morphism from C to the projective line. Our main result is that under some assumptions the minimal degree of a dominant rational map from a product of two curves to a surface with p_g=q=0 is the product of their gonalities. In particular, a product of hyperelliptic curves of general type does not admit dominant rational maps of degree less than 4 to P^2. I will finish by presenting open problems and some strategies to attack them.

SLE (Schramm-Loewner evolution) is a family of random planar curves that have some natural conformal invariance properties. They appear in a variety of planar models that exhibit conformal invariance in the scaling limit. In this talk I will introduce SLE and describe its regularity. Regarding the regularity, the optimal Hoelder and p-variation exponents are known from previous works. I will present a new approach that refines the results to the logarithmic scale.
Zoom Passcode: A*****-P**

This talk aims to introduce and give an overview of a tool for studying four manifolds which I explored in some depth at a recent workshop at the University of Nebraska this summer: Trisections. Pioneered by Rob Kirby and David Gay, the slogan "trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds" encouraged many other low-dimensional topologists to investigate this tool further. In particular, I'll define a 4-manifold trisections, present some of their basic properties, some nice results from trisection literature, and some of the numerous problems and questions going forward.

The Kuramoto model is a system of ordinary differential equations modeling the nonlinear interactions of a group phase oscillators. The Kuramoto and related models have been proposed as models for phenomenon like the synchronized flashing of fireflies and the dynamics of circadian rhythms and jet lag. We consider a number of problems in the stability of synchronized solutions to this model. Many of these problems involve some sort of index result, counting the dimension of some unstable manifold. We employ a number of different techniques including geometric, topological, and graph-theoretic ideas. [Zoom passcode distributed upon request.]

Let $S_n$ be the symmetric group. The pinnacle set of a permutation in $S_n$ is defined to be all elements of that permutation which are larger than both of their adjacent elements. Given a subset $P$ of $\{1,\ldots,n\}$ we can also ask if there exists a permutation in $S_n$ having $P$ as its pinnacle set. If so, we say $P$ is admissible. We can extend this idea further by ordering the elements of $P$ and asking if there exists a permutation in $S_n$ having pinnacle set $P$ with the elements of $P$ in the given order. If so, we say that the ordering is an admissible ordering. In this presentation, we will present an efficient recursion for counting the number of admissible orderings of a given pinnacle set.

The Chow groups of Severi--Brauer varieties associated to biquaternion division algebras were originally computed by Karpenko in the mid nineties. The main difficulty in these computations is determining whether or not CH^2, the group of codimension 2 cycles, contains nontrivial torsion; for these varieties this group is torsion-free. Since his original proof, Karpenko has given two other proofs of this result. All of these proofs involve some clever use of K-theory to determine relations between some explicit cycles. In this talk, I'll discuss a new geometric method that one can use to determine these same relations. Passcode: MSUALG

In digital signal processing, quantization is the step of converting a signal's real-valued samples into a finite string of bits. As the first step in digital processing, it plays a crucial role in determining the information conversion rate and the reconstruction accuracy. Compared to non-adaptive quantizers, the adaptive ones are known to be more efficient in quantizing bandlimited signals, especially when the bit-budget is small (e.g.,1 bit) and noises are present.
However, adaptive quantizers are currently only designed for 1D functions/signals. In this talk, I will discuss challenges in extending it to high dimensions and present our proposed solutions. Specifically, we design new adaptive quantization schemes to quantize images/videos as well as functions defined on 2D surface manifolds and general graphs, which are common objects in signal processing and machine learning. Mathematically, we start from the 1D Sigma-Delta quantization, extend them to high-dimensions and build suitable decoders. The discussed theory would be useful in natural image acquisition, medical imaging, 3D printing, and graph embedding.

For your favorite knot K, there's a braid diagram for K with n strings, and there are (n-2) ways to split the braid along a string into a pair of braids. What pairs of braids can this splitting yield? To answer this question, we take a detour into the realm of the Murasugi sum, an operation on knots and their Seifert surfaces.

The j-function, introduced by Felix Klein in 1879, is an essential ingredient in the study of elliptic curves. It is Z-periodic on the complex upper half-plane, so it admits a Fourier expansion. The original Monstrous Moonshine conjecture, due to McKay and Conway/Norton in the 1980s, relates the Fourier coefficients of the j-function around the cusp to dimensions of irreducible representations of the Monster simple group. It was proved by Borcherds in 1992.
In my talk I will try to give a rudimentary introduction to modular forms, explain Monstrous Moonshine, and discuss a new version of it obtained in joint work with Yunfan He and Shengyuan Huang. Our version involves studying the j-function around CM points (so-called Landau-Ginzburg points in the physics literature) and expanding with respect to a coordinate which arises naturally in string theory.