Video Content by Date
Speaker: Vinod Vaikuntanathan
Integer lattices play a central role in mathematics and computer science, with applications ranging from number theory and coding theory to combinatorial optimization. Over the past three decades, they have also become a cornerstone of modern cryptography.
In this talk, I will describe the evolution of lattices in cryptography: from the early use of lattices to break classical cryptosystems; to their application in designing new encryption and digital signature schemes with (conjectured) post-quantum security; and to their role in achieving long-standing cryptographic goals such as fully homomorphic encryption that allow us to compute directly on encrypted data.
The talk will not assume any prior background in cryptography.
Speaker: Kostya Druzhkov
Differential equations can be studied from a purely geometric point of view, translating many constructions from finite-dimensional differential geometry into their language. This approach helps to clarify such notions as symmetries, conservation laws, presymplectic structures, and others. However, a number of questions arise in this framework whose answers are either incomplete or currently unknown. In particular, the problem of defining the cotangent equation in terms of the intrinsic geometry of PDEs remains open. This problem is directly related to the Hamiltonian formalism for differential equations.
From an applied perspective, methods for constructing exact solutions of differential equations are of particular interest. One of the most powerful approaches is based on the study of solutions invariant under certain symmetries of the given equation. A question of practical importance in this context is how the systems describing such invariant solutions inherit geometric structures from the original system.
In this talk, I will explain how these two topics are brought together within a reduction mechanism, which in particular clarifies how Hamiltonian operators are inherited by systems describing solutions of a given equation that are invariant under some of its symmetries. To fully implement this mechanism, an interpretation of cotangent equations in intrinsic geometric terms is also required. This can be achieved in the case where the reduced system turns out to be finite-dimensional.
Speaker: Valentina Zapata Castro
Model categories provide a powerful framework for abstract homotopy theory, but their complexity often makes them difficult to classify. By focusing on finite categories, especially grids, we gain a combinatorial setting where the problem becomes explicit. In this talk, we explore model structures through weak factorization systems (WFS) on posets, which are in one-to-one correspondence with transfer systems and their duals, both introduced here. This perspective leads to a method for constructing model structures and a characterization theorem for finding weak equivalence sets in posets. Our approach offers a pathway towards classifying model structures in a controlled setting.
This is joint work with Kristen Mazur, Angélica Osorno, Constanze Roitzheim, Rekha Santhanam and Danika Van Niel.
