the mozes-shah phenomenon on homogeneous spaces of lie groups asserts that the space of ergodic measures under the action by subgroups generated by unipotents is closed. a key input to their work is ratner's fundamental rigidity theorems. beyond its intrinsic interest, this result has many applications to counting problems in number theory. the problem of counting saddle connections on flat surfaces has motivated the search for analogous phenomena for horocycle flows on moduli spaces of flat structures. in this setting, eskin, mirzakhani, and mohammadi showed that this property is enjoyed by the space of ergodic measures under the action of (the full upper triangular subgroup of) $\mathrm{sl}(2,\mathbb{r})$. we will discuss joint work with jon chaika and john smillie showing that this phenomenon fails to hold for the horocycle flow on the stratum of genus two flat surfaces with one cone point. as a corollary, we show that a dense set of horocycle flow orbits are not generic for any measure; in contrast with ratner's genericity theorem.

a bold conjecture due to connes (1980) predicts that the group von neumann algebra of an i.c.c. property (t) group completely remembers the group. the strong form of connes' conjecture, due to popa (2005) predicts that the factors arising from property (t) groups have trivial fundamental group. in this talk i shall discuss recent progress towards these conjectures, and present the first examples of property (t) factors with trivial fundamental group. this talk is based on a joint work with ionut chifan, cyril houdayer, and krishnendu khan.

hamiltonian simulation is a basic task in quantum computation. the accuracy of such simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the hamiltonian. for unbounded operators, after suitable discretization, the norm of the hamiltonian can be very large, which significantly increases the simulation cost. however, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. we demonstrate that under suitable assumptions of the hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the hamiltonian increases using trotter type methods. in this sense, our result outperforms all previous error bounds in the quantum simulation literature. our result extends that of [jahnke, lubich, bit numer. math. 2000] to the time-dependent setting. we also clarify the existence and the importance of commutator scalings of trotter and generalized trotter methods for time-dependent hamiltonian simulations.

random graphs play a central role across several branches of mathematics and applied sciences.
the phase transition of random graphs, where the component structure suddenly changes from `many small' components to `one giant' component, is an intriguing phenomenon with rich connections to percolation theory and mathematical physics.
in computer science and extremal combinatorics, random graphs also provide strong probabilistic guarantees for hard-to-answer deterministic questions, such as the construction of interesting combinatorial objects.
in this talk we illustrate these two complementary aspects of random graphs, highlighting that their analysis often brings together tools and techniques from different areas (including combinatorics, probability and differential equations).
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we will first focus on achlioptas processes, which are variants of classical erdos-renyi random graphs that are difficult to analyze. settling a number of conjectures and open problems, we show that the phase transition of so-called `bounded-size' achlioptas processes have the same key features as the erd\"o{}s-r\'{e}nyi reference model (which in the language of mathematical physics means that they are in the same `universality class').

we show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the n-dimensional torus. therefore, the semidefinite programming approach initiated by candès and fernandez-granda (commun. pure appl. math. 67(6) 906---956, 2014) in the univariate case can be applied. we extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and evaluations are sufficiently many (but not many). it also turns out that the sparse recovery lp-formulation of ℓ1-norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the restricted isometry property for exact recovery is not satisfied. (a naive sparse recovery lp approach does not offer such a guarantee.) finally, we also describe the algebraic prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. we provide two sets of numerical experiments, one in which the super-resolution technique and prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than prony's method, at the cost of an extra computational burden (i.e., a semidefinite optimization).

the directed landscape is a random `directed metric' on the spacetime plane that arises as the scaling limit of integrable models of last passage percolation. it is expected to be the universal scaling limit for all models in the kpz universality class for random growth. in this talk, i will describe its construction in terms of the airy line ensemble via an isometric property of the robinson-schensted-knuth correspondence, and discuss some surprising brownian structures that arise from this construction.
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based on joint work with m. nica, j. ortmann, b. virag, and l. zhang.

