i will prove strict comparison of c* algebras associated to free groups and then use it to solve the c* version of tarski's problem from 1945 in the negative. it is joint work with amrutam, gao and patchell and another joint work with schafhauser.

the key property of linear dispersive flows is that waves with different frequencies travel with different group velocities, which leads to the phenomena of dispersive decay. nonlinear dispersive flows also allow for interactions of linear waves, and their long time behavior is determined by the balance of linear dispersion on one hand, and nonlinear effects on the other hand. 

the first goal of this talk will be to present a new set of conjectures which aim to describe the global well-posedness and the dispersive properties of solutions in the most difficult case when the nonlinear effects are dominant, assuming only small initial data. this covers many interesting physical models, yet, as recently as a few years ago, there was no clue even as to what one might reasonably expect. the second objective of the talk will be to describe some very recent results in this direction.  this is joint work with mihaela ifrim.

deep learning has been wildly successful in practice and most state-of-the-art artificial intelligence systems are based on neural networks. lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep neural networks. in this talk, i present a new mathematical framework that provides the beginning of a deeper understanding of deep learning. this framework precisely characterizes the functional properties of trained neural networks. the key mathematical tools which support this framework include transform-domain sparse regularization, the radon transform of computed tomography, and approximation theory. this framework explains the effect of weight decay regularization in neural network training, the importance of skip connections and low-rank weight matrices in network architectures, the role of sparsity in neural networks, and explains why neural networks can perform well in high-dimensional problems. at the end of the talk we shall conclude with a number of open problems and interesting research directions.

this talk is based on work done in collaboration with rob nowak, ron devore, jonathan siegel, joe shenouda, and michael unser.

coq is a programming language and "proof assistant", where one can state and prove theorems which are checked for soundness by coq itself. looking at an example formalization of the hypernatural numbers, we'll explore what makes such a tool useful, interesting, and even fun! at the end of this talk attendees will hopefully have reasons to consider using coq or similar tools themselves, and incidentally be able to construct a non-standard model of arithmetic (whenever the need arises).

many data science problems require solving a least-squares problem min_x || a x - b ||^2. efficiently solving this problem becomes a challenge when a has millions of rows, or even higher. i am developing solutions based on randomized numerical linear algebra:

1. if a is small enough to fit in working memory, an efficient solution is conjugate gradient with randomized preconditioning.

2. if a is too large to fit in working memory but x fits in memory, an intriguing possibility is randomized kaczmarz.

3. if x is too large to fit in working memory, the final possibility is randomly sparsified richardson iteration.

two tracial von neumann algebras are elementarily equivalent if they cannot be distinguished by first-order sentences or, more algebraically, if they have isomorphic ultrapowers. the same definition can be made for (countable, discrete) groups, and it is natural to wonder whether or not there is a connection between two groups being elementarily equivalent and their corresponding group von neumann algebras being elementarily equivalent.  in the first part of the talk, i will give examples to show that, in general, there is no connection in either direction.  in the second part of the talk, i will introduce a strengthening of elementary equivalence, called back-and-forth equivalence (in the sense of computability theory) and show that back-and-forth equivalent groups have back-and-forth equivalent group von neumann algebras.  i will also discuss how the same is true for the group measure space von neumann algebra associated to the bernoulli action of a group on an arbitrary tracial von neumann algebra.  the latter half of the talk represents joint work with matthew harrison-trainor.

the superspace ring of rank $n$ is defined as the tensor product of the polynomial ring over $n$ variables and the exterior product of $n$ additional variables. this carries an action of the symmetric group, as well as the hyperoctahedral group (the group of signed permutations). for each of these actions, we define the coinvariant ideal as the ideal generated by invariants under the action with vanishing constant term. we explore some results on bases and hilbert series of the quotient rings cut out by these ideals.

there is an idea in number theory that if two objects are congruent modulo a prime p, then the congruence can also be seen for the special values of l functions attached to the objects. here is a context explicating this idea: suppose f and f' are holomorphic cuspidal eigenforms of weight k and level n, and suppose f is congruent to f' modulo p; suppose g is another cuspidal eigenform of weight l; if the difference k - l is large then the rankin-selberg l function l(s, f x g) has enough critical points; same for l(s, f' x g); one expects then that there is a congruence modulo p between the algebraic parts of l(m, f x g) and l(m, f' x g) for any critical point m. in this talk, after elaborating on this idea, i will describe the results of some computational experiments where one sees such congruences for ratios of critical values for rankin-selberg l-functions. towards the end of my talk, time-permitting, i will sketch a framework involving eisenstein cohomology for gl(4) over q which will permit us to prove such congruences. this is joint work with my student p. narayanan.

