Classical Steenrod algebra is one of the most fundamental algebraic gadgets in stable homotopy theory. It led to the theory of characteristic classes, which is key to some of the most celebrated applications of homotopy theory to geometry. The G-equivariant Steenrod algebra is not known beyond the group of order 2. In this talk, I will recall a geometric construction of the classical Steenrod algebra and generalize it to construct G-equivariant Steenrod operations. Time permitting, I will discuss potential applications to equivariant geometry.

We discuss quasi-Newton methods for minimizing a strictly convex quadratic function that may be subject to errors in the evaluation of the gradients. The methods all give identical behavior in exact arithmetic, generating minimizers of Krylov subspaces of increasing dimensions, thereby having finite termination. In exact arithmetic, the method of conjugate gradients gives identical iterates and has less computational cost. In a framework of small errors, e.g., finite precision arithmetic, the performance of the method of conjugate gradients is expected to deteriorate, whereas a BFGS quasi-Newton method is empirically known to behave very well. We discuss the behavior of limited-memory quasi-Newton methods, balancing the good performance of a BFGS method to the low computational cost of the method of conjugate gradients. We also discuss large-error scenarios, in which the expected behavior is not so clear. In particular, we are interested in the behavior of quasi-Newton matrices that differ from the identity by a low-rank matrix, such as a memoryless BFGS method. Our numerical results indicate that for large errors, a memory-less quasi-Newton method often outperforms a BFGS method. We also consider a more advanced model for generating search directions, based on solving a chance-constrained optimization problem. Our results indicate that such a model often gives a slight advantage in final accuracy, although the computational cost is significantly higher. The talk is based on joint work with Tove Odland, David Ek, Gianpiero Canessa and Shen Peng.

Gaussian measures have long been recognized as the appropriate measures to use in infinite-dimensional analysis. Their regularity properties have allowed the development of a calculus on these measure spaces that has become an invaluable tool in the analysis of stochastic processes and their applications. Gaussian measures arise naturally in the context of random diffusions, specifically as the end point distribution of Brownian motion, and one may see their regularity as arising from nice properties of the generator of the diffusion. More particularly, in finite dimensions, hypoellipticity of the generator is a standard assumption required for regularity of the associated measure. However, in infinite dimensions it has remained elusive to demonstrate that hypoellipticity is a sufficient condition for regularity. Using techniques first developed by Bruce Driver and Masha Gordina, there has been some recent success in proving regularity for some natural infinite-dimensional hypoelliptic models. These techniques rely on establishing uniform bounds on coefficients appearing in certain functional analytic inequalities for semi-groups on finite-dimensional approximations. We will discuss some of these successful applications, including more recent work studying models satisfying only a weak notion of hypoellipticity. This includes joint works with Fabrice Baudoin, Dan Dobbs, Bruce Driver, Nate Eldredge, and Masha Gordina.

We consider the classical framework of the famous De-Giorgi-Nash-Moser theorem: $div(A(x)\nabla u)=f$, where $A(x)$ is a symmetric, elliptic matrix field, $f$ is given and $u:U\subset \mathbb{R}^n\to\mathbb{R}$ is the unknown. N. Trudinger was the first one to relax the assumptions on the coefficients matrix $A(x)$. He was able to derive boundedness results if the matrix is barely integrable in the right spaces. In particular he was able to show that if $\lambda(x)|\xi|^2\leq \xi\cdot A(x)\xi\leq \Lambda(x)|\xi|^2,\quad \forall x$ and the $\lambda^{-1}\in L^p, \Lambda\in L^q$ satisfying $\frac{1}{p}+\frac{1}{q}<\frac{2}{n}$. The integrability condition had been considerably improved by P. Bella and M. Schaffner in the framework of the Moser-iteration to $\frac{1}{p}+\frac{1}{q}<\frac{2}{n-1}$. A counterexample had been constructed by Franchi, Serapioni, and Serra Cassano under $\frac{1}{p}+\frac{1}{q}>\frac{2}{n}$. The aim of this talk is to revisit De Giorgi’s original approach having in mind the question concerning the optimal integrability assumption on the coefficient field. We will present how this question is surprisingly linked to a question in linear programming with an infinite horizon. This talk will be about my ongoing project with M. Schaffner, hence about work in progress.

We present all 143 counterexamples in topology appearing in the book ``Counterexamples In Topology", by Lynn Steen and J. Arthur Seebach, Jr. For each such counterexample, we state whether or not it is compact.

