We discuss the relationship between the Borel measurable / Baire measurable combinatorics of the action of a finitely generated group on its Bernoulli shift and the discrete combinatorics of the multiplication action of that group on itself. Our focus is on various chromatic numbers of graphs generated by these actions. We show that marked groups with isomorphic Cayley graphs can have Borel measurable / Baire measurable chromatic numbers which differ by arbitrarily much. In the Borel two-ended, Baire measurable, and measurable hyperfinite settings, we show our constructions are nearly best possible (up to only a single additional color), and we discuss prospects for improving our constructions in the general Borel setting. Along the way, we will get tightness of some bounds of Conley and Miller on Baire measurable and measurable chromatic numbers of locally finite Borel graphs.

Nonlocal continuum models are in general integro-differential equations in place of the conventional partial differential equations. While nonlocal models show their effectiveness in modeling a number of anomalous and singular processes in physics and material sciences, for example, the peridynamics model of fracture mechanics, they also come with increased difficulty in computation with nonlocality involved. In this talk, we will give a review of the asymptotically compatible schemes for nonlocal models with a parameter dependence. Such numerical schemes are robust under the change of the nonlocal length parameter and are suitable for multiscale simulations where nonlocal and local models are coupled. Some open questions will also be discussed.

This is joint work with Asaf Horev and Inbar Klang. Factorization homology theories are invariants of $n$-manifolds with coefficients in suitable $E_n$-algebras. Let $G$ be a finite group and $V$ be a finite dimensional $G$-representation. The equivariant factorization homology for $V$-framed $G$-manifolds have $E_V$-algebra as coefficients. We show that when coefficient algebra $A$ is the Thom spectrum of an $E_{V+W}$-map for a large enough representation $W$, the factorization homology of $A$ can be computed by a certain Thom spectrum. With nonabelian Poincar\'e duality theorem, we are able to simplify the result in some cases. In particular, we compute $\mathrm{THR}(\mathrm{H}\mathbb{F}_{2})$, $\mathrm{THR}(\mathrm{H}\mathbb{Z}_{(2)})$, $\mathrm{THH}_{C_2}(\mathrm{H}\mathbb{F}_2)$.

The Kurosh problem can be seen as an analogue of the Burnside problem for algebras. It asks whether or not a finitely generated algebra over a field has finite dimension. While the answer is negative in general, you don't have to go all the way to \textbf{Chinatown} to find a class of algebras for which the answer is affirmative.
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In this talk, we will show that for algebras satisfying a polynomial identity (PI algebras), the Kurosh problem is true. Along the way, we will have a \textbf{(The) Conversation} about the basics of the theory of PI algebras, discussing their properties, constructing specific identities for classes of algebras, and looking at their structure. Time permitting, before we say our \textbf{(Long) Goodbyes} we will also look at another type of \textbf{Nice (Guys)} identities: central polynomials. Additionally, we will use them to prove a not so \textbf{Small (Town Crime)} result, Rowan's theorem.
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Hopefully, after this talk, as \textbf{Twilight} turns into \textbf{Night (Moves)}, you will go to \textbf{(The Big) Sleep} dreaming about PI algebras.

I will introduce and discuss some recent development of a fundamental problem in topology - the classification of continuous maps between spheres up to homotopy. These mathematical objects are called homotopy groups of spheres. I will start with some geometric background - its connection to the Generalized Poincare Conjecture for example. I will then introduce some classical and new methods of doing such computations, using certain spectral sequences. If time permits, I will discuss some recent development using motivic homotopy theory, a theory that was designed to use algebraic topology to study algebraic geometry, but has now been applied successfully in the reverse direction. Old and new open problems will be mentioned along the discussion.

In this talk, I will discuss when two probability measures on the unit sphere can be transported to one another using the Gauss map of a convex body. Here, a convex body is a compact convex subset of the Euclidean n-space with non-empty interior. Notice that the boundary of a convex body might not be smooth---in general, it can even contain a fractal structure. This problem can be viewed as the problem of reconstructing a convex body using partial data regarding its Gauss map. When smoothness is assumed, it reduces to a Monge-Ampere type equation on the sphere. However, in this talk, we will work with generic convex bodies and talk about how variational argument can work in this setting.
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This is joint work with K\'aroly B"{o}r"{o}czky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang.

We give an asymptotic formula for the number of partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over real-valued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.
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A large portion of the talk will be devoted to outlining how our method can be used to re-derive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins.

In the point location problem one is given a hyperplane arrangement and an unknown point. By making linear queries about that point one wants to determine which cell of the hyperplane arrangement it lies in. This problem has an unexpected connection to the problem in machine learning of actively learning a halfspace. We discuss these problems and their relationship and provide a new and nearly optimal algorithm for solving them.

Classically over the rational numbers, the exponential and
logarithm series converge $p$-adically within some open disc of
$\mathbb{C}_p$. For function fields, exponential and logarithm series
arise naturally from Drinfeld modules, which are objects constructed by
Drinfeld in his thesis to prove the Langlands conjecture for
$\mathrm{GL}_2$ over function fields. For a ``finite place'' $v$ on such a
curve, one can ask if the exp and log possess similar $v$-adic
convergence properties. For the most basic case, namely that of the
Carlitz module over $\mathbb{F}_q[T]$, this question has been long
understood. In this talk, we will show the $v$-adic convergence for
Drinfeld-(Hayes) modules on elliptic curves and a certain class of
hyperelliptic curves. As an application, we are then able to obtain a
formula for the $v$-adic $L$-value $L_v(1,\Psi)$ for characters in these
cases, analogous to Leopoldt's formula in the number field case.

Waiting in a queue (or a line) for some type of service is a commonplace experience. People do this at a grocery store, bank, amusement park or DMV, to give just a few examples. These customer service systems feature inherent randomness, which impacts performance. Modern computer and communications systems, manufacturing processes, transportation systems and even biological networks experience similar stochastic effects. Stochastic network theory is the study of the performance and optimal control of such systems. At the beginning of the talk, I will show a few simple examples of where mathematics plays an integral role in illuminating system behavior. Following that I will discuss some of the mathematical challenges associated with analyzing the performance of modern networks.
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Finally, I will end by discussing work-in- progress related to modern call centers that is joint with Amy Ward (U. Chicago Booth) and Yueyang Zhong (U. Chicago Booth).

Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which
are non-trivial only for genus zero and one) are defined by
Klemm-Pandharipande from Gromov-Witten theory, and their integrality
is conjectured. In this talk, I will explain how to give a sheaf
theoretical interpretation of them using counting invariants on moduli
spaces of one dimensional stable sheaves.
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Based on joint works with D. Maulik and Y. Toda.