(K. Falconer): We will introduce self-similar fractals and consider families of such fractals where the similarities are defined using certain templates. We will show how ideas from group theory can be used to enumerate the distinct fractals in these families and to classify their symmetries.
(M. Larsen): Certain counting problems in group theory can be formulated either in terms of varieties over finite fields, or (dually) in terms of irreducible character values. By comparing the two points of view, one can either use geometry to give character estimates or (what I will mostly talk about) character estimates to prove theorems in geometry.
(A. Ioana): Any countable group G gives rise to a von Neumann algebra L(G). The classification of these group von Neumann algebras is a central theme in operator algebras. I will survey recent rigidity results which provide instances when various algebraic properties of groups, such as the presence or absence of a direct product decomposition, are remembered by their von Neumann algebras.
(B. Perthame): At the individual scale, bacteria as E. coli move by performing so-called run-and-tumble movements. This means that they alternate a jump (run phase) followed by fast re-organization phase (tumble) in which they decide of a new direction for run. For this reason, the population is described by a kinetic-Botlzmann equation of scattering type. Nonlinearity occurs when one takes into account chemotaxis, the release by the individual cells of a chemical in the environment and response by the population. These models can explain experimental observations, fit precise measurements and sustain various scales. They also allow to derive, in the diffusion limit, macroscopic models (at the population scale), as the Flux-Limited-Keller-Segel system, in opposition to the traditional Keller-Segel system, this model can sustain robust traveling bands as observed in Adler's famous experiment. Furthermore, the modulation of the tumbles, can be understood using intracellular molecular pathways. Then, the kinetic-Boltzmann equation can be derived with a fast reaction scale. Long runs at the individual scale and abnormal diffusion at the population scale, can also be derived mathematically.
(J. Baez): The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a universal role in representation theory, since Schur functors act on the category of representations of any group. We can understand this as an example of categorification. A "rig" is a "ring without negatives", and the free rig on one generator is N[x], the rig of polynomials with natural number coefficients. Categorifying the concept of commutative rig we obtain the concept of "symmetric 2-rig". Here we show that the category of Schur functors is the free symmetric 2-rig on one generator.
(J. Kamnitzer): The geometric Satake correspondence links representations of a semisimple group with the geometry of the affine Grassmannian. From this perspective, it is natural to try to categorify representations using quantizations of affine Grassmannian slices. Remarkably these affine Grassmannian slices also appear in physics as Coulomb branches of 3d supersymmetric gauge theories. This connection allows us to generalize affine Grassmannian slices and achieve the desired categorification.
(L. Ji): The notion of Riemann surfaces was introduced by Riemann in his thesis in 1851, and the moduli space M_g of compact Riemann surfaces of genus g was introduced by Riemann in his masterpiece on abelian functions in 1857. When g>1, M_g is not a locally symmetric space, but shares some intriguing similarities with it. The important Jacobian map embeds M_g into the Siegel modular variety A_g, a very important locally symmetric space. The classical Schottky problem is concerned with the characterization of the image of M_g in A_g as an algebraic subvariety. In this talk, we will discuss metric properties of the image with respect to the locally symmetric metric of A_g, for example, about the metric distortion and the large scale geometry.
(G. Carlsson): Deep learning refers to a family of methods for modeling large and complex data sets. They have demonstrated remarkable capabilities on various kinds of data. It is also true, however, that the workings of the algorithms are not well understood, and that therefore the internal states of the "learner" provides an opportunity for data analysis to gain understanding. I will discuss the methods, as well as the applications of topology to them.
(P. Gressman): In the late 20th century, the pursuit of natural questions regarding the convergence of Fourier series in higher dimensions led to the remarkable discovery of important underlying phenomena of a geometric nature. Even now, aside from certain extreme cases (namely, curves and hypersurfaces), very little is known about what makes a submanifold of Euclidean space "nondegenerate'' from the standpoint of the Fourier transform or how one can easily computationally verify nondegeneracy. We will discuss some recent developments which connect this problem to affine geometry and geometric invariant theory to shed some light on what turns out to be a problem at the interface of several rather distinct mathematical areas.
(B. Viray): Let C be an algebraic curve over Q, i.e., a 1-dimensional complex manifold defined by polynomial equations with rational coefficients. A celebrated result of Faltings implies that all algebraic points on C come in families of bounded degree, with finitely many exceptions. These exceptions are known as isolated points. We explore how these isolated points behave in families of curves and deduce consequences for the arithmetic of elliptic curves. This talk is on joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.
(I. Coskun): In this talk, I will discuss the geometry of the Hilbert scheme of n points in the projective plane, which is a smooth compactification of the configuration space of n points. I will focus on the question: What is the most special codimension one position that n points can lie in? For example, three points are typically not collinear, but in codimension one they can be collinear. This simple question will lead us to a tour of some fun mathematics ranging from moduli spaces of stable sheaves on the plane to fractal curves and palindromic numbers. This talk is based on joint work with Jack Huizenga and Matthew Woolf.