Semidefinite optimization is a type of convex optimization problems over the cone of positive semidefinite (PSD) matrices, and sum-of-squares (SOS) optimization is another type of convex optimization problems concerned with the cone of SOS polynomials. Both semidefinite and SOS optimization have found a wide range of applications, including control theory, fluid dynamics, machine learning, and power systems. In theory, they can be solved in polynomial time using interior-point methods, but these methods are only practical for small- to medium-sized problem instances. In this talk, I will introduce decomposition methods for semidefinite optimization and SOS optimization with chordal sparsity, which scale more favorably to large-scale problem instances. It is known that chordal decomposition allows one to equivalently decompose a PSD cone into a set of smaller and coupled cones. In the first part, I will apply this fact to reformulate a sparse semidefinite program (SDP) into an equivalent SDP with smaller PSD constraints that is suitable for the application of first-order operator-splitting methods. The resulting algorithms have been implemented in the open-source solver CDCS. In the second part, I will extend the classical chordal decomposition to the case of sparse polynomial matrices that are positive (semi)definite globally or locally on a semi-algebraic set. The extended decomposition results can be viewed as sparsity-exploiting versions of the Hilbert-Artin, Reznick, Putinar, and Putinar-Vasilescu Positivstellensätze. This talk is based on our work: https://arxiv.org/abs/1707.05058 and https://arxiv.org/abs/2007.11410

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.

In this talk, we discuss effective versions of Ratner’s theorems in the space of affine lattices. For $d \geq 2$, let $Y=ASL_d(\mathbb{R})/ASL_d(\mathbb{Z})$, $H$ be a minimal horospherical group of $SL_d(\mathbb{R})$ embedded in $ASL_d(\mathbb{R})$, and $a_t$ be the corresponding diagonal flow. Then $(a_t)$-push-forwards of a piece of $H$-orbit become equidistributed with a polynomial error rate under certain Diophantine condition of the initial point of the orbit. This generalizes the previous results of Strömbergsson for $d = 2$ and of Prinyasart for $d = 3$.

Quaternionic automorphic representations are one attempt to generalize to other groups the special place holomorphic modular forms have among automorphic representations of $GL_2$. Like holomorphic modular forms, they are defined by having their real component be one of a particularly nice class (in this case, called quaternionic discrete series). We count quaternionic automorphic representations on the exceptional group $G_2$ by developing a $G_2$ version of the classical Eichler-Selberg trace formula for holomorphic modular forms. There are two main technical difficulties. First, quaternionic discrete series come in L-packets with non-quaternionic members and standard invariant trace formula techniques cannot easily distinguish between discrete series with real component in the same L-packet. Using the more modern stable trace formula resolves this issue. Second, quaternionic discrete series do not satisfy a technical condition of being ``regular", so the trace formula can a priori pick up unwanted contributions from automorphic representations with non-tempered components at infinity. Applying some computations of Mundy, this miraculously does not happen for our specific case of quaternionic representations on $G_2$. Finally, we are only studying level-1 forms, so we can apply some tricks of Chenevier and Taïbi to reduce the problem to counting representations on the compact form of $G_2$ and certain pairs of modular forms. This avoids involved computations on the geometric side of the trace formula.

For an advection-reaction PDE model of population, with a non-local boundary condition modeling ``birth", and with a multiplicative input whose nature is the ``harvesting rate", we design a feedback law that stabilizes a desired equilibrium profile (of population density vs. age). Without feedback the system has one eigenvalue at the origin and the remainder of its infinite spectrum has negative real parts, i.e., the systems is, as engineers call it, ``neutrally stable". Hence, a feedback is needed to move one eigenvalue to the left without making any of the other ones spill to the right of the imaginary axis. This control design objective is achieved by transforming the system into a control-theoretic canonical form consisting of a first-order ODE in which the input is present and whose eigenvalue needs to be made negative by feedback, and an infinite-dimensional input-free system called the ``zero dynamics", which we prove to be exponentially stable. The key feature of the overall PDE system and its feedback control law is the positivity of both the population density state and the harvesting rate input, which is a key element of the analysis, captured by a``control Lyapunov functional" which blows up when the population density or control approach zero.

A longstanding conjecture of T. Fujita asserts that if X is a smooth complex projective variety of dimension n and if L is an ample line bundle, then $K_X+mL$ is basepoint free for $m>=n+1$. The conjecture is known up to dimension five by work of Reider, Ein, Lazarsfeld, Kawamata, Ye and Zhu. In higher dimensions, breakthrough work of Angehrn, Siu, Helmke and others showed that the conjecture holds if m is larger than a quadratic function in n. We show that for $n>=2$ the conjecture holds for m larger than $n(loglog(n)+3)$. This is joint work with L. Ghidelli.