in this talk, we study a new recovery procedure for non-harmonic signals, or more generally for extended exponential sums y(t), which are determined by a finite number of parameters. for the reconstruction we employ a finite set of classical fourier coefficients of y(t). our new recovery method is based on the observation that the fourier coefficients of y(t) possess a special rational structure. we apply the recently proposed aaa algorithm by nakatsukasa et al. (2018) to recover this rational structure in a stable way. if a sufficiently large set of fourier coefficients of y(t) is available, then our recovery method automatically detects the correct number of terms m of the exponential sums y(t) and reconstructs all unknown parameters of the signal model. our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on prony's method.
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these results have been obtained jointly with markus petz and nadiia derevianko.

let $k$ be a number field of degree $n$ and $\mathcal{o}_k$ be
its ring of integers. an order in $\mathcal{o}_k$ is a finite index
subring that contains the identity. a major open question in arithmetic
statistics asks for the asymptotic growth of orders in $k$. in this
talk, we will give the best known lower bound for this asymptotic
growth. the main strategy is to relate orders in $\mathcal{o}_k$ to
subrings in $\mathbb{z}^n$ via zeta functions. along the way, we will
give lower bounds for the asymptotic growth of subrings in
$\mathbb{z}^n$ and for the number of index $p^e$ subrings in
$\mathbb{z}^n$. we will also discuss analytic properties of these zeta
functions.

stochastic networks arise in a variety of real world applications including telecommunications, service systems such as call centers, computer networks, health care services and biological systems. leaving aside some very simple examples, it is usually infeasible to perform an exact analysis of these networks. a useful alternative is instead to identify a suitable approximation that provides insight into performance and can be shown to be accurate in a relevant asymptotic regime. this talk looks at two different types of scaling limits of two classes of many-server stochastic networks. in both cases, the scaling limits are infinite-dimensional and require the development of new techniques for their analysis.

s. griffin introduced a quotient $r_{n,\lambda}$ of the polynomial ring $\mathbb{q}$ with $\mathbb{q}$. it simultaneously generalizes the delta conjecture coinvariant rings of haglund-rhoades-shimozono and the cohomology rings of springer fibers studied by tanisaki and garsia-procesi. we describe the space $v_{n,\lambda}$ of harmonics attached to $r_{n,\lambda}$ and produce a harmonic basis of $r_{n,\lambda}$ indexed by certain ordered set partitions $\mathcal{op}_{n,\lambda}$. our description of $v_{n,\lambda}$ involves injective tableaux and vandermonde determinants. combinatorics of our harmonic basis is governed by a new extension of the lemher code.
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this is a joint work with brendon rhoades and zehong zhao.

stochastic models of complex networks with limited resources arise in a wide variety of applications in science and engineering, e.g., in manufacturing, transportation, telecommunications, computer systems, customer service facilities, and systems biology. bottlenecks in such networks cause congestion, leading to queueing and delay. sharing of resources can lead to entrainment effects. understanding the dynamic behavior of such modern stochastic networks present challenging mathematical problems.
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this talk will describe some developments in this area. a key feature will be dimension reduction, resulting from entrainment due to resource sharing. an example of bandwidth sharing in a data network will be featured.

i will try to convince you that there is an opposite of a vector space! this talk will involve additive combinatorics and fourier analysis on finite, not infinite groups...

recently there has been great progress in constructing moduli spaces of k-stable fano varieties, and there are many questions to ask about the geometry of these moduli spaces. the question i am considering is whether they are uniruled. my goal is to show that there are components of these moduli spaces that are general type, answering that question in the negative. i wish to do this by studying moduli spaces of vector bundles on curves, showing they are k-stable, and studying the components of the moduli space of k-stable fano varieties that parametrize them.

isogeny-based cryptography is becoming an increasingly well-established subject within the post-quantum cryptography landscape. in this talk, we introduce supersingular isogeny graphs and study the hard problems in them. we also introduce some modern-day isogeny-based cryptosystems.

this is a piece of beautiful math involving basic theory of algebraic number fields and l-functions. aimed for those who did not take the 204 series.

as an analyst, i used to think i didn't like groups. and when studying generalized bernoulli actions, i came across an example of a horrible combinatorial argument to prove a fact about groups. luckily, there is another way: functional analysis, which i'd like to share with you. now i realize i just don't like finite groups, which usually aren't amenable to analytic methods.