(M. Stover): A very old, fundamental problem is classifying closed n-manifolds admitting a metric of constant curvature. The most mysterious case is constant curvature -1, that is, hyperbolic manifolds, and these divide further into "arithmetic" and "nonarithmetic" manifolds. However, it is not at all evident from the definitions that this distinction has anything to do with the differential geometry of the manifold. Recently, Uri Bader, David Fisher, Nicholas Miller and I gave a geometric characterization of arithmeticity in terms of properly immersed totally geodesic submanifolds, answering a question due independently to Alan Reid and Curtis McMullen. I will give an overview, assuming only basic differential topology, of how (non)arithmeticity and totally geodesic submanifolds are connected, then describe how this allows us to import tools from ergodic theory and homogeneous dynamics originating in groundbreaking work of Margulis to prove our characterization, along the way highlighting current and former OSU mathematicians that play important roles in the story.
(W. Slofstra): According to the rules of quantum mechanics, independent measurements in separated locations can have correlated outcomes. This phenomenon is called entanglement. The existence of entanglement has been experimentally demonstrated many times over, and these experiments are one of the reasons for the current interest in the development of quantum technologies. However, on the mathematical side, it has been a long-standing problem to characterize the set of joint correlations that can arise from entangled systems. In the last few years, we've realized that the difficulty of describing the set of quantum correlations can be explained by the fact that the membership problem for this set is undecidable. Last year, Ji, Natarajan, Vidick, Wright, and Yuen showed that even the approximate membership problem for this set is undecidable, leading to the resolution of the well-known Connes embedding problem in operator algebras. In this talk, I'll give an overview of developments in this area, as well as some new results with Honghao Fu and Carl Miller showing that the membership problem for quantum correlation sets is undecidable with a constant number of measurement settings and outcomes. This puts even stronger restrictions on possible descriptions of correlation sets.
(R. Grigorchuk): I will give an introduction to some asymptotic invariants of finitely generated groups and their applications to geometry, random walks and topology. This talk will be accessible to graduate students and non-experts.
(D. Zureick-Brown): Diophantine geometry is the study of integral solutions to a polynomial equation. For instance, for integers a,b,c \geq 2 satisfying 1/a + 1/b + 1/c < 1, Darmon and Granville proved that the individual generalized Fermat equation x^a + y^b = z^c has only finitely many coprime integer solutions. Conjecturally something stronger is true: for a,b,c \geq 3 there are no non-trivial solutions. I'll discuss various other Diophantine problems, with a focus on the underlying intuition and conjectural framework. I will especially focus on the uniformity conjecture, and will explain new ideas from tropical geometry and our recent partial proof of the uniformity conjecture.
(T. Johnson-Freyd): The Landau Paradigm classifies phases of matter by how "ordered" they are, i.e. by their symmetry groups (and symmetry breaking). The difference between liquids and solids fits into this paradigm, as does the Higgs mechanism that gives particles masses in high-energy physics. However, starting around the turn of the (21st) century, it has become clear that there are patterns of "order" or "symmetry" in quantum matter systems which cannot be described by groups. In particular, there are topological phases of matter, characterized by having no local observables whatsoever, which Landau would have thought were completely trivial but which in fact have subtle long-range topological order. To describe these topological orders requires the mathematics of fusion higher categories. In this talk, I will describe the classification of these topological orders in various dimensions and the extent to which the Landau Paradigm does or does not hold.
(C. Walton): In this talk, I will discuss "quantum symmetry" from an algebraic viewpoint, especially for symmetries of algebras. The term "quantum" is used as algebras here are usually noncommutative. I will mention some interesting results on when symmetries of algebras must factor or do not factor through symmetries of classical gadgets (such as groups or Lie algebras), that is, when we must enter the realm of quantum groups (or Hopf algebras) to understand symmetries of a given algebra. This all fits neatly into the framework of studying algebras in monoidal categories, and if time permits, I will give some recent results in this direction. I aim to keep the level of the talk down-to-earth by including many basic definitions and examples.
(I. Laba): It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered "standard" tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of M. In joint work with Itay Londner, we proved that this is true when M=(pqr)^2 is odd. (We are currently finalizing the even case.) In my talk I will discuss this problem and introduce the main ingredients in the proof.
(A. Nayak): Rigidity is a phenomenon in which optimal performance in an information processing task constrains a protocol into assuming a highly structured form. In some cases, there is essentially a *unique* optimal protocol. A prototypical example is that of the CHSH experiment in which any quantum strategy achieving the optimal violation of the eponymous inequality is identical to a canonical protocol, up to local changes of basis. This phenomenon has been at the heart of a number of applications, such as the generation of certified random bits and classical verification of quantum computation. We investigate the rigidity properties of the famous superdense coding protocol due to Bennett and Wiesner, which demonstrates that it is possible to communicate two bits of classical information by sending only one qubit and using a shared Bell state. We show that the superdense coding task is *rigid*, in that any protocol for the task that uses only one qubit of communication is *locally equivalent* to the Bennett-Wiesner protocol. We also study higher-dimensional superdense coding, where the goal is to communicate one of d^2 possible classical messages by sending a d-dimensional quantum state, for general dimensions d. Unlike the d=2 case, there are non-equivalent superdense coding protocols for higher d. We present concrete constructions of non-equivalent protocols for all d > 2. Finally, we analyze the performance of superdense coding protocols where the encoding operators are independently sampled from the Haar measure on the unitary group. The analysis involves bounding the distinguishability of random maximally entangled states, which may be of independent interest. This is joint work with Henry Yuen.
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