Mathematics

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.

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.

A classical problem in combinatorial geometry, posed by Erdös in 1946, asks to determine the maximum number of unit segments in a set of n points in the plane. Since then a great variety of extremal problems in finite point sets have been studied. Here, we look at generalizations of this question concerning regular simplices. Among others we answer the following question asked by Erdös: Given n points in R6, how many triangles can be equilateral triangles? For our proofs we use hypergraph Turán theory. This is joint work with Dumitrescu and Liu.

As an academic mathematician with a few decades of experience working with industry, the speaker has encountered many challenging problems that required the knowledge and development of a diverse collection of mathematical tools to effectively meet these challenges. This talk will present the mathematics arising in these collaborations, discussing both some technical details and why these skills might be useful to you as a young mathematician interested in an industrial career.

The talk will include work in the oil and gas sector (mathematics of imaging, partial differential equations, inverse problems, numerical methods), psychology and acoustics (Fourier transforms, digital signal processing), smart buildings (mathematical modeling and data science) and K-12 math education (mathematical visualizations and more data science).

There will be a few videos and animations to lighten up the gory technical details!

Fix a positive integer $X$ and multi-sets of complex numbers $\mathcal{I}$ and $\mathcal{J}$. We study the shifted convolution sum \[ D_{\mathcal{I},\mathcal{J}}(X,1) = \sum_{n\leq X} \tau_{\mathcal{I}}(n)\tau_{\mathcal{J}}(n+1), \] where $\tau_{\mathcal{I}}$ and $\tau_{\mathcal{J}}$ are shifted divisor functions. These sums naturally appear in the study of higher moments of the Riemann zeta function and additive problems in number theory. We review known results on $2k$-th moment of the Riemann zeta function and correlation sums associated with generalized divisor function. Assuming a conjectural bound on the averaged level of distribution of $\tau_{\mathcal{J}}(n)$ in arithmetic progressions, we present an asymptotic formula for $D_{\mathcal{I},\mathcal{J}}(X,1)$ with explicit main terms and power-saving error estimates.