This is a bit speculative, and perhaps too challenging for an undergraduate project, but I wonder if an AlphaGeometry type approach might be possible for the task of automatically upper bounding sums or integrals of nonnegative quantities up to constants, a topic which I discuss in this blog post. A typical such question is the one from this previous MathOverflow question: establish the bound
$$ \sum_{d=0}^\infty \frac{2d+1}{2h^2 (1 + \frac{d(d+1)}{h^2}) (1 + \frac{d(d+1)}{h^2m^2})^2} \ll 1 + \log(m^2)$$
for all $h, m \geq 1$. I guess to focus the situation one could pose the more specific problem class of establishing estimates of the form
$$ \sum_d F(d,m) \ll G(m)$$
for $d, m$ in various standard parameter ranges such as ${\bf Z}^n$ or ${\bf R}^n$ intersected with a polytope, and $F$ and $G$ being various nonnegative "elementary" functions that involve only rational functions, logarithms, and perhaps exponentials. (Actually, one might first want to understand the simpler problem of automatically proving estimates of the form $F(m) \ll G(m)$ first (given some reasonable constraints on the input parameters $m$, such as linear constraints), before dealing with any summations or integrations.) Generally speaking, such estimates are easier to prove than the sharp inequalities one sometimes sees for instance in Olympiad or Putnam problems, where a lot more cleverness is often needed; the freedom to lose constants in bounds allows for solutions that only use a relatively small number of simple techniques.
Somewhat analogous to how AlphaGeometry proceeds by "routine" manipulations of relations between angles and sides, together with a few "inspired" constructions, one could imagine trying to solve these sorts of problems using "obvious" upper bound techniques coupled with a few "inspired" decompositions (or changes of variable, in the case of estimating an integral, or working with multiple sums), or introducing some key quantities that appear repeatedly in the estimate (cf. my previous answer to a MO question). However, the solution methods here seem a bit more varied than they do in the geometry case, and I do not know of any large data set of such estimation problems to train on (though perhaps there are ways to generate large synthetic data sets, as was done in AlphaGeometry), so there are some challenges to getting started here. Still, this looks like a task which is close to already being feasible by existing (non-AI) algorithms (though I am not so familiar with the literature on automated estimate proving), and might only need a small amount of additional AI to obtain reasonable performance.
EDIT: Here is a very toy example of what I have in mind. Suppose one wanted to automatically prove (a weak version of) the arithmetic mean-geometric mean inequality
$$ (abc)^{1/3} \ll a+b+c$$
for non-negative $a,b,c$. If one is inspired to split into 3! cases such as $a \leq b \leq c$, then the problem can then be handled by automated methods, since for $a \leq b \leq c$ we see that $a+b+c$ is comparable to $c$, and the implication
$$ a \leq b \leq c \implies (abc)^{1/3} \ll c$$
is a linear programming problem (after taking logarithms) that can be handled by standard mathematical packages (and indeed once one has reduced to this situation, one no longer loses a constant). The only "clever" step was to do the splitting, though even in this case one could automate this step by noting that $a+b$ is comparable to $a$ when $a \geq b$ and to $b$ when $a \leq b$, and so one could tell the computer to automatically split all such sums it encounters in such a fashion. Probably all such problems are in principle decidable (it's basically the elementary theory of tropical algebra, which is "just" linear programming), but when many variables are involved, it may become computationally expensive to brute-force all the cases and some clever AI-assisted selection of case splitting may become superior (for instance in the above example, one only needs to split into three cases such as $a,b \leq c$ rather than 3! cases; this is a negligible distinction for three variables, but becomes significant for large numbers of variables). Similarly for the more complex problems proposed above where one is also performing some summation or integration over a range.
EDIT 2: A second example of an estimate that can be proven automatically after one inspired case decomposition is the weak Fenchel-Young inequality
$$ ab \ll a \log a + e^b $$
when $a,b$ are real with $a \geq 1$ and $b \geq 0$. Of course one can prove this bound (with sharp constants) using calculus or convexity methods, but let us try to prove it "automatically" by even more elementary techniques. Once one has the "inspiration" to split into the cases $b \leq 2 \log a$ and $b \geq 2 \log a$, the estimate follows automatically from linear programming (after taking logarithms) using the following "obvious" inequalities as input:
$$ a \log a \ll a \log a + e^b$$
$$ e^b \ll a \log a + e^b$$
$$ b \leq 2 \log a \implies b \ll \log a$$
$$ b \geq 2 \log a \implies e^{b/2} \gg a$$
$$ b \ll e^{b/2}$$
So the only step that could potentially benefit from AI assistance here is locating the initial case splitting (and maybe the idea to split $e^b$ as $e^{b/2} \times e^{b/2}$).
EDIT 3: Here is how I would imagine a semi-automated approach to prove the stated inequality at the top of this post would proceed. The first "inspired" step is to split into the regimes $d \leq h$, $h \leq d \leq hm$, and $d \geq hm$. In the regime $d \leq h$, come up with the bound
$$ \frac{2d+1}{2h^2 (1 + \frac{d(d+1)}{h^2}) (1 + \frac{d(d+1)}{h^2m^2})^2} \ll \frac{d+1}{h^2}$$
and prove it by automated methods. In the regime $h \leq d \leq hm$, similarly come up with the bound
$$ \frac{2d+1}{2h^2 (1 + \frac{d(d+1)}{h^2}) (1 + \frac{d(d+1)}{h^2m^2})^2} \ll \frac{1}{d}$$
and prove that by automated methods. In the regime $d \geq hm$, come up with the bound
$$ \frac{2d+1}{2h^2 (1 + \frac{d(d+1)}{h^2}) (1 + \frac{d(d+1)}{h^2m^2})^2} \ll \frac{h^4 m^4}{d^5}$$
and prove that by automated methods. Finally, evaluate these simpler sums using some standard library of summation bounds and conclude using further automated estimate-proving tools. Here I imagine the role of AI would be to discover the right splitting, as well as the right expressions to upper bound the summand by in each regime. In this particular case one could perhaps code in some non-AI automated method to achieve these tasks, but for more complex inequalities one may need a machine learning-powered algorithm to find the right splittings and bounds.
