Tuesday,
November 19 |
- 12:00–13:00 Applied and Computational Mathematics Seminar, Zoom
 
- Phase Transitions in the Output Distribution of Large Language Models
- Niels Lörch, University of Basel
- In a physical system, changing parameters such as temperature can induce a phase transition: an abrupt change from one state of matter to another. Analogous phenomena have recently been observed in large language models. Typically, the task of identifying phase transitions requires human analysis and some prior understanding of the system to narrow down which low-dimensional properties to monitor and analyze. Statistical methods for the automated detection of phase transitions from data have recently been proposed within the physics community. These methods are largely system agnostic and, as shown here, can be adapted to study the behavior of large language models. In particular, we quantify distributional changes in the generated output via statistical distances, which can be efficiently estimated with access to the probability distribution over next-tokens. This is joint work with Julian Arnold, Flemming Holtorf, and Frank Schilbach.
- 13:15–14:15 Algebra and Number Theory Seminar, Kemeny 343
- Distribution and mean values of families of multiplicative functions in arithmetic progressions
- Akash Singha Roy, University of Georgia
- The distribution of values of arithmetic functions in residue classes has been a problem of great interest in elementary and analytic number theory. The analogous question commonly studied for multiplicative functions is the distribution of their values in coprime residue classes. In work studying this problem for multiplicative functions controlled by polynomials at prime powers, Narkiewicz obtained criteria deciding when a family of such functions is jointly uniformly distributed among coprime residue classes to a fixed modulus. In the first part of this talk, we shall extend Narkiewicz's criteria to moduli that are allowed to vary in a wide range. Our results are essentially the best possible analogues of the Siegel-Walfisz theorem in this setting. One of the primary themes behind our arguments is the quantitative detection of a certain mixing (or ergodicity) phenomenon in multiplicative groups via methods belonging to the 'anatomy of integers', but we also rely heavily on more classical analytic arguments, tools from arithmetic and algebraic geometry, and from linear algebra over rings.
In the second part of this talk, we shall gain a finer understanding of these distributions, such as the second-order behavior. This shall rely on extending some of the most powerful known estimates on mean values of multiplicative functions (precisely, the Landau-Selberg-Delange method) to obtain a result that is much more uniform in certain important parameters. We will also see several applications of this extended result in other interesting settings as well.
This talk is partially based on joint work with Prof. Paul Pollack.
- 14:30–15:30 Applied and Computational Mathematics Seminar, Zoom
- A double dose of topics - responding to missed antibiotic doses and first passage times under constant immigration
- Hwai-Ray Tung, University of Utah
- This talk consists of two shorter talks. The first asks what to do when you forget to take a dose of antibiotics; in particular, should you take one dose or two for your next dose? To answer this, we examine a mathematical model that links antibiotic concentration with bacteria population dynamics through simulations, then develop an approximate model to analytically understand when double dosing is advantageous. The second asks when a target is reached by searchers that are recruited to the starting position at some constant rate. When searchers are added quickly, we are able to describe the first passage time for general classes of search problems. Both projects are work done with Sean Lawley.
- 14:45–15:45 Algebra and Number Theory Seminar, Kemeny 343
- Cyclotomic primes
- Carl Pomerance, Dartmouth College
- Mersenne primes and Fermat primes may be thought of as primes of the form Fm(2), where Fm(x) is the m-th cyclotomic polynomial. This talk discusses the more general problem of primes and composites of this form.
- 15:30 Functional Analysis Seminar, Kemeny 307
- Noncommutative metric spaces: Connes' spectral triples
- Erik van Erp, Dartmouth College
- In this talk I will explain what a spectral triple is, and discuss some interesting examples. In quantum mechanics, points in the phase space of a dynamical system are replaced by pure states. Mathematically, a state is a linear functional on an algebra of observables. In classical mechanics, the algebra of observables is commutative, while in quantum mechanics it is noncommutative. Various notions from classical geometry can be generalized to noncommutative algebras, which justifies thinking of them as "noncommutative spaces". One such notion is that of a metric, which is traditionally understood as measuring the distance between two points. Since many noncommutative spaces don't have points, it would seem that the notion of a metric doesn't make sense either. However, Alain Connes' "spectral triples" are precisely such "noncommutative metric spaces". Even a metric can be defined in the absence of points.
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