Time & Location: All talks are on Thursdays in Gibson Hall 126A at 3:30 pm unless otherwise noted. Refreshments in Gibson 426 after the talk.

Organizer: **Tommaso Buvoli and SamuelPunshon-Smith**

*February 1*

*February 1*

**Title:** Exponential generating functions and their congruences in enumerative combinatorics

**Ira Gessel - Brandeis University (Host: Amdeberhan)**

**Abstract:** Enumerative combinatorialists study sequences of integers that count things, and some of us like to find congruences for these integers. I will talk about sequences that have nice exponential generating functions, which are power series in which the numbers of interest are the coefficients of x^n/n!. An important example is the exponential generating function exp(exp(x) -1) for the Bell numbers, which count partitions of a set. I will first discuss how exponential generating functions are used in enumeration. Then I will discuss three methods for finding congruences for coefficients of exponential generating functions. The first method (which does not actually use the generating function) is the combinatorial method: Suppose that we have a finite group acting on a set S. If every element of S is in an orbit of size divisible by m, then the size of S is divisible by m. The second method, the umbral method, works with recurrences that are not so easily derived directly from generating functions. The third method uses the algebra of exponential generating functions modulo a prime, and differential operators on this algebra.

**Location:** Gibson Hall 126A

**Time:** 3:30

*February 15*

*February 15*

**Title:** DeLTA: Changing Teaching Evaluation through Departmental Action

**Paula P Lemons - University of Georgia (Host: TBA)**

**Abstract:** The University of Georgia DeLTA project works toward new core commitments in undergraduate STEM education: collaboration about teaching, basing educational decisions on evidence, and continuously improving our teaching. Modernizing our teaching evaluation is a primary way to achieve these commitments. In the DeLTA project we have achieved change in teaching evaluation by working at the departmental level. A leadership action team of department chairs convenes several times per year to learn about national models for effective teaching evaluation and to exchange ideas about the teaching evaluation practices in their units. Department chairs recruit faculty members who collaborate with faculty from other units to understand, revise, and implement new teaching evaluation practices, such as peer observation by trained peers and instructor self-reflection. The change we have achieved at the department level has been facilitated by changes in policy at the university level. DeLTA research shows that change takes place in departments at different rates and suggests factors that may contribute to departmental outcomes. This seminar will present the UGA DeLTA model, including principles and details about implementation, and will explain our research findings.

**Location:** Gibson Hall 126A

**Time:** 3:30

*February 29*

*February 29*

**Title:** A broad conjectural framework for the parity of eta-quotients

**Fabrizio Zanello - Michigan Tech (Host: Dr. Ha)**

**Abstract:**

One of the classical and most fascinating problems at the intersection between combinatorics

and number theory is the study of the parity of the partition function. Even though p(n) is widely

believed to be equidistributed modulo 2, progress in this area has always proven exceptionally hard. The

best results we have today, obtained incrementally over several decades by Serre, Soundararajan, Ono

and many others, do not even guarantee that, asymptotically, p(n) is odd for √x values of n ≤ x.

In this colloquium talk, we present a new, general conjectural framework that naturally places the

parity of p(n) into the much broader, number-theoretic context of eta-quotients. We discuss the history of

this problem as well as recent progress on our “master conjecture,” which includes novel results on multiand

regular partitions. We then show how seemingly unrelated classes of eta-quotients carry surprising

(and surprisingly deep) connections modulo 2. One instance is the following striking result: If any tmultipartition

function, with t ̸≡ 0 (mod 3), is odd with positive density, then so is p(n). (Note that

proving either fact unconditionally seems entirely out of reach with current methods.)

Throughout our talk, we will also try to give a sense of the many interesting mathematical techniques

that come into play in this area. They include a variety of algebraic and combinatorial ideas, as well as

tools from modular forms and number theory.

Much of this work is in collaboration with my former Ph.D. student S. Judge or with W.J. Keith (see

my papers in the J. Number Theory, 2015, 2018, 2021, 2022, and 2023; Annals of Comb., 2018; Int. J.

Number Theory, 2021 and 2023).

