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4:00pm - RH 440R - Logic Set Theory Alex Berenstein - (Universidad de los Andes) Expansions of quasiminimal classes by dense condense predicates Quasiminimal classes form an abstract analogue of strongly minimal theories. Following what can be done in the strongly minimal case, we consider two expansions of quasiminimal classes with a unary predicate: beautiful pairs and H-structures. We show each of these expansions can be axiomatized with a single Lω1ω (Q)-sentence and that both expansions are ω-stable. We will explain why these expansions are natural in the strongly minimal context and how to extrapolate some results to the new setting. Conversely, we show how to produce new examples of quasiminimal classes using beautiful pairs. This is joint work with E. Vassiliev. |
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3:00pm to 4:00pm - RH 306 - Nonlinear PDEs Bryan Dimler - (UC Irvine) Minimal submanifolds with multiple isolated singularities Dating back to L\'evy (1948) and Courant (1950), the \emph{bridge principle} is the idea that it should be possible to join two minimal submanifolds along their boundaries by a thin bridge and perturb the new configuration (i.e. the \emph{approximate solution}) so that it is minimal. In 1987, Smale proved the bridge principle for smooth (possibly unstable) minimal submanifolds in Euclidean space having arbitrary dimension and codimension by solving a fixed point problem for the stability operator $L$ on the normal bundle of the approximate solution. Two years later, Smale constructed the first examples of minimal hypersurfaces with multiple isolated singularities by extending their bridge principle to strictly stable (i.e. $L$ positive definite) minimal hypercones in $\mathbb{R}^{n+1}$ ($n \geq 7$). In this talk, we discuss a recent extension of Smale's singular bridge principle to strictly stable minimal cones in $\mathbb{R}^{n+m+1}$ ($n \geq 3$) having arbitrary codimension $m +1 = 1,2, \ldots$. As an application, we demonstrate that the bridge principle can be used to produce a four dimensional Lipschitz minimal graph in $\mathbb{R}^7$ with any finite number of isolated singularities. Since $n$-dimensional Lipschitz minimal graphs are smooth when $n \leq 3$ (Fischer-Colbrie, 1980) and the singular set of any such Lipschitz graph has Hausdorff dimension at most $n-4$ (Dimler, 2023), this construction is sharp in $n$. |
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3:00pm to 4:00pm - 510R Rowland Hall - Combinatorics and Probability Rayan Saab - (UCSD) Compressing neural networks: sparsity, quantization, and low-rank approximation We will discuss recent advances in the compression of pre-trained neural networks using both novel and existing computationally efficient algorithms. The approaches we consider leverage sparsity, low-rank approximations of weight matrices, and weight quantization to achieve significant reductions in model size, while maintaining performance. We provide rigorous theoretical error guarantees as well as numerical experiments. |
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1:00pm to 2:00pm - RH 510R - Algebra Laura Cossu - (University of Cagliari) From a classical problem in matrix theory to a novel approach to factorization theory In the second half of the 1960s, Erdős proved that every singular matrix over a field can be expressed as a product of idempotent matrices. Since then, the characterization of integral domains satisfying the same property has become a widely investigated problem in ring theory. Notably, this problem is connected to other significant open questions, such as the characterization of integral domains whose general linear groups are generated by elementary matrices and those satisfying variations of the Euclidean algorithm. This seminar provides an informal overview of classical results regarding the idempotent factorization of matrices, as well as recent advancements in the field. Furthermore, it explores how the natural question "Can we study the (non-)uniqueness (in some sense) of idempotent matrix decompositions?" has led to a novel approach to factorization theory, significantly broadening the scope of the classical theory. |
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3:00pm to 4:00pm - RH 306 - Number Theory Daniele Garzoni - (USC) Characteristic polynomial of random tridiagonal matrices In the talk, we will discuss the irreducibility and the Galois group of random polynomials over the integers. After giving motivation (coming from work of Breuillard--Varjú, Eberhard, Ferber--Jain--Sah--Sawhney, and others), I will present a result, conditional on the extended Riemann hypothesis, showing that the characteristic polynomial of certain random tridiagonal matrices is irreducible, with probability tending to 1 as the size of the matrices tends to infinity. The proof involves random walks in direct products of SL_2(p), where we use results of Breuillard--Gamburd and Golsefidy--Srinivas. Joint work with Lior Bary-Soroker and Sasha Sodin. |
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3:00pm to 3:50pm - RH 306 - Analysis Xiaojun Huang - (Rutgers University) Bounding a Levi-flat Hypersurface in a Stein Manifold Let M be a smooth real codimension two compact submanifold in a Stein manifold. We will prove the following theorem: Suppose that M has two elliptic complex tangents and that CR points are non-minimal. Assume further that M is contained in a bounded strongly pseudoconvex domain. Then M bounds a unique smoothly up to M Levi-flat hypersurface \widehat{M} that is foliated by Stein hyper-surfaces diffeomorphic to the ball. Moreover, \widehat{M} is the hull of holomorphy of M . This subject has a long history of investigation dating back to E. Bishop and Harvey-Lawson. I will discuss both the historical context and the techniques used in the proof of the aforementioned theorem. |