in this talk, we present a unified framework for accelerated gradient methods through the variable and operator splitting. the operator splitting decouples the optimization process into simpler subproblems, and more importantly, the variable splitting leads to acceleration. the key contributions include the development of strong lyapunov functions to analyze stability and convergence rates, as well as advanced discretization techniques like accelerated over-relaxation (aor) and extrapolation by the predictor-corrector (epc) methods. the framework effectively handles a wide range of optimization problems, including convex problems, composite convex optimization, and saddle point systems with bilinear coupling. a dynamic updating parameter, which serves as a rescaling of time, is introduced to handle the weak convex cases.

many stochastic processes (such as the full family of lévy processes) can be linearly and deterministically transformed into a white noise process. consequently these processes can be viewed as the deterministic "mixing" of a white noise process. this formulation is the so-called "innovation model" of bode, shannon, and kailath (ca. 1950-1970), where the white noise represents the stochastic part of the process, called its innovation. this allows for a conceptual decoupling between the correlation properties of a process (which are imposed by the whitening operator l) and its underlying randomness, which is determined by its innovation. this reduces the study of a stochastic process to the study of its underlying innovation. in this talk, we will introduce the innovations approach to studying stochastic processes and adopt the beautiful formalism of generalized stochastic processes of gelfand (ca. 1955), where stochastic processes are viewed as random tempered distributions (more generally, random variables that take values in the dual of a nuclear space). this formulation uses the so-called characteristic functional (infinite-dimensional analog of the characteristic function of a random variable) of a stochastic process in lieu of more traditional concepts such as random measures and itô integrals. a by-product of this formulation is that the characteristic functional of any stochastic process that satisfies the innovation model can be readily derived, providing a complete description of its law. we will then discuss some of my recent work where we have derived the characteristic functional of random neural networks to study their properties. this setting will reveal the true power of the characteristic functional: any property of a stochastic process can be derived with short and simple proofs. for example, we will show that, as the "width" of these random neural networks tends to infinity, these processes can converge in law not only to gaussian processes, but also to non-gaussian processes depending on the law of the parameters. our asymptotic results provide a new take on several classical results that have appeared in the machine learning community (wide networks converge to gaussian processes) as well as some new ones (wide networks can converge to non-gaussian processes). this talk is based on joint work with pakshal bohra, ayoub el biari, mehrsa pourya, and michael unser from our recent preprint arxiv:2405.10229.

to a smooth projective curve over a finite field, we associate rigid-analytic moduli stacks of isocrystals together with the verschiebung endomorphism. we develop relevant foundations of rigid-analytic stacks, and discuss the examples and properties of such moduli stacks. we also illustrate how such moduli can be used to count p-adic coefficient objects on the curve of rank one.

the main talk will be given by oh. in the pre-talk, shimizu will introduce integrable connections and isocrystals, which will be the key objects in the main talk.

[pre-talk at 1:00pm]

tba

an endomorphism on a normal projective variety x is said to be polarized if the pullback of an ample divisor a is linearly equivalent to qa, for some integer q>1. examples of these endomorphisms are naturally found in toric varieties and abelian varieties. indeed, it is conjectured that if x admits a polarized endomorphism, then x is a finite quotient of a toric fibration over an abelian variety. in this talk, we will restrict to the case of log calabi-yau pairs (x,b). we prove that if (x,b) admits a polarized endomorphism that preserves the boundary structure, then (x,b) is a finite quotient of a toric log calabi-yau fibration over an abelian variety. this is joint work with joaquin moraga and wern yeong.

in this talk, i will present a full resolution of the question of whether the regularity problem for the parabolic pde $-\partial_tu + {\rm div}(a\nabla u)=0$ on a lipschitz cylinder $\mathcal o\times\mathbb r$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $a$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a very natural carleson condition (a parabolic analog of the so-called dkp-condition).

we prove that for some $p_0>1$ the regularity problem is solvable in the range $(1,p_0)$. we note that answer to this question was not known previously even in the  "small carleson case", that is, when the carleson norm of coefficients is sufficiently small.

in the elliptic case the analogous question was only fully resolved recently (2022) independently by two groups using two very different methods; one involving s. hofmann, j. pipher and the presenter, the second by m. mourgoglou, b. poggi and x. tolsa. our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges. the result is a joint work with l. li and j. pipher.

tba

the plate bending or babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. we provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. we show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem. the results are relevant for the construction of curved folding devices.