We describe a new and elementary proof of the fact that many groups of piecewise-projective transformation of the line are non-amenable by constructing an explicit action without fixed points. One the one hand, such groups provide explicit counter-examples to the Day-von Neumann problem. On the other hand, they illustrate that we can distinguish many "layers" of relative non-amenability between nested subgroups.

The "boundaries" of the title refer to group action on measure spaces that enjoy the amenability property discovered by Furstenberg, Margulis and Zimmer. Over the last half-century these boundaries have been invaluable for group theory, rigidity, and more recently "bounded cohomology" -- which is the same as the cohomology of convolution algebras.

We will see that many familiar groups admit surprisingly strong ergodicity properties for boundaries. This applies to lamplighters, Thompson groups and many transformation groups. As a consequence, we determine the bounded cohomology of some of these groups.

Regularity lemmas are a cornerstone of combinatorics, with many applications in math and CS.  The most famous one is Szemeredi's regularity lemma. It shows that any graph can be partitioned into a "few" parts that mostly "look random". However, there is a caveat - "few" is really a huge number, a tower of exponentials in the error parameter.

Motivated by this, Frieze and Kannan designed a "weak" regularity lemma, sufficient for some applications, where the number of parts is much smaller, only exponential in the error parameter. In this work, we develop an even weaker regularity lemma, called "spread regularity", where the number of parts is even smaller - quasi-polynomial in the error parameter.

I will describe our new notion of regularity and some applications:
1. new lower bound technique in communication complexity, where players partially share information
2. new combinatorial algorithm for boolean matrix multiplication
3. improved bounds for variants of the corners problem in additive combinatorics

Based on joint works with Amir Abboud, Nick Fischer, Michael Jaber, Zander Kelley, Raghu Meka and Anthony Ostuni

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus's (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(C)×SL2(C)→SO4(C), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.

The Birch and Swinnerton-Dyer conjecture relates the leading coefficient of the L-function of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. Beilinson’s conjectures generalize the BSD conjecture to formulas for values of motivic L-functions at non-critical points. In this talk, I will relate motivic cohomology classes, with non-trivial coefficients, of Picard modular surfaces to a non-critical value of the motivic L-function of certain automorphic representations of the group GU(2,1).

We discuss work in progress regarding the following assertion: Let $G$ be a simple higher rank Lie group, then any closed manifold $M$, admitting a smooth locally free action of $G$, with codimension one orbits, is finitely and equivariantly covered by $G/\Gamma \times S^1$, for some cocompact lattice $\Gamma$ of $G$, where $G$ acts by left translations on the first factor, and trivially on $S^1$. This result is in the spirit of the Zimmer program. We will focus on the case $G = \mathrm{SL}(3,\mathbb{R})$ for the talk.

One method to generate random permutations involves using Gaussian elimination with partial pivoting (GEPP) on a random matrix A and storing the permutation matrix factor P from the resulting GEPP factorization PA=LU. We are interested in exploring properties of random butterfly permutations, which are generated using GEPP on specific random butterfly matrices. We address the questions of the longest increasing subsequence (LIS) and number of cycles for particular uniform butterfly permutations, with full distributional descriptions and limit theorems for simple butterfly permutations. We also establish scaling limit results and limit theorems for nonsimple butterfly permutations, which include certain p-Sylow subgroups of the symmetric group of N=pⁿ elements. For the LIS, we establish power-law bounds on the first moment N^αₚ ≤ E(LIS) ≤ N^βₚ where 1/2 < αₚ < βₚ < 1 and αₚ = 1 - oₚ(1), showing distinction from the typical O(N¹ᐟ²) expected LIS frequently encountered in the study of random permutations (e.g., uniform permutations, colored permutations, recent wreath product permutations studied by Diaconis). For the number of cycles scaled by (2-1/p)ⁿ, we establish a full CLT to a new limiting distribution depending on p with positive support we introduce that is uniquely determined by its positive moments that satisfy explicit recursive formulas; this thus determines a number of cycles CLT for any uniform p-Sylow subgroups of Sₚn. This work is joint with Chenyang Zhong, and continues from a UCSD probability seminar talk I gave last May to now include novel results and proof techniques from our recent preprint arXiv:2410.20952.