**Location:** Gibson Hall 126A

**Time:** 3:30

*March 14*

*March 14*

**Title:** *The fractional Yamabe equation on homogeneous groups*

**Dimiter Vassilev - University of New Mexico (Host: Dr. Can)**

**Abstract:** The general themes of the talk are Dirichlet forms, fractional (non-local) operators and associated Sobolev type spaces on groups of homogeneous type. I will recall some general motivating examples for considering non-local operators and particular equations before focusing on the respective questions in the setting of homogenous groups. The considered groups are not assumed to be Carnot groups or to satisfy a Hörmander type condition. I will describe a result on sharp asymptotic decay of solutions to non-linear equations modeled on the fractional Yamabe equation.

**Location:** Gibson Hall 126A

**Time:** 3:30

*March 21*

*March 21*

**Title:** *Mathematics of magic angles for twisted bilayer graphene*

**Maciej Zworski - UC Berkley (Host: Punshon-Smith)**

**Abstract: **Magic angles refer to a remarkable theoretical (Bistritzer--MacDonald, 2011) and experimental (Jarillo-Herrero et al 2018) discovery, that two sheets of graphene twisted by a certain (magic) angle display unusual electronic properties such as superconductivity.

Mathematically, this is related to having flat bands of nontrivial topology for the corresponding periodic Hamiltonian and their existence be shown for the chiral model of twisted bilayer graphene (Tarnopolsky-Kruchkov-Vishwanath, 2019). A spectral characterization of magic angles (Becker--Embree--Wittsten--Z, 2021, Galkowski--Z, 2023) also produces complex values and the distribution of their reciprocals looks remarkably like a distribution of scattering resonances for a two-dimensional problem, with the real magic angles corresponding to anti-bound states. I will review various results on that distribution as well as on the properties of the associated eigenstates.

The talk is based on joint works with S Becker, M Embree, J Galkowski, M Hitrik, T Humbert and J Wittsten.

**Location:** Gibson Hall 126A

**Time:** 3:30

*April 4*

*April 4*

**Title:** Peeling high-dimensional oranges

**Anton Dochtermann - Texas State University (Host: Dr. Ha)**

**Abstract:** A `simpicial complex' is a space that one can obtain by gluing together triangles, tetrahedra, and higher dimensional analogues called `simplices'. Simplicial complexes model a wide variety of topological spaces in a way that is accessible to calculation, and also define the Stanley-Reisner rings in commutative algebra. One way to study a simplicial complex X is via a `shelling': an ordering of the top dimensional faces of X that is similar to the way one would (un)peel an orange. The existence of a shelling has important consequences for its topological and algebraic properties of X. A well-known conjecture of Simon posits that one can always extend a given shelling of a shellable complex to the full skeleton of a simplex. We will discuss ideas and new results surrounding Simon's conjecture, including a proof for the special case of `vertex decomposable' complexes, connections to chordal graphs, and certain extremal cases in our search for counterexamples.

**Location:** Gibson Hall 126A

**Time:** 3:30

*April 18*

*April 18*

**Title:** Stabilizing phenomenon for incompressible fluids

**Jiahong Wu - Notre Dame (Host: Zhao)**

**Abstract:** This talk presents several examples of a remarkable stabilizing phenomenon. The results of T. Elgindi and T. Hou's group show that the 3D incompressible Euler equation can blow up in a finite time. Even small data would not help. But when the 3D Euler is coupled with the non-Newtonian stress tensor in the Oldroyd-B model, small smooth data always lead to global and stable solutions.The 3D incompressible Navier-Stokes equation with dissipation in only one direction is not known to always have global solutions even when the initial data are small. However, when this Navier-Stokes is coupled with the magnetic field in the magneto-hydrodynamic system, solutions near a background magnetic field are shown to be always global in time. The magnetic field stabilizes the fluid. Solutions of the 2D Navier-Stokes in R^2 with dissipation in only one direction are not known to be stable, but the Boussinesq system involving this Navier-Stokes is always stable near the hydrostatic equilibrium. The buoyancy forcing helps stabilize the fluid. In all these examples the systems governing the perturbations can be converted to damped wave equations, which reveal the smoothing and stabilizing effect.

**Location:** Gibson Hall 126A

**Time:** 3:30