Non-rigid point cloud registration plays a crucial role in various fields, including robotics, neuroscience, computer graphics, and medical imaging. This process involves determining spatial relationships between different sets of points, typically within a 3D space. In real-world scenarios, complexities arise from non-rigid movements and partial visibility, such as occlusions or sensor noise, making non-rigid registration a challenging problem. Classic non-rigid registration methods are often computationally demanding, suffer from unstable performance, and, importantly, have limited theoretical guarantees.

The talk will focus primarily on a new way to understand non-rigid alignment and manifold learning of point clouds in Euclidean space using Whitney Extensions machinery developed by the author and his collaborators over the last few years. We will possibly  explore relationships of our work to topological data analysis, optimal transport, neural networks and neuroscience.

In this seminar I aim to show you what happens to a theoretical physicist, who gets to work in Theoretical Immunology and Virology. During the journey the physicist gets to learn mathematics she did not know at the time, and of course, explores the biological universe at different scales.

Since we only have a limited amount of time, I would like to introduce you to a problem that I have recently become interested in, and which has received NIH funding. Co-infection of a single host by different pathogens is ubiquitous in nature. We consider a population of hosts (e.g., small or large vertebrates) and a population of ticks, both of them susceptible to  infection with two different strains of a given virus.  We note that for the purposes of our models, we have Crimean-Congo hemorrhagic fever virus (a segmented Bunyavirus) in mind, as the application system.

First, we focus on the dynamics of a single infection, proposing  a deterministic  model to understand the role of co-feeding in the transmission of the virus.  We then compute the basic reproduction number by making use of the next generation matrix approach.  When considering co-infection by two distinct strains (one resident and one invasive), we make use of differential equations to model the dynamics of susceptible, infected and co-infected species, and we compute the invasion reproduction number of the invasive strain.  I discuss some problems with the calculation, and the solution proposed by Samuel Alizon and Marc Lipsitch. I conclude with a perspective on how the co-infection model can be applied to HIV, and plans for future work and work in progress for tick-borne pathogens. To end the talk, I would like to showcase a number of problems in immunology I have worked on, and which have required, for instance, the theory of stochastic processes, probability generating functions, or the use of a Gröbner basis.

Hopf algebras provide a rich framework for studying symmetry in noncommutative settings, generalizing both group algebras and enveloping algebras of Lie algebras. If one has any hope of understanding the structure of noetherian Hopf algebras, one should first understand when group algebras and enveloping algebras are noetherian. These two famously difficult problems are open to this day. In this talk, I will give a survey of what is known about the noetherianity of group algebras and enveloping algebras, mainly focusing on the Lie algebra side.

Elliptic Curves have been one of the many objects of studying due to their rich structure. We will discuss the group law on elliptic curves and the staggering fact that this group is abelian and finitely generated. Knowing the structure of finitely generated abelian groups, we ask what the torsion piece of the group can be. Mazur’s theorem will give us our answer! In discussion we will encounter modular curves, rational points on a variety of objects and attempt to get a sense of arithmetic geometry. This talk should be accessible to anyone but knowledge of the classification of finitely generated abelian groups and a little complex analysis may be useful in some sections.

We will discuss a natural perspective on stable reduction that extends Deligne--Mumford's stable reduction for curves to higher dimensions. From this perspective, we will outline a new proof of the Hacon--Kovács theorem on the properness of the moduli stack $\overline{\mathscr{M}}_{2,v,k}$ of stable surfaces of volume $v$ defined over $k=\overline{k}$, provided that $\operatorname{char}k>C(v)$, a constant depending only on $v$.